Direct product of Z4 and Z2 in M16

Definition
We consider the group:

$$G = M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$$

with $$e$$ denoting the identity element.

This is a group of order 16, with elements:

$$\{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$$

We are interested in the subgroup:

$$H = \{ e, a^2, a^4, a^6, x, a^2x, a^4x, a^6x \} = \langle a^2,x \rangle$$

This is a subgroup of order eight isomorphic to direct product of Z4 and Z2, where the cyclic four-subgroup is $$\langle a^2 \rangle$$ and the cyclic two-subgroup is $$\langle x \rangle$$.

Cosets
The subgroup is a subgroup of index two, hence it is a normal subgroup (see index two implies normal) and in particular its left cosets coincide with its right cosets. The two cosets are:

$$H =\{ e, a^2, a^4, a^6, x, a^2x, a^4x, a^6x \}, G \setminus H = \{ a, a^3, a^5, a^7, ax, a^3x, a^5x, a^7x \}$$

Complements
The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.