Subgroup structure of direct product of D8 and Z2

This article gives specific information, namely, subgroup structure, about a particular group, namely: direct product of D8 and Z2.
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This article discusses the subgroup structure of the direct product of D8 and Z2.

A presentation for the group that we use is:

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

The group has the following subgroups:

1. The trivial group. (1)
2. The cyclic group $\langle a^2 \rangle$ of order two. This equals the commutator subgroup, is central, and is also the set of squares. Isomorphic to cyclic group:Z2. (1)
3. The subgroups $\langle y \rangle$ and $\langle a^2y \rangle$. These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
4. The subgroups $\langle x \rangle$, $\langle ax \rangle$, $\langle a^2x \rangle$, $\langle a^3x \rangle$, $\langle xy \rangle$, $\langle axy \rangle$, $\langle a^2xy$, and $\langle a^3xy \rangle$. These are all related by automorphisms, and are all 2-subnormal subgroups. They come in four conjugacy classes, namely the class $\langle x \rangle, \langle a^2x \rangle$, the class $\langle ax \rangle, \langle a^3x \rangle$, the class $\langle xy \rangle, \langle a^2xy$, and the class $\langle axy \rangle, \langle a^3xy \rangle$. Isomorphic to cyclic group:Z2. (8)
5. The subgroup $\langle a^2, y \rangle$. This is the center, hence is a characteristic subgroup. Isomorphic to Klein four-group. (1)
6. The subgroups $\langle a^2, x \rangle$, $\langle a^2, ax \rangle$, $\langle a^2, xy \rangle$, and $\langle a^2, axy \rangle$. These are all normal subgroups, but are related by outer automorphisms. Isomorphic to Klein four-group. (4)
7. The subgroups $\langle y, x \rangle$, $\langle y, ax \rangle$, $\langle y, a^2x \rangle$, $\langle y, a^3x$, $\langle a^2y, x \rangle$, $\langle a^2y, ax$, $\langle a^2y, a^2x \rangle$, $\langle a^2y, a^3x \rangle$. These subgroups are all 2-subnormal subgroups and are related by outer automorphisms, and they come in four conjugacy classes of size two. Isomorphic to Klein four-group. (8)
8. The subgroups $\langle a \rangle$ and $\langle ay \rangle$. They are both normal and are related via an outer automorphism. Isomorphic to cyclic group:Z4. (2)
9. The subgroups $\langle a^2,x,y \rangle$ and $\langle a^2, ax, y \rangle$. These are normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (2)
10. The subgroups $\langle a,x \rangle$, $\langle a, xy \rangle$, $\langle ay, x \rangle$ and $\langle ay, xy \rangle$. These are all normal and are related by an outer automorphism. Isomorphic to dihedral group:D8. (4)
11. The subgroup $\langle a, y \rangle$. This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
12. The whole group. (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	$(2,1)$	11	7	3	1
Cyclic group:Z4	$(4,1)$	2	2	2	0
Klein four-group	$(4,2)$	13	9	5	1
Direct product of Z4 and Z2	$(8,2)$	1	1	1	1
Dihedral group:D8	$(8,3)$	4	4	4	0
Elementary abelian group of order eight	$(8,5)$	2	2	2	0
Direct product of D8 and Z2	$(16,11)$	1	1	1	1
Total	--	35	27	19	5

Table listing number of subgroups by order

Group name	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
$1$	1	1	1	1
$2$	11	7	3	1
$4$	15	11	7	1
$8$	7	7	7	1
$16$	1	1	1	1
Total	35	27	19	5

The commutator subgroup (type (2))

This is the two-element subgroup generated by $a^2$.

Subgroup-defining functions yielding this subgroup

• first agemo subgroup
• commutator subgroup
• Frattini subgroup

Subgroup properties satisfied by this subgroup

On account of being a commutator subgroup as well as an agemo subgroup, this subgroup is a verbal subgroup. Thus, it satisfies the following subgroup properties:

• Fully invariant subgroup
• Image-closed fully invariant subgroup
• Characteristic subgroup
• Image-closed characteristic subgroup

It also satisfies the following properties:

• Central subgroup, Central factor

Subgroup properties not satisfied by this subgroup

• Intermediately characteristic subgroup
• Normal-isomorph-free subgroup: There are other normal subgroups isomorphic to it.
• Complemented normal subgroup, lattice-complemented subgroup, permutably complemented subgroup: This follows from the general phenomenon that nilpotent implies center is normality-large.

The center (type (5))

This is a Klein four-subgroup comprising the identity, $a^2$, $y$ and $a^2y$.

Subgroup-defining functions yielding this subgroup

• Center
• Baer norm

Subgroup properties satisfied by this subgroup

• Characteristic subgroup
• Central subgroup, central factor, transitively normal subgroup
• Characteristic central factor
• Normality-large subgroup: This is more general, since nilpotent implies center is normality-large.

Subgroup properties not satisfied by this subgroup

• Verbal subgroup
• Fully invariant subgroup
• Normal-isomorph-free subgroup
• Lattice-complemented subgroup, permutably complemented subgroup: This follows from the fact that nilpotent implies center is normality-large.

The unique characteristic subgroup of order eight (type (10))

This is a subgroup generated by $a$ and $y$, and is the direct product of a cyclic group of order four generated by $a$ and a cyclic group of order two generated by $y$.

Subgroup-defining functions yielding this subgroup

• Maximal among abelian characteristic subgroups: It is the unique maximal among abelian characteristic subgroups. In general, there need not be a unique subgroup maximal among abelian characteristic subgroups. Further information: Maximal among abelian characteristic subgroups may be multiple and isomorphic
• The unique constructibly critical subgroup.

Subgroup properties satisfied by this subgroup

• Isomorph-free subgroup
• Intermediately characteristic subgroup
• Maximal characteristic subgroup
• Characteristic subgroup
• Self-centralizing subgroup
• Maximal among abelian characteristic subgroups
• Maximal among abelian normal subgroups
• Abelian critical subgroup, constructibly critical subgroup, critical subgroup

Subgroup properties not satisfied by this subgroup

• Fully invariant subgroup
• Verbal subgroup
• Homomorph-containing subgroup