# Uniform 8-polytope

Graphs of three regular and related uniform polytopes. :
| 8-simplex | 8-simplex | 8-simplex | 8-simplex | Rectified 8-simplex | Rectified 8-simplex | Rectified 8-simplex | Rectified 8-simplex | Truncated 8-simplex | Truncated 8-simplex | Truncated 8-simplex | Truncated 8-simplex |
| Cantellated 8-simplex | Cantellated 8-simplex | Cantellated 8-simplex | Cantellated 8-simplex | Runcinated 8-simplex | Runcinated 8-simplex | Runcinated 8-simplex | Runcinated 8-simplex | Stericated 8-simplex | Stericated 8-simplex | Stericated 8-simplex | Stericated 8-simplex |
| Pentellated 8-simplex | Pentellated 8-simplex | Pentellated 8-simplex | Pentellated 8-simplex | Hexicated 8-simplex | Hexicated 8-simplex | Hexicated 8-simplex | Hexicated 8-simplex | Heptellated 8-simplex | Heptellated 8-simplex | Heptellated 8-simplex | Heptellated 8-simplex |
| 8-orthoplex | 8-orthoplex | 8-orthoplex | 8-orthoplex | Rectified 8-orthoplex | Rectified 8-orthoplex | Rectified 8-orthoplex | Rectified 8-orthoplex | Truncated 8-orthoplex | Truncated 8-orthoplex | Truncated 8-orthoplex | Truncated 8-orthoplex |
| Cantellated 8-orthoplex | Cantellated 8-orthoplex | Cantellated 8-orthoplex | Cantellated 8-orthoplex | Cantellated 8-orthoplex | Cantellated 8-orthoplex | Runcinated 8-orthoplex | Runcinated 8-orthoplex | Runcinated 8-orthoplex | Runcinated 8-orthoplex | Runcinated 8-orthoplex | Runcinated 8-orthoplex |
| Hexicated 8-orthoplex | Hexicated 8-orthoplex | Hexicated 8-orthoplex | Hexicated 8-orthoplex | Hexicated 8-orthoplex | Hexicated 8-orthoplex | Cantellated 8-cube | Cantellated 8-cube | Cantellated 8-cube | Cantellated 8-cube | Cantellated 8-cube | Cantellated 8-cube |
| Runcinated 8-cube | Runcinated 8-cube | Runcinated 8-cube | Runcinated 8-cube | Stericated 8-cube | Stericated 8-cube | Stericated 8-cube | Stericated 8-cube | Pentellated 8-cube | Pentellated 8-cube | Pentellated 8-cube | Pentellated 8-cube |
| Hexicated 8-cube | Hexicated 8-cube | Hexicated 8-cube | Hexicated 8-cube | Hexicated 8-cube | Hexicated 8-cube | Heptellated 8-cube | Heptellated 8-cube | Heptellated 8-cube | Heptellated 8-cube | Heptellated 8-cube | Heptellated 8-cube |
| 8-cube | 8-cube | 8-cube | 8-cube | Rectified 8-cube | Rectified 8-cube | Rectified 8-cube | Rectified 8-cube | Truncated 8-cube | Truncated 8-cube | Truncated 8-cube | Truncated 8-cube |
| 8-demicube | 8-demicube | 8-demicube | 8-demicube | Truncated 8-demicube | Truncated 8-demicube | Truncated 8-demicube | Truncated 8-demicube | Cantellated 8-demicube | Cantellated 8-demicube | Cantellated 8-demicube | Cantellated 8-demicube |
| Runcinated 8-demicube | Runcinated 8-demicube | Runcinated 8-demicube | Runcinated 8-demicube | Runcinated 8-demicube | Runcinated 8-demicube | Stericated 8-demicube | Stericated 8-demicube | Stericated 8-demicube | Stericated 8-demicube | Stericated 8-demicube | Stericated 8-demicube |
| Pentellated 8-demicube | Pentellated 8-demicube | Pentellated 8-demicube | Pentellated 8-demicube | Pentellated 8-demicube | Pentellated 8-demicube | Hexicated 8-demicube | Hexicated 8-demicube | Hexicated 8-demicube | Hexicated 8-demicube | Hexicated 8-demicube | Hexicated 8-demicube |
| 4$_{21}$ | 4$_{21}$ | 4$_{21}$ | 4$_{21}$ | 1$_{42}$ | 1$_{42}$ | 1$_{42}$ | 1$_{42}$ | 2$_{41}$ | 2$_{41}$ | 2$_{41}$ | 2$_{41}$ |
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets.

## Regular 8-polytopes

Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.
There are exactly three such convex regular 8-polytopes:
* {3,3,3,3,3,3,3} - 8-simplex
* {4,3,3,3,3,3,3} - 8-cube
* {3,3,3,3,3,3,4} - 8-orthoplex
There are no nonconvex regular 8-polytopes.

## Characteristics

The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

## Uniform 8-polytopes by fundamental Coxeter groups

Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
| # | Coxeter group | Coxeter group | Forms |
| 1 | A$_8$ | [3$^7$] | 135 |
| 2 | BC$_8$ | [4,3$^6$] | 255 |
| 3 | D$_8$ | [3$^{5,1,1}$] | 191 (64 unique) |
| 4 | E$_8$ | [3$^{4,2,1}$] | 255 |
Selected regular and uniform 8-polytopes from each family include:
* Simplex family: A$_8$ [3$^7$] -
** 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
*** {3$^7$} - 8-simplex or ennea-9-tope or enneazetton -
* Hypercube/orthoplex family: B$_8$ [4,3$^6$] -
** 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
*** {4,3$^6$} - 8-cube or octeract-
*** {3$^6$,4} - 8-orthoplex or octacross -
* Demihypercube D$_8$ family: [3$^{5,1,1}$] -
** 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
*** {3,3$^{5,1}$} - 8-demicube or demiocteract, 1$_{51}$ - ; also as h{4,3$^6$} .
*** {3,3,3,3,3,3$^{1,1}$} - 8-orthoplex, 5$_{11}$ -
* E-polytope family E$_8$ family: [3$^{4,1,1}$] -
** 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
*** {3,3,3,3,3$^{2,1}$} - Thorold Gosset's semiregular 4$_{21}$,
*** {3,3$^{4,2}$} - the uniform 1$_{42}$, ,
*** {3,3,3$^{4,1}$} - the uniform 2$_{41}$,

### Uniform prismatic forms

There are many uniform prismatic families, including:
Uniform 8-polytope prism families :
| # | Coxeter group | Coxeter group |
| 7+1 ||||
| 1 | A$_7$A$_1$ | [3,3,3,3,3,3]×[ ] |
| 2 | B$_7$A$_1$ | [4,3,3,3,3,3]×[ ] |
| 3 | D$_7$A$_1$ | [3$^{4,1,1}$]×[ ] |
| 4 | E$_7$A$_1$ | [3$^{3,2,1}$]×[ ] |
| 6+2 ||||
| 1 | A$_6$I$_2$(p) | [3,3,3,3,3]×[p] |
| 2 | B$_6$I$_2$(p) | [4,3,3,3,3]×[p] |
| 3 | D$_6$I$_2$(p) | [3$^{3,1,1}$]×[p] |
| 4 | E$_6$I$_2$(p) | [3,3,3,3,3]×[p] |
| 6+1+1 ||||
| 1 | A$_6$A$_1$A$_1$ | [3,3,3,3,3]×[ ]x[ ] |
| 2 | B$_6$A$_1$A$_1$ | [4,3,3,3,3]×[ ]x[ ] |
| 3 | D$_6$A$_1$A$_1$ | [3$^{3,1,1}$]×[ ]x[ ] |
| 4 | E$_6$A$_1$A$_1$ | [3,3,3,3,3]×[ ]x[ ] |
| 5+3 ||||
| 1 | A$_5$A$_3$ | [3$^4$]×[3,3] |
| 2 | B$_5$A$_3$ | [4,3$^3$]×[3,3] |
| 3 | D$_5$A$_3$ | [3$^{2,1,1}$]×[3,3] |
| 4 | A$_5$B$_3$ | [3$^4$]×[4,3] |
| 5 | B$_5$B$_3$ | [4,3$^3$]×[4,3] |
| 6 | D$_5$B$_3$ | [3$^{2,1,1}$]×[4,3] |
| 7 | A$_5$H$_3$ | [3$^4$]×[5,3] |
| 8 | B$_5$H$_3$ | [4,3$^3$]×[5,3] |
| 9 | D$_5$H$_3$ | [3$^{2,1,1}$]×[5,3] |
| 5+2+1 ||||
| 1 | A$_5$I$_2$(p)A$_1$ | [3,3,3]×[p]×[ ] |
| 2 | B$_5$I$_2$(p)A$_1$ | [4,3,3]×[p]×[ ] |
| 3 | D$_5$I$_2$(p)A$_1$ | [3$^{2,1,1}$]×[p]×[ ] |
| 5+1+1+1 ||||
| 1 | A$_5$A$_1$A$_1$A$_1$ | [3,3,3]×[ ]×[ ]×[ ] |
| 2 | B$_5$A$_1$A$_1$A$_1$ | [4,3,3]×[ ]×[ ]×[ ] |
| 3 | D$_5$A$_1$A$_1$A$_1$ | [3$^{2,1,1}$]×[ ]×[ ]×[ ] |
| 4+4 ||||
| 1 | A$_4$A$_4$ | [3,3,3]×[3,3,3] |
| 2 | B$_4$A$_4$ | [4,3,3]×[3,3,3] |
| 3 | D$_4$A$_4$ | [3$^{1,1,1}$]×[3,3,3] |
| 4 | F$_4$A$_4$ | [3,4,3]×[3,3,3] |
| 5 | H$_4$A$_4$ | [5,3,3]×[3,3,3] |
| 6 | B$_4$B$_4$ | [4,3,3]×[4,3,3] |
| 7 | D$_4$B$_4$ | [3$^{1,1,1}$]×[4,3,3] |
| 8 | F$_4$B$_4$ | [3,4,3]×[4,3,3] |
| 9 | H$_4$B$_4$ | [5,3,3]×[4,3,3] |
| 10 | D$_4$D$_4$ | [3$^{1,1,1}$]×[3$^{1,1,1}$] |
| 11 | F$_4$D$_4$ | [3,4,3]×[3$^{1,1,1}$] |
| 12 | H$_4$D$_4$ | [5,3,3]×[3$^{1,1,1}$] |
| 13 | F$_4$×F$_4$ | [3,4,3]×[3,4,3] |
| 14 | H$_4$×F$_4$ | [5,3,3]×[3,4,3] |
| 15 | H$_4$H$_4$ | [5,3,3]×[5,3,3] |
| 4+3+1 ||||
| 1 | A$_4$A$_3$A$_1$ | [3,3,3]×[3,3]×[ ] |
| 2 | A$_4$B$_3$A$_1$ | [3,3,3]×[4,3]×[ ] |
| 3 | A$_4$H$_3$A$_1$ | [3,3,3]×[5,3]×[ ] |
| 4 | B$_4$A$_3$A$_1$ | [4,3,3]×[3,3]×[ ] |
| 5 | B$_4$B$_3$A$_1$ | [4,3,3]×[4,3]×[ ] |
| 6 | B$_4$H$_3$A$_1$ | [4,3,3]×[5,3]×[ ] |
| 7 | H$_4$A$_3$A$_1$ | [5,3,3]×[3,3]×[ ] |
| 8 | H$_4$B$_3$A$_1$ | [5,3,3]×[4,3]×[ ] |
| 9 | H$_4$H$_3$A$_1$ | [5,3,3]×[5,3]×[ ] |
| 10 | F$_4$A$_3$A$_1$ | [3,4,3]×[3,3]×[ ] |
| 11 | F$_4$B$_3$A$_1$ | [3,4,3]×[4,3]×[ ] |
| 12 | F$_4$H$_3$A$_1$ | [3,4,3]×[5,3]×[ ] |
| 13 | D$_4$A$_3$A$_1$ | [3$^{1,1,1}$]×[3,3]×[ ] |
| 14 | D$_4$B$_3$A$_1$ | [3$^{1,1,1}$]×[4,3]×[ ] |
| 15 | D$_4$H$_3$A$_1$ | [3$^{1,1,1}$]×[5,3]×[ ] |
| 4+2+2 ||||
| 4+2+1+1 ||||
| 4+1+1+1+1 ||||
| 3+3+2 ||||
| 1 | A$_3$A$_3$I$_2$(p) | [3,3]×[3,3]×[p] |
| 2 | B$_3$A$_3$I$_2$(p) | [4,3]×[3,3]×[p] |
| 3 | H$_3$A$_3$I$_2$(p) | [5,3]×[3,3]×[p] |
| 4 | B$_3$B$_3$I$_2$(p) | [4,3]×[4,3]×[p] |
| 5 | H$_3$B$_3$I$_2$(p) | [5,3]×[4,3]×[p] |
| 6 | H$_3$H$_3$I$_2$(p) | [5,3]×[5,3]×[p] |
| 3+3+1+1 ||||
| 1 | A$_3$$^2$A$_1$$^2$ | [3,3]×[3,3]×[ ]×[ ] |
| 2 | B$_3$A$_3$A$_1$$^2$ | [4,3]×[3,3]×[ ]×[ ] |
| 3 | H$_3$A$_3$A$_1$$^2$ | [5,3]×[3,3]×[ ]×[ ] |
| 4 | B$_3$B$_3$A$_1$$^2$ | [4,3]×[4,3]×[ ]×[ ] |
| 5 | H$_3$B$_3$A$_1$$^2$ | [5,3]×[4,3]×[ ]×[ ] |
| 6 | H$_3$H$_3$A$_1$$^2$ | [5,3]×[5,3]×[ ]×[ ] |
| 3+2+2+1 ||||
| 1 | A$_3$I$_2$(p)I$_2$(q)A$_1$ | [3,3]×[p]×[q]×[ ] |
| 2 | B$_3$I$_2$(p)I$_2$(q)A$_1$ | [4,3]×[p]×[q]×[ ] |
| 3 | H$_3$I$_2$(p)I$_2$(q)A$_1$ | [5,3]×[p]×[q]×[ ] |
| 3+2+1+1+1 ||||
| 1 | A$_3$I$_2$(p)A$_1$$^3$ | [3,3]×[p]×[ ]x[ ]×[ ] |
| 2 | B$_3$I$_2$(p)A$_1$$^3$ | [4,3]×[p]×[ ]x[ ]×[ ] |
| 3 | H$_3$I$_2$(p)A$_1$$^3$ | [5,3]×[p]×[ ]x[ ]×[ ] |
| 3+1+1+1+1+1 ||||
| 1 | A$_3$A$_1$$^5$ | [3,3]×[ ]x[ ]×[ ]x[ ]×[ ] |
| 2 | B$_3$A$_1$$^5$ | [4,3]×[ ]x[ ]×[ ]x[ ]×[ ] |
| 3 | H$_3$A$_1$$^5$ | [5,3]×[ ]x[ ]×[ ]x[ ]×[ ] |
| 2+2+2+2 ||||
| 1 | I$_2$(p)I$_2$(q)I$_2$(r)I$_2$(s) | [p]×[q]×[r]×[s] |
| 2+2+2+1+1 ||||
| 1 | I$_2$(p)I$_2$(q)I$_2$(r)A$_1$$^2$ | [p]×[q]×[r]×[ ]×[ ] |
| 2+2+1+1+1+1 ||||
| 2 | I$_2$(p)I$_2$(q)A$_1$$^4$ | [p]×[q]×[ ]×[ ]×[ ]×[ ] |
| 2+1+1+1+1+1+1 ||||
| 1 | I$_2$(p)A$_1$$^6$ | [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] |
| 1+1+1+1+1+1+1+1 ||||
| 1 | A$_1$$^8$ | [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] |

### The A$_8$ family

The A$_8$ family has symmetry of order 362880 (9 factorial).
There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (128+8-1 cases) These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.
See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes.
A$_8$ uniform polytopes :
| # | Coxeter-Dynkin diagram | Truncation indices | Johnson name | Basepoint | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram | Truncation indices | Johnson name | Basepoint | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | | t$_0$ | 8-simplex (ene) | (0,0,0,0,0,0,0,0,1) | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 |
| 2 | | t$_1$ | Rectified 8-simplex (rene) | (0,0,0,0,0,0,0,1,1) | 18 | 108 | 336 | 630 | 576 | 588 | 252 | 36 |
| 3 | | t$_2$ | Birectified 8-simplex (bene) | (0,0,0,0,0,0,1,1,1) | 18 | 144 | 588 | 1386 | 2016 | 1764 | 756 | 84 |
| 4 | | t$_3$ | Trirectified 8-simplex (trene) | (0,0,0,0,0,1,1,1,1) | | | | | | | 1260 | 126 |
| 5 | | t$_{0,1}$ | Truncated 8-simplex (tene) | (0,0,0,0,0,0,0,1,2) | | | | | | | 288 | 72 |
| 6 | | t$_{0,2}$ | Cantellated 8-simplex | (0,0,0,0,0,0,1,1,2) | | | | | | | 1764 | 252 |
| 7 | | t$_{1,2}$ | Bitruncated 8-simplex | (0,0,0,0,0,0,1,2,2) | | | | | | | 1008 | 252 |
| 8 | | t$_{0,3}$ | Runcinated 8-simplex | (0,0,0,0,0,1,1,1,2) | | | | | | | 4536 | 504 |
| 9 | | t$_{1,3}$ | Bicantellated 8-simplex | (0,0,0,0,0,1,1,2,2) | | | | | | | 5292 | 756 |
| 10 | | t$_{2,3}$ | Tritruncated 8-simplex | (0,0,0,0,0,1,2,2,2) | | | | | | | 2016 | 504 |
| 11 | | t$_{0,4}$ | Stericated 8-simplex | (0,0,0,0,1,1,1,1,2) | | | | | | | 6300 | 630 |
| 12 | | t$_{1,4}$ | Biruncinated 8-simplex | (0,0,0,0,1,1,1,2,2) | | | | | | | 11340 | 1260 |
| 13 | | t$_{2,4}$ | Tricantellated 8-simplex | (0,0,0,0,1,1,2,2,2) | | | | | | | 8820 | 1260 |
| 14 | | t$_{3,4}$ | Quadritruncated 8-simplex | (0,0,0,0,1,2,2,2,2) | | | | | | | 2520 | 630 |
| 15 | | t$_{0,5}$ | Pentellated 8-simplex | (0,0,0,1,1,1,1,1,2) | | | | | | | 5040 | 504 |
| 16 | | t$_{1,5}$ | Bistericated 8-simplex | (0,0,0,1,1,1,1,2,2) | | | | | | | 12600 | 1260 |
| 17 | | t$_{2,5}$ | Triruncinated 8-simplex | (0,0,0,1,1,1,2,2,2) | | | | | | | 15120 | 1680 |
| 18 | | t$_{0,6}$ | Hexicated 8-simplex | (0,0,1,1,1,1,1,1,2) | | | | | | | 2268 | 252 |
| 19 | | t$_{1,6}$ | Bipentellated 8-simplex | (0,0,1,1,1,1,1,2,2) | | | | | | | 7560 | 756 |
| 20 | | t$_{0,7}$ | Heptellated 8-simplex | (0,1,1,1,1,1,1,1,2) | | | | | | | 504 | 72 |
| 21 | | t$_{0,1,2}$ | Cantitruncated 8-simplex | (0,0,0,0,0,0,1,2,3) | | | | | | | 2016 | 504 |
| 22 | | t$_{0,1,3}$ | Runcitruncated 8-simplex | (0,0,0,0,0,1,1,2,3) | | | | | | | 9828 | 1512 |
| 23 | | t$_{0,2,3}$ | Runcicantellated 8-simplex | (0,0,0,0,0,1,2,2,3) | | | | | | | 6804 | 1512 |
| 24 | | t$_{1,2,3}$ | Bicantitruncated 8-simplex | (0,0,0,0,0,1,2,3,3) | | | | | | | 6048 | 1512 |
| 25 | | t$_{0,1,4}$ | Steritruncated 8-simplex | (0,0,0,0,1,1,1,2,3) | | | | | | | 20160 | 2520 |
| 26 | | t$_{0,2,4}$ | Stericantellated 8-simplex | (0,0,0,0,1,1,2,2,3) | | | | | | | 26460 | 3780 |
| 27 | | t$_{1,2,4}$ | Biruncitruncated 8-simplex | (0,0,0,0,1,1,2,3,3) | | | | | | | 22680 | 3780 |
| 28 | | t$_{0,3,4}$ | Steriruncinated 8-simplex | (0,0,0,0,1,2,2,2,3) | | | | | | | 12600 | 2520 |
| 29 | | t$_{1,3,4}$ | Biruncicantellated 8-simplex | (0,0,0,0,1,2,2,3,3) | | | | | | | 18900 | 3780 |
| 30 | | t$_{2,3,4}$ | Tricantitruncated 8-simplex | (0,0,0,0,1,2,3,3,3) | | | | | | | 10080 | 2520 |
| 31 | | t$_{0,1,5}$ | Pentitruncated 8-simplex | (0,0,0,1,1,1,1,2,3) | | | | | | | 21420 | 2520 |
| 32 | | t$_{0,2,5}$ | Penticantellated 8-simplex | (0,0,0,1,1,1,2,2,3) | | | | | | | 42840 | 5040 |
| 33 | | t$_{1,2,5}$ | Bisteritruncated 8-simplex | (0,0,0,1,1,1,2,3,3) | | | | | | | 35280 | 5040 |
| 34 | | t$_{0,3,5}$ | Pentiruncinated 8-simplex | (0,0,0,1,1,2,2,2,3) | | | | | | | 37800 | 5040 |
| 35 | | t$_{1,3,5}$ | Bistericantellated 8-simplex | (0,0,0,1,1,2,2,3,3) | | | | | | | 52920 | 7560 |
| 36 | | t$_{2,3,5}$ | Triruncitruncated 8-simplex | (0,0,0,1,1,2,3,3,3) | | | | | | | 27720 | 5040 |
| 37 | | t$_{0,4,5}$ | Pentistericated 8-simplex | (0,0,0,1,2,2,2,2,3) | | | | | | | 13860 | 2520 |
| 38 | | t$_{1,4,5}$ | Bisteriruncinated 8-simplex | (0,0,0,1,2,2,2,3,3) | | | | | | | 30240 | 5040 |
| 39 | | t$_{0,1,6}$ | Hexitruncated 8-simplex | (0,0,1,1,1,1,1,2,3) | | | | | | | 12096 | 1512 |
| 40 | | t$_{0,2,6}$ | Hexicantellated 8-simplex | (0,0,1,1,1,1,2,2,3) | | | | | | | 34020 | 3780 |
| 41 | | t$_{1,2,6}$ | Bipentitruncated 8-simplex | (0,0,1,1,1,1,2,3,3) | | | | | | | 26460 | 3780 |
| 42 | | t$_{0,3,6}$ | Hexiruncinated 8-simplex | (0,0,1,1,1,2,2,2,3) | | | | | | | 45360 | 5040 |
| 43 | | t$_{1,3,6}$ | Bipenticantellated 8-simplex | (0,0,1,1,1,2,2,3,3) | | | | | | | 60480 | 7560 |
| 44 | | t$_{0,4,6}$ | Hexistericated 8-simplex | (0,0,1,1,2,2,2,2,3) | | | | | | | 30240 | 3780 |
| 45 | | t$_{0,5,6}$ | Hexipentellated 8-simplex | (0,0,1,2,2,2,2,2,3) | | | | | | | 9072 | 1512 |
| 46 | | t$_{0,1,7}$ | Heptitruncated 8-simplex | (0,1,1,1,1,1,1,2,3) | | | | | | | 3276 | 504 |
| 47 | | t$_{0,2,7}$ | Hepticantellated 8-simplex | (0,1,1,1,1,1,2,2,3) | | | | | | | 12852 | 1512 |
| 48 | | t$_{0,3,7}$ | Heptiruncinated 8-simplex | (0,1,1,1,1,2,2,2,3) | | | | | | | 23940 | 2520 |
| 49 | | t$_{0,1,2,3}$ | Runcicantitruncated 8-simplex | (0,0,0,0,0,1,2,3,4) | | | | | | | 12096 | 3024 |
| 50 | | t$_{0,1,2,4}$ | Stericantitruncated 8-simplex | (0,0,0,0,1,1,2,3,4) | | | | | | | 45360 | 7560 |
| 51 | | t$_{0,1,3,4}$ | Steriruncitruncated 8-simplex | (0,0,0,0,1,2,2,3,4) | | | | | | | 34020 | 7560 |
| 52 | | t$_{0,2,3,4}$ | Steriruncicantellated 8-simplex | (0,0,0,0,1,2,3,3,4) | | | | | | | 34020 | 7560 |
| 53 | | t$_{1,2,3,4}$ | Biruncicantitruncated 8-simplex | (0,0,0,0,1,2,3,4,4) | | | | | | | 30240 | 7560 |
| 54 | | t$_{0,1,2,5}$ | Penticantitruncated 8-simplex | (0,0,0,1,1,1,2,3,4) | | | | | | | 70560 | 10080 |
| 55 | | t$_{0,1,3,5}$ | Pentiruncitruncated 8-simplex | (0,0,0,1,1,2,2,3,4) | | | | | | | 98280 | 15120 |
| 56 | | t$_{0,2,3,5}$ | Pentiruncicantellated 8-simplex | (0,0,0,1,1,2,3,3,4) | | | | | | | 90720 | 15120 |
| 57 | | t$_{1,2,3,5}$ | Bistericantitruncated 8-simplex | (0,0,0,1,1,2,3,4,4) | | | | | | | 83160 | 15120 |
| 58 | | t$_{0,1,4,5}$ | Pentisteritruncated 8-simplex | (0,0,0,1,2,2,2,3,4) | | | | | | | 50400 | 10080 |
| 59 | | t$_{0,2,4,5}$ | Pentistericantellated 8-simplex | (0,0,0,1,2,2,3,3,4) | | | | | | | 83160 | 15120 |
| 60 | | t$_{1,2,4,5}$ | Bisteriruncitruncated 8-simplex | (0,0,0,1,2,2,3,4,4) | | | | | | | 68040 | 15120 |
| 61 | | t$_{0,3,4,5}$ | Pentisteriruncinated 8-simplex | (0,0,0,1,2,3,3,3,4) | | | | | | | 50400 | 10080 |
| 62 | | t$_{1,3,4,5}$ | Bisteriruncicantellated 8-simplex | (0,0,0,1,2,3,3,4,4) | | | | | | | 75600 | 15120 |
| 63 | | t$_{2,3,4,5}$ | Triruncicantitruncated 8-simplex | (0,0,0,1,2,3,4,4,4) | | | | | | | 40320 | 10080 |
| 64 | | t$_{0,1,2,6}$ | Hexicantitruncated 8-simplex | (0,0,1,1,1,1,2,3,4) | | | | | | | 52920 | 7560 |
| 65 | | t$_{0,1,3,6}$ | Hexiruncitruncated 8-simplex | (0,0,1,1,1,2,2,3,4) | | | | | | | 113400 | 15120 |
| 66 | | t$_{0,2,3,6}$ | Hexiruncicantellated 8-simplex | (0,0,1,1,1,2,3,3,4) | | | | | | | 98280 | 15120 |
| 67 | | t$_{1,2,3,6}$ | Bipenticantitruncated 8-simplex | (0,0,1,1,1,2,3,4,4) | | | | | | | 90720 | 15120 |
| 68 | | t$_{0,1,4,6}$ | Hexisteritruncated 8-simplex | (0,0,1,1,2,2,2,3,4) | | | | | | | 105840 | 15120 |
| 69 | | t$_{0,2,4,6}$ | Hexistericantellated 8-simplex | (0,0,1,1,2,2,3,3,4) | | | | | | | 158760 | 22680 |
| 70 | | t$_{1,2,4,6}$ | Bipentiruncitruncated 8-simplex | (0,0,1,1,2,2,3,4,4) | | | | | | | 136080 | 22680 |
| 71 | | t$_{0,3,4,6}$ | Hexisteriruncinated 8-simplex | (0,0,1,1,2,3,3,3,4) | | | | | | | 90720 | 15120 |
| 72 | | t$_{1,3,4,6}$ | Bipentiruncicantellated 8-simplex | (0,0,1,1,2,3,3,4,4) | | | | | | | 136080 | 22680 |
| 73 | | t$_{0,1,5,6}$ | Hexipentitruncated 8-simplex | (0,0,1,2,2,2,2,3,4) | | | | | | | 41580 | 7560 |
| 74 | | t$_{0,2,5,6}$ | Hexipenticantellated 8-simplex | (0,0,1,2,2,2,3,3,4) | | | | | | | 98280 | 15120 |
| 75 | | t$_{1,2,5,6}$ | Bipentisteritruncated 8-simplex | (0,0,1,2,2,2,3,4,4) | | | | | | | 75600 | 15120 |
| 76 | | t$_{0,3,5,6}$ | Hexipentiruncinated 8-simplex | (0,0,1,2,2,3,3,3,4) | | | | | | | 98280 | 15120 |
| 77 | | t$_{0,4,5,6}$ | Hexipentistericated 8-simplex | (0,0,1,2,3,3,3,3,4) | | | | | | | 41580 | 7560 |
| 78 | | t$_{0,1,2,7}$ | Hepticantitruncated 8-simplex | (0,1,1,1,1,1,2,3,4) | | | | | | | 18144 | 3024 |
| 79 | | t$_{0,1,3,7}$ | Heptiruncitruncated 8-simplex | (0,1,1,1,1,2,2,3,4) | | | | | | | 56700 | 7560 |
| 80 | | t$_{0,2,3,7}$ | Heptiruncicantellated 8-simplex | (0,1,1,1,1,2,3,3,4) | | | | | | | 45360 | 7560 |
| 81 | | t$_{0,1,4,7}$ | Heptisteritruncated 8-simplex | (0,1,1,1,2,2,2,3,4) | | | | | | | 80640 | 10080 |
| 82 | | t$_{0,2,4,7}$ | Heptistericantellated 8-simplex | (0,1,1,1,2,2,3,3,4) | | | | | | | 113400 | 15120 |
| 83 | | t$_{0,3,4,7}$ | Heptisteriruncinated 8-simplex | (0,1,1,1,2,3,3,3,4) | | | | | | | 60480 | 10080 |
| 84 | | t$_{0,1,5,7}$ | Heptipentitruncated 8-simplex | (0,1,1,2,2,2,2,3,4) | | | | | | | 56700 | 7560 |
| 85 | | t$_{0,2,5,7}$ | Heptipenticantellated 8-simplex | (0,1,1,2,2,2,3,3,4) | | | | | | | 120960 | 15120 |
| 86 | | t$_{0,1,6,7}$ | Heptihexitruncated 8-simplex | (0,1,2,2,2,2,2,3,4) | | | | | | | 18144 | 3024 |
| 87 | | t$_{0,1,2,3,4}$ | Steriruncicantitruncated 8-simplex | (0,0,0,0,1,2,3,4,5) | | | | | | | 60480 | 15120 |
| 88 | | t$_{0,1,2,3,5}$ | Pentiruncicantitruncated 8-simplex | (0,0,0,1,1,2,3,4,5) | | | | | | | 166320 | 30240 |
| 89 | | t$_{0,1,2,4,5}$ | Pentistericantitruncated 8-simplex | (0,0,0,1,2,2,3,4,5) | | | | | | | 136080 | 30240 |
| 90 | | t$_{0,1,3,4,5}$ | Pentisteriruncitruncated 8-simplex | (0,0,0,1,2,3,3,4,5) | | | | | | | 136080 | 30240 |
| 91 | | t$_{0,2,3,4,5}$ | Pentisteriruncicantellated 8-simplex | (0,0,0,1,2,3,4,4,5) | | | | | | | 136080 | 30240 |
| 92 | | t$_{1,2,3,4,5}$ | Bisteriruncicantitruncated 8-simplex | (0,0,0,1,2,3,4,5,5) | | | | | | | 120960 | 30240 |
| 93 | | t$_{0,1,2,3,6}$ | Hexiruncicantitruncated 8-simplex | (0,0,1,1,1,2,3,4,5) | | | | | | | 181440 | 30240 |
| 94 | | t$_{0,1,2,4,6}$ | Hexistericantitruncated 8-simplex | (0,0,1,1,2,2,3,4,5) | | | | | | | 272160 | 45360 |
| 95 | | t$_{0,1,3,4,6}$ | Hexisteriruncitruncated 8-simplex | (0,0,1,1,2,3,3,4,5) | | | | | | | 249480 | 45360 |
| 96 | | t$_{0,2,3,4,6}$ | Hexisteriruncicantellated 8-simplex | (0,0,1,1,2,3,4,4,5) | | | | | | | 249480 | 45360 |
| 97 | | t$_{1,2,3,4,6}$ | Bipentiruncicantitruncated 8-simplex | (0,0,1,1,2,3,4,5,5) | | | | | | | 226800 | 45360 |
| 98 | | t$_{0,1,2,5,6}$ | Hexipenticantitruncated 8-simplex | (0,0,1,2,2,2,3,4,5) | | | | | | | 151200 | 30240 |
| 99 | | t$_{0,1,3,5,6}$ | Hexipentiruncitruncated 8-simplex | (0,0,1,2,2,3,3,4,5) | | | | | | | 249480 | 45360 |
| 100 | | t$_{0,2,3,5,6}$ | Hexipentiruncicantellated 8-simplex | (0,0,1,2,2,3,4,4,5) | | | | | | | 226800 | 45360 |
| 101 | | t$_{1,2,3,5,6}$ | Bipentistericantitruncated 8-simplex | (0,0,1,2,2,3,4,5,5) | | | | | | | 204120 | 45360 |
| 102 | | t$_{0,1,4,5,6}$ | Hexipentisteritruncated 8-simplex | (0,0,1,2,3,3,3,4,5) | | | | | | | 151200 | 30240 |
| 103 | | t$_{0,2,4,5,6}$ | Hexipentistericantellated 8-simplex | (0,0,1,2,3,3,4,4,5) | | | | | | | 249480 | 45360 |
| 104 | | t$_{0,3,4,5,6}$ | Hexipentisteriruncinated 8-simplex | (0,0,1,2,3,4,4,4,5) | | | | | | | 151200 | 30240 |
| 105 | | t$_{0,1,2,3,7}$ | Heptiruncicantitruncated 8-simplex | (0,1,1,1,1,2,3,4,5) | | | | | | | 83160 | 15120 |
| 106 | | t$_{0,1,2,4,7}$ | Heptistericantitruncated 8-simplex | (0,1,1,1,2,2,3,4,5) | | | | | | | 196560 | 30240 |
| 107 | | t$_{0,1,3,4,7}$ | Heptisteriruncitruncated 8-simplex | (0,1,1,1,2,3,3,4,5) | | | | | | | 166320 | 30240 |
| 108 | | t$_{0,2,3,4,7}$ | Heptisteriruncicantellated 8-simplex | (0,1,1,1,2,3,4,4,5) | | | | | | | 166320 | 30240 |
| 109 | | t$_{0,1,2,5,7}$ | Heptipenticantitruncated 8-simplex | (0,1,1,2,2,2,3,4,5) | | | | | | | 196560 | 30240 |
| 110 | | t$_{0,1,3,5,7}$ | Heptipentiruncitruncated 8-simplex | (0,1,1,2,2,3,3,4,5) | | | | | | | 294840 | 45360 |
| 111 | | t$_{0,2,3,5,7}$ | Heptipentiruncicantellated 8-simplex | (0,1,1,2,2,3,4,4,5) | | | | | | | 272160 | 45360 |
| 112 | | t$_{0,1,4,5,7}$ | Heptipentisteritruncated 8-simplex | (0,1,1,2,3,3,3,4,5) | | | | | | | 166320 | 30240 |
| 113 | | t$_{0,1,2,6,7}$ | Heptihexicantitruncated 8-simplex | (0,1,2,2,2,2,3,4,5) | | | | | | | 83160 | 15120 |
| 114 | | t$_{0,1,3,6,7}$ | Heptihexiruncitruncated 8-simplex | (0,1,2,2,2,3,3,4,5) | | | | | | | 196560 | 30240 |
| 115 | | t$_{0,1,2,3,4,5}$ | Pentisteriruncicantitruncated 8-simplex | (0,0,0,1,2,3,4,5,6) | | | | | | | 241920 | 60480 |
| 116 | | t$_{0,1,2,3,4,6}$ | Hexisteriruncicantitruncated 8-simplex | (0,0,1,1,2,3,4,5,6) | | | | | | | 453600 | 90720 |
| 117 | | t$_{0,1,2,3,5,6}$ | Hexipentiruncicantitruncated 8-simplex | (0,0,1,2,2,3,4,5,6) | | | | | | | 408240 | 90720 |
| 118 | | t$_{0,1,2,4,5,6}$ | Hexipentistericantitruncated 8-simplex | (0,0,1,2,3,3,4,5,6) | | | | | | | 408240 | 90720 |
| 119 | | t$_{0,1,3,4,5,6}$ | Hexipentisteriruncitruncated 8-simplex | (0,0,1,2,3,4,4,5,6) | | | | | | | 408240 | 90720 |
| 120 | | t$_{0,2,3,4,5,6}$ | Hexipentisteriruncicantellated 8-simplex | (0,0,1,2,3,4,5,5,6) | | | | | | | 408240 | 90720 |
| 121 | | t$_{1,2,3,4,5,6}$ | Bipentisteriruncicantitruncated 8-simplex | (0,0,1,2,3,4,5,6,6) | | | | | | | 362880 | 90720 |
| 122 | | t$_{0,1,2,3,4,7}$ | Heptisteriruncicantitruncated 8-simplex | (0,1,1,1,2,3,4,5,6) | | | | | | | 302400 | 60480 |
| 123 | | t$_{0,1,2,3,5,7}$ | Heptipentiruncicantitruncated 8-simplex | (0,1,1,2,2,3,4,5,6) | | | | | | | 498960 | 90720 |
| 124 | | t$_{0,1,2,4,5,7}$ | Heptipentistericantitruncated 8-simplex | (0,1,1,2,3,3,4,5,6) | | | | | | | 453600 | 90720 |
| 125 | | t$_{0,1,3,4,5,7}$ | Heptipentisteriruncitruncated 8-simplex | (0,1,1,2,3,4,4,5,6) | | | | | | | 453600 | 90720 |
| 126 | | t$_{0,2,3,4,5,7}$ | Heptipentisteriruncicantellated 8-simplex | (0,1,1,2,3,4,5,5,6) | | | | | | | 453600 | 90720 |
| 127 | | t$_{0,1,2,3,6,7}$ | Heptihexiruncicantitruncated 8-simplex | (0,1,2,2,2,3,4,5,6) | | | | | | | 302400 | 60480 |
| 128 | | t$_{0,1,2,4,6,7}$ | Heptihexistericantitruncated 8-simplex | (0,1,2,2,3,3,4,5,6) | | | | | | | 498960 | 90720 |
| 129 | | t$_{0,1,3,4,6,7}$ | Heptihexisteriruncitruncated 8-simplex | (0,1,2,2,3,4,4,5,6) | | | | | | | 453600 | 90720 |
| 130 | | t$_{0,1,2,5,6,7}$ | Heptihexipenticantitruncated 8-simplex | (0,1,2,3,3,3,4,5,6) | | | | | | | 302400 | 60480 |
| 131 | | t$_{0,1,2,3,4,5,6}$ | Hexipentisteriruncicantitruncated 8-simplex | (0,0,1,2,3,4,5,6,7) | | | | | | | 725760 | 181440 |
| 132 | | t$_{0,1,2,3,4,5,7}$ | Heptipentisteriruncicantitruncated 8-simplex | (0,1,1,2,3,4,5,6,7) | | | | | | | 816480 | 181440 |
| 133 | | t$_{0,1,2,3,4,6,7}$ | Heptihexisteriruncicantitruncated 8-simplex | (0,1,2,2,3,4,5,6,7) | | | | | | | 816480 | 181440 |
| 134 | | t$_{0,1,2,3,5,6,7}$ | Heptihexipentiruncicantitruncated 8-simplex | (0,1,2,3,3,4,5,6,7) | | | | | | | 816480 | 181440 |
| 135 | | t$_{0,1,2,3,4,5,6,7}$ | Omnitruncated 8-simplex | (0,1,2,3,4,5,6,7,8) | | | | | | | 1451520 | 362880 |

### The B$_8$ family

The B$_8$ family has symmetry of order 10321920 (8 factorial x 2$^8$). There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes.
B$_8$ uniform polytopes :
| # | Coxeter-Dynkin diagram | Schläfli symbol | Name | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram | Schläfli symbol | Name | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | | t$_0${3$^6$,4} | 8-orthoplex Diacosipentacontahexazetton (ek) | 256 | 1024 | 1792 | 1792 | 1120 | 448 | 112 | 16 |
| 2 | | t$_1${3$^6$,4} | Rectified 8-orthoplex Rectified diacosipentacontahexazetton (rek) | 272 | 3072 | 8960 | 12544 | 10080 | 4928 | 1344 | 112 |
| 3 | | t$_2${3$^6$,4} | Birectified 8-orthoplex Birectified diacosipentacontahexazetton (bark) | 272 | 3184 | 16128 | 34048 | 36960 | 22400 | 6720 | 448 |
| 4 | | t$_3${3$^6$,4} | Trirectified 8-orthoplex Trirectified diacosipentacontahexazetton (tark) | 272 | 3184 | 16576 | 48384 | 71680 | 53760 | 17920 | 1120 |
| 5 | | t$_3${4,3$^6$} | Trirectified 8-cube Trirectified octeract (tro) | 272 | 3184 | 16576 | 47712 | 80640 | 71680 | 26880 | 1792 |
| 6 | | t$_2${4,3$^6$} | Birectified 8-cube Birectified octeract (bro) | 272 | 3184 | 14784 | 36960 | 55552 | 50176 | 21504 | 1792 |
| 7 | | t$_1${4,3$^6$} | Rectified 8-cube Rectified octeract (recto) | 272 | 2160 | 7616 | 15456 | 19712 | 16128 | 7168 | 1024 |
| 8 | | t$_0${4,3$^6$} | 8-cube Octeract (octo) | 16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 |
| 9 | | t$_{0,1}${3$^6$,4} | Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek) | | | | | | | 1456 | 224 |
| 10 | | t$_{0,2}${3$^6$,4} | Cantellated 8-orthoplex Small rhombated diacosipentacontahexazetton (srek) | | | | | | | 14784 | 1344 |
| 11 | | t$_{1,2}${3$^6$,4} | Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek) | | | | | | | 8064 | 1344 |
| 12 | | t$_{0,3}${3$^6$,4} | Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek) | | | | | | | 60480 | 4480 |
| 13 | | t$_{1,3}${3$^6$,4} | Bicantellated 8-orthoplex Small birhombated diacosipentacontahexazetton (sabork) | | | | | | | 67200 | 6720 |
| 14 | | t$_{2,3}${3$^6$,4} | Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek) | | | | | | | 24640 | 4480 |
| 15 | | t$_{0,4}${3$^6$,4} | Stericated 8-orthoplex Small cellated diacosipentacontahexazetton (scak) | | | | | | | 125440 | 8960 |
| 16 | | t$_{1,4}${3$^6$,4} | Biruncinated 8-orthoplex Small biprismated diacosipentacontahexazetton (sabpek) | | | | | | | 215040 | 17920 |
| 17 | | t$_{2,4}${3$^6$,4} | Tricantellated 8-orthoplex Small trirhombated diacosipentacontahexazetton (satrek) | | | | | | | 161280 | 17920 |
| 18 | | t$_{3,4}${4,3$^6$} | Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke) | | | | | | | 44800 | 8960 |
| 19 | | t$_{0,5}${3$^6$,4} | Pentellated 8-orthoplex Small terated diacosipentacontahexazetton (setek) | | | | | | | 134400 | 10752 |
| 20 | | t$_{1,5}${3$^6$,4} | Bistericated 8-orthoplex Small bicellated diacosipentacontahexazetton (sibcak) | | | | | | | 322560 | 26880 |
| 21 | | t$_{2,5}${4,3$^6$} | Triruncinated 8-cube Small triprismato-octeractidiacosipentacontahexazetton (sitpoke) | | | | | | | 376320 | 35840 |
| 22 | | t$_{2,4}${4,3$^6$} | Tricantellated 8-cube Small trirhombated octeract (satro) | | | | | | | 215040 | 26880 |
| 23 | | t$_{2,3}${4,3$^6$} | Tritruncated 8-cube Tritruncated octeract (tato) | | | | | | | 48384 | 10752 |
| 24 | | t$_{0,6}${3$^6$,4} | Hexicated 8-orthoplex Small petated diacosipentacontahexazetton (supek) | | | | | | | 64512 | 7168 |
| 25 | | t$_{1,6}${4,3$^6$} | Bipentellated 8-cube Small biteri-octeractidiacosipentacontahexazetton (sabtoke) | | | | | | | 215040 | 21504 |
| 26 | | t$_{1,5}${4,3$^6$} | Bistericated 8-cube Small bicellated octeract (sobco) | | | | | | | 358400 | 35840 |
| 27 | | t$_{1,4}${4,3$^6$} | Biruncinated 8-cube Small biprismated octeract (sabepo) | | | | | | | 322560 | 35840 |
| 28 | | t$_{1,3}${4,3$^6$} | Bicantellated 8-cube Small birhombated octeract (subro) | | | | | | | 150528 | 21504 |
| 29 | | t$_{1,2}${4,3$^6$} | Bitruncated 8-cube Bitruncated octeract (bato) | | | | | | | 28672 | 7168 |
| 30 | | t$_{0,7}${4,3$^6$} | Heptellated 8-cube Small exi-octeractidiacosipentacontahexazetton (saxoke) | | | | | | | 14336 | 2048 |
| 31 | | t$_{0,6}${4,3$^6$} | Hexicated 8-cube Small petated octeract (supo) | | | | | | | 64512 | 7168 |
| 32 | | t$_{0,5}${4,3$^6$} | Pentellated 8-cube Small terated octeract (soto) | | | | | | | 143360 | 14336 |
| 33 | | t$_{0,4}${4,3$^6$} | Stericated 8-cube Small cellated octeract (soco) | | | | | | | 179200 | 17920 |
| 34 | | t$_{0,3}${4,3$^6$} | Runcinated 8-cube Small prismated octeract (sopo) | | | | | | | 129024 | 14336 |
| 35 | | t$_{0,2}${4,3$^6$} | Cantellated 8-cube Small rhombated octeract (soro) | | | | | | | 50176 | 7168 |
| 36 | | t$_{0,1}${4,3$^6$} | Truncated 8-cube Truncated octeract (tocto) | | | | | | | 8192 | 2048 |
| 37 | | t$_{0,1,2}${3$^6$,4} | Cantitruncated 8-orthoplex Great rhombated diacosipentacontahexazetton | | | | | | | 16128 | 2688 |
| 38 | | t$_{0,1,3}${3$^6$,4} | Runcitruncated 8-orthoplex Prismatotruncated diacosipentacontahexazetton | | | | | | | 127680 | 13440 |
| 39 | | t$_{0,2,3}${3$^6$,4} | Runcicantellated 8-orthoplex Prismatorhombated diacosipentacontahexazetton | | | | | | | 80640 | 13440 |
| 40 | | t$_{1,2,3}${3$^6$,4} | Bicantitruncated 8-orthoplex Great birhombated diacosipentacontahexazetton | | | | | | | 73920 | 13440 |
| 41 | | t$_{0,1,4}${3$^6$,4} | Steritruncated 8-orthoplex Cellitruncated diacosipentacontahexazetton | | | | | | | 394240 | 35840 |
| 42 | | t$_{0,2,4}${3$^6$,4} | Stericantellated 8-orthoplex Cellirhombated diacosipentacontahexazetton | | | | | | | 483840 | 53760 |
| 43 | | t$_{1,2,4}${3$^6$,4} | Biruncitruncated 8-orthoplex Biprismatotruncated diacosipentacontahexazetton | | | | | | | 430080 | 53760 |
| 44 | | t$_{0,3,4}${3$^6$,4} | Steriruncinated 8-orthoplex Celliprismated diacosipentacontahexazetton | | | | | | | 215040 | 35840 |
| 45 | | t$_{1,3,4}${3$^6$,4} | Biruncicantellated 8-orthoplex Biprismatorhombated diacosipentacontahexazetton | | | | | | | 322560 | 53760 |
| 46 | | t$_{2,3,4}${3$^6$,4} | Tricantitruncated 8-orthoplex Great trirhombated diacosipentacontahexazetton | | | | | | | 179200 | 35840 |
| 47 | | t$_{0,1,5}${3$^6$,4} | Pentitruncated 8-orthoplex Teritruncated diacosipentacontahexazetton | | | | | | | 564480 | 53760 |
| 48 | | t$_{0,2,5}${3$^6$,4} | Penticantellated 8-orthoplex Terirhombated diacosipentacontahexazetton | | | | | | | 1075200 | 107520 |
| 49 | | t$_{1,2,5}${3$^6$,4} | Bisteritruncated 8-orthoplex Bicellitruncated diacosipentacontahexazetton | | | | | | | 913920 | 107520 |
| 50 | | t$_{0,3,5}${3$^6$,4} | Pentiruncinated 8-orthoplex Teriprismated diacosipentacontahexazetton | | | | | | | 913920 | 107520 |
| 51 | | t$_{1,3,5}${3$^6$,4} | Bistericantellated 8-orthoplex Bicellirhombated diacosipentacontahexazetton | | | | | | | 1290240 | 161280 |
| 52 | | t$_{2,3,5}${3$^6$,4} | Triruncitruncated 8-orthoplex Triprismatotruncated diacosipentacontahexazetton | | | | | | | 698880 | 107520 |
| 53 | | t$_{0,4,5}${3$^6$,4} | Pentistericated 8-orthoplex Tericellated diacosipentacontahexazetton | | | | | | | 322560 | 53760 |
| 54 | | t$_{1,4,5}${3$^6$,4} | Bisteriruncinated 8-orthoplex Bicelliprismated diacosipentacontahexazetton | | | | | | | 698880 | 107520 |
| 55 | | t$_{2,3,5}${4,3$^6$} | Triruncitruncated 8-cube Triprismatotruncated octeract | | | | | | | 645120 | 107520 |
| 56 | | t$_{2,3,4}${4,3$^6$} | Tricantitruncated 8-cube Great trirhombated octeract | | | | | | | 241920 | 53760 |
| 57 | | t$_{0,1,6}${3$^6$,4} | Hexitruncated 8-orthoplex Petitruncated diacosipentacontahexazetton | | | | | | | 344064 | 43008 |
| 58 | | t$_{0,2,6}${3$^6$,4} | Hexicantellated 8-orthoplex Petirhombated diacosipentacontahexazetton | | | | | | | 967680 | 107520 |
| 59 | | t$_{1,2,6}${3$^6$,4} | Bipentitruncated 8-orthoplex Biteritruncated diacosipentacontahexazetton | | | | | | | 752640 | 107520 |
| 60 | | t$_{0,3,6}${3$^6$,4} | Hexiruncinated 8-orthoplex Petiprismated diacosipentacontahexazetton | | | | | | | 1290240 | 143360 |
| 61 | | t$_{1,3,6}${3$^6$,4} | Bipenticantellated 8-orthoplex Biterirhombated diacosipentacontahexazetton | | | | | | | 1720320 | 215040 |
| 62 | | t$_{1,4,5}${4,3$^6$} | Bisteriruncinated 8-cube Bicelliprismated octeract | | | | | | | 860160 | 143360 |
| 63 | | t$_{0,4,6}${3$^6$,4} | Hexistericated 8-orthoplex Peticellated diacosipentacontahexazetton | | | | | | | 860160 | 107520 |
| 64 | | t$_{1,3,6}${4,3$^6$} | Bipenticantellated 8-cube Biterirhombated octeract | | | | | | | 1720320 | 215040 |
| 65 | | t$_{1,3,5}${4,3$^6$} | Bistericantellated 8-cube Bicellirhombated octeract | | | | | | | 1505280 | 215040 |
| 66 | | t$_{1,3,4}${4,3$^6$} | Biruncicantellated 8-cube Biprismatorhombated octeract | | | | | | | 537600 | 107520 |
| 67 | | t$_{0,5,6}${3$^6$,4} | Hexipentellated 8-orthoplex Petiterated diacosipentacontahexazetton | | | | | | | 258048 | 43008 |
| 68 | | t$_{1,2,6}${4,3$^6$} | Bipentitruncated 8-cube Biteritruncated octeract | | | | | | | 752640 | 107520 |
| 69 | | t$_{1,2,5}${4,3$^6$} | Bisteritruncated 8-cube Bicellitruncated octeract | | | | | | | 1003520 | 143360 |
| 70 | | t$_{1,2,4}${4,3$^6$} | Biruncitruncated 8-cube Biprismatotruncated octeract | | | | | | | 645120 | 107520 |
| 71 | | t$_{1,2,3}${4,3$^6$} | Bicantitruncated 8-cube Great birhombated octeract | | | | | | | 172032 | 43008 |
| 72 | | t$_{0,1,7}${3$^6$,4} | Heptitruncated 8-orthoplex Exitruncated diacosipentacontahexazetton | | | | | | | 93184 | 14336 |
| 73 | | t$_{0,2,7}${3$^6$,4} | Hepticantellated 8-orthoplex Exirhombated diacosipentacontahexazetton | | | | | | | 365568 | 43008 |
| 74 | | t$_{0,5,6}${4,3$^6$} | Hexipentellated 8-cube Petiterated octeract | | | | | | | 258048 | 43008 |
| 75 | | t$_{0,3,7}${3$^6$,4} | Heptiruncinated 8-orthoplex Exiprismated diacosipentacontahexazetton | | | | | | | 680960 | 71680 |
| 76 | | t$_{0,4,6}${4,3$^6$} | Hexistericated 8-cube Peticellated octeract | | | | | | | 860160 | 107520 |
| 77 | | t$_{0,4,5}${4,3$^6$} | Pentistericated 8-cube Tericellated octeract | | | | | | | 394240 | 71680 |
| 78 | | t$_{0,3,7}${4,3$^6$} | Heptiruncinated 8-cube Exiprismated octeract | | | | | | | 680960 | 71680 |
| 79 | | t$_{0,3,6}${4,3$^6$} | Hexiruncinated 8-cube Petiprismated octeract | | | | | | | 1290240 | 143360 |
| 80 | | t$_{0,3,5}${4,3$^6$} | Pentiruncinated 8-cube Teriprismated octeract | | | | | | | 1075200 | 143360 |
| 81 | | t$_{0,3,4}${4,3$^6$} | Steriruncinated 8-cube Celliprismated octeract | | | | | | | 358400 | 71680 |
| 82 | | t$_{0,2,7}${4,3$^6$} | Hepticantellated 8-cube Exirhombated octeract | | | | | | | 365568 | 43008 |
| 83 | | t$_{0,2,6}${4,3$^6$} | Hexicantellated 8-cube Petirhombated octeract | | | | | | | 967680 | 107520 |
| 84 | | t$_{0,2,5}${4,3$^6$} | Penticantellated 8-cube Terirhombated octeract | | | | | | | 1218560 | 143360 |
| 85 | | t$_{0,2,4}${4,3$^6$} | Stericantellated 8-cube Cellirhombated octeract | | | | | | | 752640 | 107520 |
| 86 | | t$_{0,2,3}${4,3$^6$} | Runcicantellated 8-cube Prismatorhombated octeract | | | | | | | 193536 | 43008 |
| 87 | | t$_{0,1,7}${4,3$^6$} | Heptitruncated 8-cube Exitruncated octeract | | | | | | | 93184 | 14336 |
| 88 | | t$_{0,1,6}${4,3$^6$} | Hexitruncated 8-cube Petitruncated octeract | | | | | | | 344064 | 43008 |
| 89 | | t$_{0,1,5}${4,3$^6$} | Pentitruncated 8-cube Teritruncated octeract | | | | | | | 609280 | 71680 |
| 90 | | t$_{0,1,4}${4,3$^6$} | Steritruncated 8-cube Cellitruncated octeract | | | | | | | 573440 | 71680 |
| 91 | | t$_{0,1,3}${4,3$^6$} | Runcitruncated 8-cube Prismatotruncated octeract | | | | | | | 279552 | 43008 |
| 92 | | t$_{0,1,2}${4,3$^6$} | Cantitruncated 8-cube Great rhombated octeract | | | | | | | 57344 | 14336 |
| 93 | | t$_{0,1,2,3}${3$^6$,4} | Runcicantitruncated 8-orthoplex Great prismated diacosipentacontahexazetton | | | | | | | 147840 | 26880 |
| 94 | | t$_{0,1,2,4}${3$^6$,4} | Stericantitruncated 8-orthoplex Celligreatorhombated diacosipentacontahexazetton | | | | | | | 860160 | 107520 |
| 95 | | t$_{0,1,3,4}${3$^6$,4} | Steriruncitruncated 8-orthoplex Celliprismatotruncated diacosipentacontahexazetton | | | | | | | 591360 | 107520 |
| 96 | | t$_{0,2,3,4}${3$^6$,4} | Steriruncicantellated 8-orthoplex Celliprismatorhombated diacosipentacontahexazetton | | | | | | | 591360 | 107520 |
| 97 | | t$_{1,2,3,4}${3$^6$,4} | Biruncicantitruncated 8-orthoplex Great biprismated diacosipentacontahexazetton | | | | | | | 537600 | 107520 |
| 98 | | t$_{0,1,2,5}${3$^6$,4} | Penticantitruncated 8-orthoplex Terigreatorhombated diacosipentacontahexazetton | | | | | | | 1827840 | 215040 |
| 99 | | t$_{0,1,3,5}${3$^6$,4} | Pentiruncitruncated 8-orthoplex Teriprismatotruncated diacosipentacontahexazetton | | | | | | | 2419200 | 322560 |
| 100 | | t$_{0,2,3,5}${3$^6$,4} | Pentiruncicantellated 8-orthoplex Teriprismatorhombated diacosipentacontahexazetton | | | | | | | 2257920 | 322560 |
| 101 | | t$_{1,2,3,5}${3$^6$,4} | Bistericantitruncated 8-orthoplex Bicelligreatorhombated diacosipentacontahexazetton | | | | | | | 2096640 | 322560 |
| 102 | | t$_{0,1,4,5}${3$^6$,4} | Pentisteritruncated 8-orthoplex Tericellitruncated diacosipentacontahexazetton | | | | | | | 1182720 | 215040 |
| 103 | | t$_{0,2,4,5}${3$^6$,4} | Pentistericantellated 8-orthoplex Tericellirhombated diacosipentacontahexazetton | | | | | | | 1935360 | 322560 |
| 104 | | t$_{1,2,4,5}${3$^6$,4} | Bisteriruncitruncated 8-orthoplex Bicelliprismatotruncated diacosipentacontahexazetton | | | | | | | 1612800 | 322560 |
| 105 | | t$_{0,3,4,5}${3$^6$,4} | Pentisteriruncinated 8-orthoplex Tericelliprismated diacosipentacontahexazetton | | | | | | | 1182720 | 215040 |
| 106 | | t$_{1,3,4,5}${3$^6$,4} | Bisteriruncicantellated 8-orthoplex Bicelliprismatorhombated diacosipentacontahexazetton | | | | | | | 1774080 | 322560 |
| 107 | | t$_{2,3,4,5}${4,3$^6$} | Triruncicantitruncated 8-cube Great triprismato-octeractidiacosipentacontahexazetton | | | | | | | 967680 | 215040 |
| 108 | | t$_{0,1,2,6}${3$^6$,4} | Hexicantitruncated 8-orthoplex Petigreatorhombated diacosipentacontahexazetton | | | | | | | 1505280 | 215040 |
| 109 | | t$_{0,1,3,6}${3$^6$,4} | Hexiruncitruncated 8-orthoplex Petiprismatotruncated diacosipentacontahexazetton | | | | | | | 3225600 | 430080 |
| 110 | | t$_{0,2,3,6}${3$^6$,4} | Hexiruncicantellated 8-orthoplex Petiprismatorhombated diacosipentacontahexazetton | | | | | | | 2795520 | 430080 |
| 111 | | t$_{1,2,3,6}${3$^6$,4} | Bipenticantitruncated 8-orthoplex Biterigreatorhombated diacosipentacontahexazetton | | | | | | | 2580480 | 430080 |
| 112 | | t$_{0,1,4,6}${3$^6$,4} | Hexisteritruncated 8-orthoplex Peticellitruncated diacosipentacontahexazetton | | | | | | | 3010560 | 430080 |
| 113 | | t$_{0,2,4,6}${3$^6$,4} | Hexistericantellated 8-orthoplex Peticellirhombated diacosipentacontahexazetton | | | | | | | 4515840 | 645120 |
| 114 | | t$_{1,2,4,6}${3$^6$,4} | Bipentiruncitruncated 8-orthoplex Biteriprismatotruncated diacosipentacontahexazetton | | | | | | | 3870720 | 645120 |
| 115 | | t$_{0,3,4,6}${3$^6$,4} | Hexisteriruncinated 8-orthoplex Peticelliprismated diacosipentacontahexazetton | | | | | | | 2580480 | 430080 |
| 116 | | t$_{1,3,4,6}${4,3$^6$} | Bipentiruncicantellated 8-cube Biteriprismatorhombi-octeractidiacosipentacontahexazetton | | | | | | | 3870720 | 645120 |
| 117 | | t$_{1,3,4,5}${4,3$^6$} | Bisteriruncicantellated 8-cube Bicelliprismatorhombated octeract | | | | | | | 2150400 | 430080 |
| 118 | | t$_{0,1,5,6}${3$^6$,4} | Hexipentitruncated 8-orthoplex Petiteritruncated diacosipentacontahexazetton | | | | | | | 1182720 | 215040 |
| 119 | | t$_{0,2,5,6}${3$^6$,4} | Hexipenticantellated 8-orthoplex Petiterirhombated diacosipentacontahexazetton | | | | | | | 2795520 | 430080 |
| 120 | | t$_{1,2,5,6}${4,3$^6$} | Bipentisteritruncated 8-cube Bitericellitrunki-octeractidiacosipentacontahexazetton | | | | | | | 2150400 | 430080 |
| 121 | | t$_{0,3,5,6}${3$^6$,4} | Hexipentiruncinated 8-orthoplex Petiteriprismated diacosipentacontahexazetton | | | | | | | 2795520 | 430080 |
| 122 | | t$_{1,2,4,6}${4,3$^6$} | Bipentiruncitruncated 8-cube Biteriprismatotruncated octeract | | | | | | | 3870720 | 645120 |
| 123 | | t$_{1,2,4,5}${4,3$^6$} | Bisteriruncitruncated 8-cube Bicelliprismatotruncated octeract | | | | | | | 1935360 | 430080 |
| 124 | | t$_{0,4,5,6}${3$^6$,4} | Hexipentistericated 8-orthoplex Petitericellated diacosipentacontahexazetton | | | | | | | 1182720 | 215040 |
| 125 | | t$_{1,2,3,6}${4,3$^6$} | Bipenticantitruncated 8-cube Biterigreatorhombated octeract | | | | | | | 2580480 | 430080 |
| 126 | | t$_{1,2,3,5}${4,3$^6$} | Bistericantitruncated 8-cube Bicelligreatorhombated octeract | | | | | | | 2365440 | 430080 |
| 127 | | t$_{1,2,3,4}${4,3$^6$} | Biruncicantitruncated 8-cube Great biprismated octeract | | | | | | | 860160 | 215040 |
| 128 | | t$_{0,1,2,7}${3$^6$,4} | Hepticantitruncated 8-orthoplex Exigreatorhombated diacosipentacontahexazetton | | | | | | | 516096 | 86016 |
| 129 | | t$_{0,1,3,7}${3$^6$,4} | Heptiruncitruncated 8-orthoplex Exiprismatotruncated diacosipentacontahexazetton | | | | | | | 1612800 | 215040 |
| 130 | | t$_{0,2,3,7}${3$^6$,4} | Heptiruncicantellated 8-orthoplex Exiprismatorhombated diacosipentacontahexazetton | | | | | | | 1290240 | 215040 |
| 131 | | t$_{0,4,5,6}${4,3$^6$} | Hexipentistericated 8-cube Petitericellated octeract | | | | | | | 1182720 | 215040 |
| 132 | | t$_{0,1,4,7}${3$^6$,4} | Heptisteritruncated 8-orthoplex Exicellitruncated diacosipentacontahexazetton | | | | | | | 2293760 | 286720 |
| 133 | | t$_{0,2,4,7}${3$^6$,4} | Heptistericantellated 8-orthoplex Exicellirhombated diacosipentacontahexazetton | | | | | | | 3225600 | 430080 |
| 134 | | t$_{0,3,5,6}${4,3$^6$} | Hexipentiruncinated 8-cube Petiteriprismated octeract | | | | | | | 2795520 | 430080 |
| 135 | | t$_{0,3,4,7}${4,3$^6$} | Heptisteriruncinated 8-cube Exicelliprismato-octeractidiacosipentacontahexazetton | | | | | | | 1720320 | 286720 |
| 136 | | t$_{0,3,4,6}${4,3$^6$} | Hexisteriruncinated 8-cube Peticelliprismated octeract | | | | | | | 2580480 | 430080 |
| 137 | | t$_{0,3,4,5}${4,3$^6$} | Pentisteriruncinated 8-cube Tericelliprismated octeract | | | | | | | 1433600 | 286720 |
| 138 | | t$_{0,1,5,7}${3$^6$,4} | Heptipentitruncated 8-orthoplex Exiteritruncated diacosipentacontahexazetton | | | | | | | 1612800 | 215040 |
| 139 | | t$_{0,2,5,7}${4,3$^6$} | Heptipenticantellated 8-cube Exiterirhombi-octeractidiacosipentacontahexazetton | | | | | | | 3440640 | 430080 |
| 140 | | t$_{0,2,5,6}${4,3$^6$} | Hexipenticantellated 8-cube Petiterirhombated octeract | | | | | | | 2795520 | 430080 |
| 141 | | t$_{0,2,4,7}${4,3$^6$} | Heptistericantellated 8-cube Exicellirhombated octeract | | | | | | | 3225600 | 430080 |
| 142 | | t$_{0,2,4,6}${4,3$^6$} | Hexistericantellated 8-cube Peticellirhombated octeract | | | | | | | 4515840 | 645120 |
| 143 | | t$_{0,2,4,5}${4,3$^6$} | Pentistericantellated 8-cube Tericellirhombated octeract | | | | | | | 2365440 | 430080 |
| 144 | | t$_{0,2,3,7}${4,3$^6$} | Heptiruncicantellated 8-cube Exiprismatorhombated octeract | | | | | | | 1290240 | 215040 |
| 145 | | t$_{0,2,3,6}${4,3$^6$} | Hexiruncicantellated 8-cube Petiprismatorhombated octeract | | | | | | | 2795520 | 430080 |
| 146 | | t$_{0,2,3,5}${4,3$^6$} | Pentiruncicantellated 8-cube Teriprismatorhombated octeract | | | | | | | 2580480 | 430080 |
| 147 | | t$_{0,2,3,4}${4,3$^6$} | Steriruncicantellated 8-cube Celliprismatorhombated octeract | | | | | | | 967680 | 215040 |
| 148 | | t$_{0,1,6,7}${4,3$^6$} | Heptihexitruncated 8-cube Exipetitrunki-octeractidiacosipentacontahexazetton | | | | | | | 516096 | 86016 |
| 149 | | t$_{0,1,5,7}${4,3$^6$} | Heptipentitruncated 8-cube Exiteritruncated octeract | | | | | | | 1612800 | 215040 |
| 150 | | t$_{0,1,5,6}${4,3$^6$} | Hexipentitruncated 8-cube Petiteritruncated octeract | | | | | | | 1182720 | 215040 |
| 151 | | t$_{0,1,4,7}${4,3$^6$} | Heptisteritruncated 8-cube Exicellitruncated octeract | | | | | | | 2293760 | 286720 |
| 152 | | t$_{0,1,4,6}${4,3$^6$} | Hexisteritruncated 8-cube Peticellitruncated octeract | | | | | | | 3010560 | 430080 |
| 153 | | t$_{0,1,4,5}${4,3$^6$} | Pentisteritruncated 8-cube Tericellitruncated octeract | | | | | | | 1433600 | 286720 |
| 154 | | t$_{0,1,3,7}${4,3$^6$} | Heptiruncitruncated 8-cube Exiprismatotruncated octeract | | | | | | | 1612800 | 215040 |
| 155 | | t$_{0,1,3,6}${4,3$^6$} | Hexiruncitruncated 8-cube Petiprismatotruncated octeract | | | | | | | 3225600 | 430080 |
| 156 | | t$_{0,1,3,5}${4,3$^6$} | Pentiruncitruncated 8-cube Teriprismatotruncated octeract | | | | | | | 2795520 | 430080 |
| 157 | | t$_{0,1,3,4}${4,3$^6$} | Steriruncitruncated 8-cube Celliprismatotruncated octeract | | | | | | | 967680 | 215040 |
| 158 | | t$_{0,1,2,7}${4,3$^6$} | Hepticantitruncated 8-cube Exigreatorhombated octeract | | | | | | | 516096 | 86016 |
| 159 | | t$_{0,1,2,6}${4,3$^6$} | Hexicantitruncated 8-cube Petigreatorhombated octeract | | | | | | | 1505280 | 215040 |
| 160 | | t$_{0,1,2,5}${4,3$^6$} | Penticantitruncated 8-cube Terigreatorhombated octeract | | | | | | | 2007040 | 286720 |
| 161 | | t$_{0,1,2,4}${4,3$^6$} | Stericantitruncated 8-cube Celligreatorhombated octeract | | | | | | | 1290240 | 215040 |
| 162 | | t$_{0,1,2,3}${4,3$^6$} | Runcicantitruncated 8-cube Great prismated octeract | | | | | | | 344064 | 86016 |
| 163 | | t$_{0,1,2,3,4}${3$^6$,4} | Steriruncicantitruncated 8-orthoplex Great cellated diacosipentacontahexazetton | | | | | | | 1075200 | 215040 |
| 164 | | t$_{0,1,2,3,5}${3$^6$,4} | Pentiruncicantitruncated 8-orthoplex Terigreatoprismated diacosipentacontahexazetton | | | | | | | 4193280 | 645120 |
| 165 | | t$_{0,1,2,4,5}${3$^6$,4} | Pentistericantitruncated 8-orthoplex Tericelligreatorhombated diacosipentacontahexazetton | | | | | | | 3225600 | 645120 |
| 166 | | t$_{0,1,3,4,5}${3$^6$,4} | Pentisteriruncitruncated 8-orthoplex Tericelliprismatotruncated diacosipentacontahexazetton | | | | | | | 3225600 | 645120 |
| 167 | | t$_{0,2,3,4,5}${3$^6$,4} | Pentisteriruncicantellated 8-orthoplex Tericelliprismatorhombated diacosipentacontahexazetton | | | | | | | 3225600 | 645120 |
| 168 | | t$_{1,2,3,4,5}${3$^6$,4} | Bisteriruncicantitruncated 8-orthoplex Great bicellated diacosipentacontahexazetton | | | | | | | 2903040 | 645120 |
| 169 | | t$_{0,1,2,3,6}${3$^6$,4} | Hexiruncicantitruncated 8-orthoplex Petigreatoprismated diacosipentacontahexazetton | | | | | | | 5160960 | 860160 |
| 170 | | t$_{0,1,2,4,6}${3$^6$,4} | Hexistericantitruncated 8-orthoplex Peticelligreatorhombated diacosipentacontahexazetton | | | | | | | 7741440 | 1290240 |
| 171 | | t$_{0,1,3,4,6}${3$^6$,4} | Hexisteriruncitruncated 8-orthoplex Peticelliprismatotruncated diacosipentacontahexazetton | | | | | | | 7096320 | 1290240 |
| 172 | | t$_{0,2,3,4,6}${3$^6$,4} | Hexisteriruncicantellated 8-orthoplex Peticelliprismatorhombated diacosipentacontahexazetton | | | | | | | 7096320 | 1290240 |
| 173 | | t$_{1,2,3,4,6}${3$^6$,4} | Bipentiruncicantitruncated 8-orthoplex Biterigreatoprismated diacosipentacontahexazetton | | | | | | | 6451200 | 1290240 |
| 174 | | t$_{0,1,2,5,6}${3$^6$,4} | Hexipenticantitruncated 8-orthoplex Petiterigreatorhombated diacosipentacontahexazetton | | | | | | | 4300800 | 860160 |
| 175 | | t$_{0,1,3,5,6}${3$^6$,4} | Hexipentiruncitruncated 8-orthoplex Petiteriprismatotruncated diacosipentacontahexazetton | | | | | | | 7096320 | 1290240 |
| 176 | | t$_{0,2,3,5,6}${3$^6$,4} | Hexipentiruncicantellated 8-orthoplex Petiteriprismatorhombated diacosipentacontahexazetton | | | | | | | 6451200 | 1290240 |
| 177 | | t$_{1,2,3,5,6}${3$^6$,4} | Bipentistericantitruncated 8-orthoplex Bitericelligreatorhombated diacosipentacontahexazetton | | | | | | | 5806080 | 1290240 |
| 178 | | t$_{0,1,4,5,6}${3$^6$,4} | Hexipentisteritruncated 8-orthoplex Petitericellitruncated diacosipentacontahexazetton | | | | | | | 4300800 | 860160 |
| 179 | | t$_{0,2,4,5,6}${3$^6$,4} | Hexipentistericantellated 8-orthoplex Petitericellirhombated diacosipentacontahexazetton | | | | | | | 7096320 | 1290240 |
| 180 | | t$_{1,2,3,5,6}${4,3$^6$} | Bipentistericantitruncated 8-cube Bitericelligreatorhombated octeract | | | | | | | 5806080 | 1290240 |
| 181 | | t$_{0,3,4,5,6}${3$^6$,4} | Hexipentisteriruncinated 8-orthoplex Petitericelliprismated diacosipentacontahexazetton | | | | | | | 4300800 | 860160 |
| 182 | | t$_{1,2,3,4,6}${4,3$^6$} | Bipentiruncicantitruncated 8-cube Biterigreatoprismated octeract | | | | | | | 6451200 | 1290240 |
| 183 | | t$_{1,2,3,4,5}${4,3$^6$} | Bisteriruncicantitruncated 8-cube Great bicellated octeract | | | | | | | 3440640 | 860160 |
| 184 | | t$_{0,1,2,3,7}${3$^6$,4} | Heptiruncicantitruncated 8-orthoplex Exigreatoprismated diacosipentacontahexazetton | | | | | | | 2365440 | 430080 |
| 185 | | t$_{0,1,2,4,7}${3$^6$,4} | Heptistericantitruncated 8-orthoplex Exicelligreatorhombated diacosipentacontahexazetton | | | | | | | 5591040 | 860160 |
| 186 | | t$_{0,1,3,4,7}${3$^6$,4} | Heptisteriruncitruncated 8-orthoplex Exicelliprismatotruncated diacosipentacontahexazetton | | | | | | | 4730880 | 860160 |
| 187 | | t$_{0,2,3,4,7}${3$^6$,4} | Heptisteriruncicantellated 8-orthoplex Exicelliprismatorhombated diacosipentacontahexazetton | | | | | | | 4730880 | 860160 |
| 188 | | t$_{0,3,4,5,6}${4,3$^6$} | Hexipentisteriruncinated 8-cube Petitericelliprismated octeract | | | | | | | 4300800 | 860160 |
| 189 | | t$_{0,1,2,5,7}${3$^6$,4} | Heptipenticantitruncated 8-orthoplex Exiterigreatorhombated diacosipentacontahexazetton | | | | | | | 5591040 | 860160 |
| 190 | | t$_{0,1,3,5,7}${3$^6$,4} | Heptipentiruncitruncated 8-orthoplex Exiteriprismatotruncated diacosipentacontahexazetton | | | | | | | 8386560 | 1290240 |
| 191 | | t$_{0,2,3,5,7}${3$^6$,4} | Heptipentiruncicantellated 8-orthoplex Exiteriprismatorhombated diacosipentacontahexazetton | | | | | | | 7741440 | 1290240 |
| 192 | | t$_{0,2,4,5,6}${4,3$^6$} | Hexipentistericantellated 8-cube Petitericellirhombated octeract | | | | | | | 7096320 | 1290240 |
| 193 | | t$_{0,1,4,5,7}${3$^6$,4} | Heptipentisteritruncated 8-orthoplex Exitericellitruncated diacosipentacontahexazetton | | | | | | | 4730880 | 860160 |
| 194 | | t$_{0,2,3,5,7}${4,3$^6$} | Heptipentiruncicantellated 8-cube Exiteriprismatorhombated octeract | | | | | | | 7741440 | 1290240 |
| 195 | | t$_{0,2,3,5,6}${4,3$^6$} | Hexipentiruncicantellated 8-cube Petiteriprismatorhombated octeract | | | | | | | 6451200 | 1290240 |
| 196 | | t$_{0,2,3,4,7}${4,3$^6$} | Heptisteriruncicantellated 8-cube Exicelliprismatorhombated octeract | | | | | | | 4730880 | 860160 |
| 197 | | t$_{0,2,3,4,6}${4,3$^6$} | Hexisteriruncicantellated 8-cube Peticelliprismatorhombated octeract | | | | | | | 7096320 | 1290240 |
| 198 | | t$_{0,2,3,4,5}${4,3$^6$} | Pentisteriruncicantellated 8-cube Tericelliprismatorhombated octeract | | | | | | | 3870720 | 860160 |
| 199 | | t$_{0,1,2,6,7}${3$^6$,4} | Heptihexicantitruncated 8-orthoplex Exipetigreatorhombated diacosipentacontahexazetton | | | | | | | 2365440 | 430080 |
| 200 | | t$_{0,1,3,6,7}${3$^6$,4} | Heptihexiruncitruncated 8-orthoplex Exipetiprismatotruncated diacosipentacontahexazetton | | | | | | | 5591040 | 860160 |
| 201 | | t$_{0,1,4,5,7}${4,3$^6$} | Heptipentisteritruncated 8-cube Exitericellitruncated octeract | | | | | | | 4730880 | 860160 |
| 202 | | t$_{0,1,4,5,6}${4,3$^6$} | Hexipentisteritruncated 8-cube Petitericellitruncated octeract | | | | | | | 4300800 | 860160 |
| 203 | | t$_{0,1,3,6,7}${4,3$^6$} | Heptihexiruncitruncated 8-cube Exipetiprismatotruncated octeract | | | | | | | 5591040 | 860160 |
| 204 | | t$_{0,1,3,5,7}${4,3$^6$} | Heptipentiruncitruncated 8-cube Exiteriprismatotruncated octeract | | | | | | | 8386560 | 1290240 |
| 205 | | t$_{0,1,3,5,6}${4,3$^6$} | Hexipentiruncitruncated 8-cube Petiteriprismatotruncated octeract | | | | | | | 7096320 | 1290240 |
| 206 | | t$_{0,1,3,4,7}${4,3$^6$} | Heptisteriruncitruncated 8-cube Exicelliprismatotruncated octeract | | | | | | | 4730880 | 860160 |
| 207 | | t$_{0,1,3,4,6}${4,3$^6$} | Hexisteriruncitruncated 8-cube Peticelliprismatotruncated octeract | | | | | | | 7096320 | 1290240 |
| 208 | | t$_{0,1,3,4,5}${4,3$^6$} | Pentisteriruncitruncated 8-cube Tericelliprismatotruncated octeract | | | | | | | 3870720 | 860160 |
| 209 | | t$_{0,1,2,6,7}${4,3$^6$} | Heptihexicantitruncated 8-cube Exipetigreatorhombated octeract | | | | | | | 2365440 | 430080 |
| 210 | | t$_{0,1,2,5,7}${4,3$^6$} | Heptipenticantitruncated 8-cube Exiterigreatorhombated octeract | | | | | | | 5591040 | 860160 |
| 211 | | t$_{0,1,2,5,6}${4,3$^6$} | Hexipenticantitruncated 8-cube Petiterigreatorhombated octeract | | | | | | | 4300800 | 860160 |
| 212 | | t$_{0,1,2,4,7}${4,3$^6$} | Heptistericantitruncated 8-cube Exicelligreatorhombated octeract | | | | | | | 5591040 | 860160 |
| 213 | | t$_{0,1,2,4,6}${4,3$^6$} | Hexistericantitruncated 8-cube Peticelligreatorhombated octeract | | | | | | | 7741440 | 1290240 |
| 214 | | t$_{0,1,2,4,5}${4,3$^6$} | Pentistericantitruncated 8-cube Tericelligreatorhombated octeract | | | | | | | 3870720 | 860160 |
| 215 | | t$_{0,1,2,3,7}${4,3$^6$} | Heptiruncicantitruncated 8-cube Exigreatoprismated octeract | | | | | | | 2365440 | 430080 |
| 216 | | t$_{0,1,2,3,6}${4,3$^6$} | Hexiruncicantitruncated 8-cube Petigreatoprismated octeract | | | | | | | 5160960 | 860160 |
| 217 | | t$_{0,1,2,3,5}${4,3$^6$} | Pentiruncicantitruncated 8-cube Terigreatoprismated octeract | | | | | | | 4730880 | 860160 |
| 218 | | t$_{0,1,2,3,4}${4,3$^6$} | Steriruncicantitruncated 8-cube Great cellated octeract | | | | | | | 1720320 | 430080 |
| 219 | | t$_{0,1,2,3,4,5}${3$^6$,4} | Pentisteriruncicantitruncated 8-orthoplex Great terated diacosipentacontahexazetton | | | | | | | 5806080 | 1290240 |
| 220 | | t$_{0,1,2,3,4,6}${3$^6$,4} | Hexisteriruncicantitruncated 8-orthoplex Petigreatocellated diacosipentacontahexazetton | | | | | | | 12902400 | 2580480 |
| 221 | | t$_{0,1,2,3,5,6}${3$^6$,4} | Hexipentiruncicantitruncated 8-orthoplex Petiterigreatoprismated diacosipentacontahexazetton | | | | | | | 11612160 | 2580480 |
| 222 | | t$_{0,1,2,4,5,6}${3$^6$,4} | Hexipentistericantitruncated 8-orthoplex Petitericelligreatorhombated diacosipentacontahexazetton | | | | | | | 11612160 | 2580480 |
| 223 | | t$_{0,1,3,4,5,6}${3$^6$,4} | Hexipentisteriruncitruncated 8-orthoplex Petitericelliprismatotruncated diacosipentacontahexazetton | | | | | | | 11612160 | 2580480 |
| 224 | | t$_{0,2,3,4,5,6}${3$^6$,4} | Hexipentisteriruncicantellated 8-orthoplex Petitericelliprismatorhombated diacosipentacontahexazetton | | | | | | | 11612160 | 2580480 |
| 225 | | t$_{1,2,3,4,5,6}${4,3$^6$} | Bipentisteriruncicantitruncated 8-cube Great biteri-octeractidiacosipentacontahexazetton | | | | | | | 10321920 | 2580480 |
| 226 | | t$_{0,1,2,3,4,7}${3$^6$,4} | Heptisteriruncicantitruncated 8-orthoplex Exigreatocellated diacosipentacontahexazetton | | | | | | | 8601600 | 1720320 |
| 227 | | t$_{0,1,2,3,5,7}${3$^6$,4} | Heptipentiruncicantitruncated 8-orthoplex Exiterigreatoprismated diacosipentacontahexazetton | | | | | | | 14192640 | 2580480 |
| 228 | | t$_{0,1,2,4,5,7}${3$^6$,4} | Heptipentistericantitruncated 8-orthoplex Exitericelligreatorhombated diacosipentacontahexazetton | | | | | | | 12902400 | 2580480 |
| 229 | | t$_{0,1,3,4,5,7}${3$^6$,4} | Heptipentisteriruncitruncated 8-orthoplex Exitericelliprismatotruncated diacosipentacontahexazetton | | | | | | | 12902400 | 2580480 |
| 230 | | t$_{0,2,3,4,5,7}${4,3$^6$} | Heptipentisteriruncicantellated 8-cube Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton | | | | | | | 12902400 | 2580480 |
| 231 | | t$_{0,2,3,4,5,6}${4,3$^6$} | Hexipentisteriruncicantellated 8-cube Petitericelliprismatorhombated octeract | | | | | | | 11612160 | 2580480 |
| 232 | | t$_{0,1,2,3,6,7}${3$^6$,4} | Heptihexiruncicantitruncated 8-orthoplex Exipetigreatoprismated diacosipentacontahexazetton | | | | | | | 8601600 | 1720320 |
| 233 | | t$_{0,1,2,4,6,7}${3$^6$,4} | Heptihexistericantitruncated 8-orthoplex Exipeticelligreatorhombated diacosipentacontahexazetton | | | | | | | 14192640 | 2580480 |
| 234 | | t$_{0,1,3,4,6,7}${4,3$^6$} | Heptihexisteriruncitruncated 8-cube Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton | | | | | | | 12902400 | 2580480 |
| 235 | | t$_{0,1,3,4,5,7}${4,3$^6$} | Heptipentisteriruncitruncated 8-cube Exitericelliprismatotruncated octeract | | | | | | | 12902400 | 2580480 |
| 236 | | t$_{0,1,3,4,5,6}${4,3$^6$} | Hexipentisteriruncitruncated 8-cube Petitericelliprismatotruncated octeract | | | | | | | 11612160 | 2580480 |
| 237 | | t$_{0,1,2,5,6,7}${4,3$^6$} | Heptihexipenticantitruncated 8-cube Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton | | | | | | | 8601600 | 1720320 |
| 238 | | t$_{0,1,2,4,6,7}${4,3$^6$} | Heptihexistericantitruncated 8-cube Exipeticelligreatorhombated octeract | | | | | | | 14192640 | 2580480 |
| 239 | | t$_{0,1,2,4,5,7}${4,3$^6$} | Heptipentistericantitruncated 8-cube Exitericelligreatorhombated octeract | | | | | | | 12902400 | 2580480 |
| 240 | | t$_{0,1,2,4,5,6}${4,3$^6$} | Hexipentistericantitruncated 8-cube Petitericelligreatorhombated octeract | | | | | | | 11612160 | 2580480 |
| 241 | | t$_{0,1,2,3,6,7}${4,3$^6$} | Heptihexiruncicantitruncated 8-cube Exipetigreatoprismated octeract | | | | | | | 8601600 | 1720320 |
| 242 | | t$_{0,1,2,3,5,7}${4,3$^6$} | Heptipentiruncicantitruncated 8-cube Exiterigreatoprismated octeract | | | | | | | 14192640 | 2580480 |
| 243 | | t$_{0,1,2,3,5,6}${4,3$^6$} | Hexipentiruncicantitruncated 8-cube Petiterigreatoprismated octeract | | | | | | | 11612160 | 2580480 |
| 244 | | t$_{0,1,2,3,4,7}${4,3$^6$} | Heptisteriruncicantitruncated 8-cube Exigreatocellated octeract | | | | | | | 8601600 | 1720320 |
| 245 | | t$_{0,1,2,3,4,6}${4,3$^6$} | Hexisteriruncicantitruncated 8-cube Petigreatocellated octeract | | | | | | | 12902400 | 2580480 |
| 246 | | t$_{0,1,2,3,4,5}${4,3$^6$} | Pentisteriruncicantitruncated 8-cube Great terated octeract | | | | | | | 6881280 | 1720320 |
| 247 | | t$_{0,1,2,3,4,5,6}${3$^6$,4} | Hexipentisteriruncicantitruncated 8-orthoplex Great petated diacosipentacontahexazetton | | | | | | | 20643840 | 5160960 |
| 248 | | t$_{0,1,2,3,4,5,7}${3$^6$,4} | Heptipentisteriruncicantitruncated 8-orthoplex Exigreatoterated diacosipentacontahexazetton | | | | | | | 23224320 | 5160960 |
| 249 | | t$_{0,1,2,3,4,6,7}${3$^6$,4} | Heptihexisteriruncicantitruncated 8-orthoplex Exipetigreatocellated diacosipentacontahexazetton | | | | | | | 23224320 | 5160960 |
| 250 | | t$_{0,1,2,3,5,6,7}${3$^6$,4} | Heptihexipentiruncicantitruncated 8-orthoplex Exipetiterigreatoprismated diacosipentacontahexazetton | | | | | | | 23224320 | 5160960 |
| 251 | | t$_{0,1,2,3,5,6,7}${4,3$^6$} | Heptihexipentiruncicantitruncated 8-cube Exipetiterigreatoprismated octeract | | | | | | | 23224320 | 5160960 |
| 252 | | t$_{0,1,2,3,4,6,7}${4,3$^6$} | Heptihexisteriruncicantitruncated 8-cube Exipetigreatocellated octeract | | | | | | | 23224320 | 5160960 |
| 253 | | t$_{0,1,2,3,4,5,7}${4,3$^6$} | Heptipentisteriruncicantitruncated 8-cube Exigreatoterated octeract | | | | | | | 23224320 | 5160960 |
| 254 | | t$_{0,1,2,3,4,5,6}${4,3$^6$} | Hexipentisteriruncicantitruncated 8-cube Great petated octeract | | | | | | | 20643840 | 5160960 |
| 255 | | t$_{0,1,2,3,4,5,6,7}${4,3$^6$} | Omnitruncated 8-cube Great exi-octeractidiacosipentacontahexazetton | | | | | | | 41287680 | 10321920 |

### The D$_8$ family

The D$_8$ family has symmetry of order 5,160,960 (8 factorial x 2$^7$).
This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D$_8$ Coxeter-Dynkin diagram with one or more rings. 127 (2x64-1) are repeated from the B$_8$ family and 64 are unique to this family, all listed below.
See list of D8 polytopes for Coxeter plane graphs of these polytopes.
D$_8$ uniform polytopes :
| # | Coxeter-Dynkin diagram | Name | Base point (Alternately signed) | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Circumrad |
| # | Coxeter-Dynkin diagram | Name | Base point (Alternately signed) | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | Circumrad |
| 1 | = | 8-demicube h{4,3,3,3,3,3,3} | (1,1,1,1,1,1,1,1) | 144 | 1136 | 4032 | 8288 | 10752 | 7168 | 1792 | 128 | 1.0000000 |
| 2 | = | cantic 8-cube h$_2${4,3,3,3,3,3,3} | (1,1,3,3,3,3,3,3) | | | | | | | 23296 | 3584 | 2.6457512 |
| 3 | = | runcic 8-cube h$_3${4,3,3,3,3,3,3} | (1,1,1,3,3,3,3,3) | | | | | | | 64512 | 7168 | 2.4494896 |
| 4 | = | steric 8-cube h$_4${4,3,3,3,3,3,3} | (1,1,1,1,3,3,3,3) | | | | | | | 98560 | 8960 | 2.2360678 |
| 5 | = | pentic 8-cube h$_5${4,3,3,3,3,3,3} | (1,1,1,1,1,3,3,3) | | | | | | | 89600 | 7168 | 1.9999999 |
| 6 | = | hexic 8-cube h$_6${4,3,3,3,3,3,3} | (1,1,1,1,1,1,3,3) | | | | | | | 48384 | 3584 | 1.7320508 |
| 7 | = | heptic 8-cube h$_7${4,3,3,3,3,3,3} | (1,1,1,1,1,1,1,3) | | | | | | | 14336 | 1024 | 1.4142135 |
| 8 | = | runcicantic 8-cube h$_{2,3}${4,3,3,3,3,3,3} | (1,1,3,5,5,5,5,5) | | | | | | | 86016 | 21504 | 4.1231055 |
| 9 | = | stericantic 8-cube h$_{2,4}${4,3,3,3,3,3,3} | (1,1,3,3,5,5,5,5) | | | | | | | 349440 | 53760 | 3.8729835 |
| 10 | = | steriruncic 8-cube h$_{3,4}${4,3,3,3,3,3,3} | (1,1,1,3,5,5,5,5) | | | | | | | 179200 | 35840 | 3.7416575 |
| 11 | = | penticantic 8-cube h$_{2,5}${4,3,3,3,3,3,3} | (1,1,3,3,3,5,5,5) | | | | | | | 573440 | 71680 | 3.6055512 |
| 12 | = | pentiruncic 8-cube h$_{3,5}${4,3,3,3,3,3,3} | (1,1,1,3,3,5,5,5) | | | | | | | 537600 | 71680 | 3.4641016 |
| 13 | = | pentisteric 8-cube h$_{4,5}${4,3,3,3,3,3,3} | (1,1,1,1,3,5,5,5) | | | | | | | 232960 | 35840 | 3.3166249 |
| 14 | = | hexicantic 8-cube h$_{2,6}${4,3,3,3,3,3,3} | (1,1,3,3,3,3,5,5) | | | | | | | 456960 | 53760 | 3.3166249 |
| 15 | = | hexicruncic 8-cube h$_{3,6}${4,3,3,3,3,3,3} | (1,1,1,3,3,3,5,5) | | | | | | | 645120 | 71680 | 3.1622777 |
| 16 | = | hexisteric 8-cube h$_{4,6}${4,3,3,3,3,3,3} | (1,1,1,1,3,3,5,5) | | | | | | | 483840 | 53760 | 3 |
| 17 | = | hexipentic 8-cube h$_{5,6}${4,3,3,3,3,3,3} | (1,1,1,1,1,3,5,5) | | | | | | | 182784 | 21504 | 2.8284271 |
| 18 | = | hepticantic 8-cube h$_{2,7}${4,3,3,3,3,3,3} | (1,1,3,3,3,3,3,5) | | | | | | | 172032 | 21504 | 3 |
| 19 | = | heptiruncic 8-cube h$_{3,7}${4,3,3,3,3,3,3} | (1,1,1,3,3,3,3,5) | | | | | | | 340480 | 35840 | 2.8284271 |
| 20 | = | heptsteric 8-cube h$_{4,7}${4,3,3,3,3,3,3} | (1,1,1,1,3,3,3,5) | | | | | | | 376320 | 35840 | 2.6457512 |
| 21 | = | heptipentic 8-cube h$_{5,7}${4,3,3,3,3,3,3} | (1,1,1,1,1,3,3,5) | | | | | | | 236544 | 21504 | 2.4494898 |
| 22 | = | heptihexic 8-cube h$_{6,7}${4,3,3,3,3,3,3} | (1,1,1,1,1,1,3,5) | | | | | | | 78848 | 7168 | 2.236068 |
| 23 | = | steriruncicantic 8-cube h$_{2,3,4}${4,3$^6$} | (1,1,3,5,7,7,7,7) | | | | | | | 430080 | 107520 | 5.3851647 |
| 24 | = | pentiruncicantic 8-cube h$_{2,3,5}${4,3$^6$} | (1,1,3,5,5,7,7,7) | | | | | | | 1182720 | 215040 | 5.0990195 |
| 25 | = | pentistericantic 8-cube h$_{2,4,5}${4,3$^6$} | (1,1,3,3,5,7,7,7) | | | | | | | 1075200 | 215040 | 4.8989797 |
| 26 | = | pentisterirunic 8-cube h$_{3,4,5}${4,3$^6$} | (1,1,1,3,5,7,7,7) | | | | | | | 716800 | 143360 | 4.7958317 |
| 27 | = | hexiruncicantic 8-cube h$_{2,3,6}${4,3$^6$} | (1,1,3,5,5,5,7,7) | | | | | | | 1290240 | 215040 | 4.7958317 |
| 28 | = | hexistericantic 8-cube h$_{2,4,6}${4,3$^6$} | (1,1,3,3,5,5,7,7) | | | | | | | 2096640 | 322560 | 4.5825758 |
| 29 | = | hexisterirunic 8-cube h$_{3,4,6}${4,3$^6$} | (1,1,1,3,5,5,7,7) | | | | | | | 1290240 | 215040 | 4.472136 |
| 30 | = | hexipenticantic 8-cube h$_{2,5,6}${4,3$^6$} | (1,1,3,3,3,5,7,7) | | | | | | | 1290240 | 215040 | 4.3588991 |
| 31 | = | hexipentirunic 8-cube h$_{3,5,6}${4,3$^6$} | (1,1,1,3,3,5,7,7) | | | | | | | 1397760 | 215040 | 4.2426405 |
| 32 | = | hexipentisteric 8-cube h$_{4,5,6}${4,3$^6$} | (1,1,1,1,3,5,7,7) | | | | | | | 698880 | 107520 | 4.1231055 |
| 33 | = | heptiruncicantic 8-cube h$_{2,3,7}${4,3$^6$} | (1,1,3,5,5,5,5,7) | | | | | | | 591360 | 107520 | 4.472136 |
| 34 | = | heptistericantic 8-cube h$_{2,4,7}${4,3$^6$} | (1,1,3,3,5,5,5,7) | | | | | | | 1505280 | 215040 | 4.2426405 |
| 35 | = | heptisterruncic 8-cube h$_{3,4,7}${4,3$^6$} | (1,1,1,3,5,5,5,7) | | | | | | | 860160 | 143360 | 4.1231055 |
| 36 | = | heptipenticantic 8-cube h$_{2,5,7}${4,3$^6$} | (1,1,3,3,3,5,5,7) | | | | | | | 1612800 | 215040 | 4 |
| 37 | = | heptipentiruncic 8-cube h$_{3,5,7}${4,3$^6$} | (1,1,1,3,3,5,5,7) | | | | | | | 1612800 | 215040 | 3.8729835 |
| 38 | = | heptipentisteric 8-cube h$_{4,5,7}${4,3$^6$} | (1,1,1,1,3,5,5,7) | | | | | | | 752640 | 107520 | 3.7416575 |
| 39 | = | heptihexicantic 8-cube h$_{2,6,7}${4,3$^6$} | (1,1,3,3,3,3,5,7) | | | | | | | 752640 | 107520 | 3.7416575 |
| 40 | = | heptihexiruncic 8-cube h$_{3,6,7}${4,3$^6$} | (1,1,1,3,3,3,5,7) | | | | | | | 1146880 | 143360 | 3.6055512 |
| 41 | = | heptihexisteric 8-cube h$_{4,6,7}${4,3$^6$} | (1,1,1,1,3,3,5,7) | | | | | | | 913920 | 107520 | 3.4641016 |
| 42 | = | heptihexipentic 8-cube h$_{5,6,7}${4,3$^6$} | (1,1,1,1,1,3,5,7) | | | | | | | 365568 | 43008 | 3.3166249 |
| 43 | = | pentisteriruncicantic 8-cube h$_{2,3,4,5}${4,3$^6$} | (1,1,3,5,7,9,9,9) | | | | | | | 1720320 | 430080 | 6.4031243 |
| 44 | = | hexisteriruncicantic 8-cube h$_{2,3,4,6}${4,3$^6$} | (1,1,3,5,7,7,9,9) | | | | | | | 3225600 | 645120 | 6.0827627 |
| 45 | = | hexipentiruncicantic 8-cube h$_{2,3,5,6}${4,3$^6$} | (1,1,3,5,5,7,9,9) | | | | | | | 2903040 | 645120 | 5.8309517 |
| 46 | = | hexipentistericantic 8-cube h$_{2,4,5,6}${4,3$^6$} | (1,1,3,3,5,7,9,9) | | | | | | | 3225600 | 645120 | 5.6568542 |
| 47 | = | hexipentisteriruncic 8-cube h$_{3,4,5,6}${4,3$^6$} | (1,1,1,3,5,7,9,9) | | | | | | | 2150400 | 430080 | 5.5677648 |
| 48 | = | heptsteriruncicantic 8-cube h$_{2,3,4,7}${4,3$^6$} | (1,1,3,5,7,7,7,9) | | | | | | | 2150400 | 430080 | 5.7445626 |
| 49 | = | heptipentiruncicantic 8-cube h$_{2,3,5,7}${4,3$^6$} | (1,1,3,5,5,7,7,9) | | | | | | | 3548160 | 645120 | 5.4772258 |
| 50 | = | heptipentistericantic 8-cube h$_{2,4,5,7}${4,3$^6$} | (1,1,3,3,5,7,7,9) | | | | | | | 3548160 | 645120 | 5.291503 |
| 51 | = | heptipentisteriruncic 8-cube h$_{3,4,5,7}${4,3$^6$} | (1,1,1,3,5,7,7,9) | | | | | | | 2365440 | 430080 | 5.1961527 |
| 52 | = | heptihexiruncicantic 8-cube h$_{2,3,6,7}${4,3$^6$} | (1,1,3,5,5,5,7,9) | | | | | | | 2150400 | 430080 | 5.1961527 |
| 53 | = | heptihexistericantic 8-cube h$_{2,4,6,7}${4,3$^6$} | (1,1,3,3,5,5,7,9) | | | | | | | 3870720 | 645120 | 5 |
| 54 | = | heptihexisteriruncic 8-cube h$_{3,4,6,7}${4,3$^6$} | (1,1,1,3,5,5,7,9) | | | | | | | 2365440 | 430080 | 4.8989797 |
| 55 | = | heptihexipenticantic 8-cube h$_{2,5,6,7}${4,3$^6$} | (1,1,3,3,3,5,7,9) | | | | | | | 2580480 | 430080 | 4.7958317 |
| 56 | = | heptihexipentiruncic 8-cube h$_{3,5,6,7}${4,3$^6$} | (1,1,1,3,3,5,7,9) | | | | | | | 2795520 | 430080 | 4.6904159 |
| 57 | = | heptihexipentisteric 8-cube h$_{4,5,6,7}${4,3$^6$} | (1,1,1,1,3,5,7,9) | | | | | | | 1397760 | 215040 | 4.5825758 |
| 58 | = | hexipentisteriruncicantic 8-cube h$_{2,3,4,5,6}${4,3$^6$} | (1,1,3,5,7,9,11,11) | | | | | | | 5160960 | 1290240 | 7.1414285 |
| 59 | = | heptipentisteriruncicantic 8-cube h$_{2,3,4,5,7}${4,3$^6$} | (1,1,3,5,7,9,9,11) | | | | | | | 5806080 | 1290240 | 6.78233 |
| 60 | = | heptihexisteriruncicantic 8-cube h$_{2,3,4,6,7}${4,3$^6$} | (1,1,3,5,7,7,9,11) | | | | | | | 5806080 | 1290240 | 6.480741 |
| 61 | = | heptihexipentiruncicantic 8-cube h$_{2,3,5,6,7}${4,3$^6$} | (1,1,3,5,5,7,9,11) | | | | | | | 5806080 | 1290240 | 6.244998 |
| 62 | = | heptihexipentistericantic 8-cube h$_{2,4,5,6,7}${4,3$^6$} | (1,1,3,3,5,7,9,11) | | | | | | | 6451200 | 1290240 | 6.0827627 |
| 63 | = | heptihexipentisteriruncic 8-cube h$_{3,4,5,6,7}${4,3$^6$} | (1,1,1,3,5,7,9,11) | | | | | | | 4300800 | 860160 | 6.0000000 |
| 64 | = | heptihexipentisteriruncicantic 8-cube h$_{2,3,4,5,6,7}${4,3$^6$} | (1,1,3,5,7,9,11,13) | | | | | | | 2580480 | 10321920 | 7.5498347 |

### The E$_8$ family

The E$_8$ family has symmetry order 696,729,600.
There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Eight forms are shown below, 4 single-ringed, 3 truncations (2 rings), and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing.
See also list of E8 polytopes for Coxeter plane graphs of this family.
E$_8$ uniform polytopes :
| # | Coxeter-Dynkin diagram | Names | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram | Names | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices |
| 1 | | 4$_{21}$ (fy) | 19440 | 207360 | 483840 | 483840 | 241920 | 60480 | 6720 | 240 |
| 2 | | Truncated 4$_{21}$ (tiffy) | | | | | | | 188160 | 13440 |
| 3 | | Rectified 4$_{21}$ (riffy) | 19680 | 375840 | 1935360 | 3386880 | 2661120 | 1028160 | 181440 | 6720 |
| 4 | | Birectified 4$_{21}$ (borfy) | 19680 | 382560 | 2600640 | 7741440 | 9918720 | 5806080 | 1451520 | 60480 |
| 5 | | Trirectified 4$_{21}$ (torfy) | 19680 | 382560 | 2661120 | 9313920 | 16934400 | 14515200 | 4838400 | 241920 |
| 6 | | Rectified 1$_{42}$ (buffy) | 19680 | 382560 | 2661120 | 9072000 | 16934400 | 16934400 | 7257600 | 483840 |
| 7 | | Rectified 2$_{41}$ (robay) | 19680 | 313440 | 1693440 | 4717440 | 7257600 | 5322240 | 1451520 | 69120 |
| 8 | | 2$_{41}$ (bay) | 17520 | 144960 | 544320 | 1209600 | 1209600 | 483840 | 69120 | 2160 |
| 9 | | Truncated 2$_{41}$ | | | | | | | | 138240 |
| 10 | | 1$_{42}$ (bif) | 2400 | 106080 | 725760 | 2298240 | 3628800 | 2419200 | 483840 | 17280 |
| 11 | | Truncated 1$_{42}$ | | | | | | | | 967680 |
| 12 | | Omnitruncated 4$_{21}$ | | | | | | | | 696729600 |

## Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 7-space:
| # | Coxeter group | Coxeter group | Forms |
| 1 | ${\tilde {A}}_{7}$ | [3$^{[8]}$] | 29 |
| 2 | ${\tilde {C}}_{7}$ | [4,3$^5$,4] | 135 |
| 3 | ${\tilde {B}}_{7}$ | [4,3$^4$,3$^{1,1}$] | 191 (64 new) |
| 4 | ${\tilde {D}}_{7}$ | [3$^{1,1}$,3$^3$,3$^{1,1}$] | 77 (10 new) |
| 5 | ${\tilde {E}}_{7}$ | [3$^{3,3,1}$] | 143 |
Regular and uniform tessellations include:
* ${\tilde {A}}_{7}$ 29 uniquely ringed forms, including:
** 7-simplex honeycomb: {3$^{[8]}$}
* ${\tilde {C}}_{7}$ 135 uniquely ringed forms, including:
** Regular 7-cube honeycomb: {4,3$^4$,4} = {4,3$^4$,3$^{1,1}$}, =
* ${\tilde {B}}_{7}$ 191 uniquely ringed forms, 127 shared with ${\tilde {C}}_{7}$, and 64 new, including:
** 7-demicube honeycomb: h{4,3$^4$,4} = {3$^{1,1}$,3$^4$,4}, =
* ${\tilde {D}}_{7}$, [3$^{1,1}$,3$^3$,3$^{1,1}$]: 77 unique ring permutations, and 10 are new, the first Coxeter called a quarter 7-cubic honeycomb.
* ${\tilde {E}}_{7}$ 143 uniquely ringed forms, including:
** 1$_{33}$ honeycomb: {3,3$^{3,3}$},
** 3$_{31}$ honeycomb: {3,3,3,3$^{3,1}$},

### Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 8, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 paracompact hyperbolic Coxeter groups of rank 8, each generating uniform honeycombs in 7-space as permutations of rings of the Coxeter diagrams.
| ${\bar {P}}_{7}$ = [3,3$^{[7]}$]: | ${\bar {Q}}_{7}$ = [3$^{1,1}$,3$^2$,3$^{2,1}$]: | ${\bar {S}}_{7}$ = [4,3$^3$,3$^{2,1}$]: | ${\bar {T}}_{7}$ = [3$^{3,2,2}$]: |