Subgroup structure of binary octahedral group

This article discusses the subgroup structure of the binary octahedral group, which is given by the presentation:

$$\langle a,b,c \mid a^4 = b^3 = c^2 = abc \rangle$$.

Table classifying subgroups up to automorphisms
(remaining items to be transcribed into table later)


 * 1) The quaternion group of the form . Isomorphic to quaternion group.
 * 2) The cyclic groups conjugate to $$\langle a \rangle$$. Isomorphic to cyclic group:Z8. (3)
 * 3) The dicyclic group of order $$12$$, i.e., the binary von Dyck group $$\Gamma(3,2,2)$$. Isomorphic to dicyclic group:Dic12. (4)
 * 4) The generalized quaternion group of order $$16$$. Isomorphic to generalized quaternion group. (3)
 * 5) A unique subgroup of order $$24$$, isomorphic to special linear group:SL(2,3). (1)
 * 6) The whole group. (1)