Subgroup structure of direct product of D8 and Z2

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)

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.
 * 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