page_title: Binary octahedral group
section_title: Subgroups
hierarchical_section_title: Binary octahedral group / Properties / Subgroups
caption_attribution_description: English: en:Binary octahedral group subgroups
context_page_description: In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩ is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin of order 48.
  The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp is the multiplicative group of unit quaternions.
context_section_description: The binary tetrahedral group, 2T, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. The quaternion group, Q8, consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6. The quotient group is isomorphic to S₃ (the symmetric group on 3 letters). These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2O.
  The generalized quaternion group, Q16, also forms a subgroup of 2O, index 3. This subgroup is self-normalizing so its conjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups Q8 and Q12 in 2O.
  All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).
caption: The binary octahedral group 2O=⟨2,3,4⟩ order 48, has 3 primary subgroups: 2T=⟨2,3,3⟩, index 2, Q16=⟨2,2,4⟩ index 3, and Q12=⟨2,2,3⟩ index 4.
   ⟨l,m,n⟩=binary polyhedral group
   ⟨p⟩≃Z2p, (p)≃Zp (cyclic groups)
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