Subgroup structure of central product of D8 and Z4

This article gives specific information, namely, subgroup structure, about a particular group, namely: central product of D8 and Z4.
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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 $\langle x\rangle$, $\langle ax\rangle$, $\langle a^{2}x\rangle$, $\langle a^{3}x\rangle$. These come in two conjugacy classes of 2-subnormal subgroups, one comprising $\langle x\rangle$ and $\langle a^{2}x\rangle$ and the other comprising $\langle ax\rangle$ and $\langle a^{3}x\rangle$. However, they are all automorphic subgroups. Isomorphic to cyclic group:Z2. (4)
4. The subgroups $\langle ay\rangle$ and $\langle a^{3}y\rangle$. 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. (1)
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)

Tables for quick information

Table classifying isomorphism types of subgroups

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	$(1,1)$	1	1	1	1
Cyclic group:Z2	$(2,1)$	7	4	1	1
Cyclic group:Z4	$(4,1)$	4	4	4	1
Klein four-group	$(4,2)$	3	3	3	0
Direct product of Z4 and Z2	$(8,2)$	3	3	3	0
Dihedral group:D8	$(8,3)$	3	3	3	0
Quaternion group	$(8,4)$	1	1	1	1
Central product of D8 and Z4	$(16,13)$	1	1	1	1
Total	--	23	20	17	5

Table listing number of subgroups by order

Group order	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
$1$	1	1	1	1
$2$	7	4	1	1
$4$	7	7	7	1
$8$	7	7	7	1
$16$	1	1	1	1
Total	23	20	17	5