Subgroup structure of central product of D8 and Z4

The central product of the dihedral group of order eight and cyclic group of order four is a central product of these groups, over a common central subgroup of order two.

It is given by the presentation:

$$G := \langle a,x,y \mid a^4 = x^2 = e, a^2 = y^2, xax = a^{-1}, ay = ya, xy = yx \rangle$$.

Here, $$\langle a,x \rangle$$ is the dihedral group of order eight and $$\langle y \rangle$$ is the cyclic group of order four.

Here is a quick description of all the subgroups of this group:


 * 1) The trivial subgroup. Isomorphic to trivial group. (1)
 * 2) The subgroup $$\langle a^2 \rangle$$. This is the unique normal subgroup of order two, and is contained in the center. Isomorphic to cyclic group:Z2. (1)
 * 3) The subgroups $$\langle x \rangle$$, $$\langle ax \rangle$$, $$\langle a^2x \rangle$$, $$\langle a^3x \rangle$$. These come in two conjugacy classes of 2-subnormal subgroups, one comprising $$\langle x \rangle$$ and $$\langle a^2x \rangle$$ and the other comprising $$\langle ax \rangle$$ and $$\langle a^3x \rangle$$. However, they are all automorphic subgroups. Isomorphic to cyclic group:Z2. (4)
 * 4) The subgroups $$\langle ay \rangle$$ and $$\langle a^3y \rangle$$. These form a single conjugacy class of 2-subnormal subgroups. Isomorphic to cyclic group:Z2. (2)
 * 5) The subgroup $$\langle y \rangle$$ of order four. This is the center. Isomorphic to cyclic group:Z4. (1)
 * 6) The subgroups $$\langle a \rangle$$, $$\langle xy \rangle$$ and $$\langle axy \rangle$$. These are  normal subgroups but are automorphic subgroups: they are related by outer automorphisms. Isomorphic to cyclic group:Z4. (3)
 * 7) The subgroup $$\langle a^2, x \rangle$$, $$\langle a^2, ax \rangle$$ and $$\langle a^2, ay \rangle$$. These are all normal subgroups but are related by outer automorphisms. Isomorphic to Klein four-group. (3)
 * 8) The subgroup $$\langle a, xy \rangle$$. This is an isomorph-free subgroup of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to quaternion group. (3)
 * 9) The subgroups $$\langle a,y \rangle$$, $$\langle x, y \rangle$$ and $$\langle ax, y \rangle$$. These are all normal and related by outer automorphisms. Isomorphic to direct product of Z4 and Z2. (3)
 * 10) The subgroups $$\langle a,x \rangle$$, $$\langle xy, ay \rangle$$ and $$\langle axy, ay \rangle$$. These are all normal and are related by outer automorphisms. Isomorphic to dihedral group:D8. (3)
 * 11) The whole group. Isomorphic to central product of D8 and Z4. (1)