Before revision:  A recent observation at least one more pattern exists raise the question of the actual number of subdivisions  of the hexahedron into tetrahedra. In this articleanswers we enumerate these subdivisions by exhausting all possible ways to combine tetrahedra into hexahedra. We introduce a combinatorial algorithm that (1) generates all the combinations of tetrahedra that can be built from eight vertices and (2) tests if they subdivide a hexahedron or not. We use topological arguments and an efficient pruning strategy to drastically reduce this number. Our main result is that there are 6,966 combinatorialtriangulations of the hexahedron which can be classified into 174 equivalence classes.
After revision:  A recent observation at least one more pattern exists raise the question of the actual number of subdivisions of the hexahedron into tetrahedra is our main result. Each of the 174 equivalence classes.