Klein four-subgroup of 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:

$$\! H = \{ e, x, a^4, a^4x \}$$

This is a subgroup of order four isomorphic to the Klein four-group, i.e., it is an elementary abelian group of prime-square order for the prime 2: all its non-identity elements have order 2.

Cosets
The subgroup has order 4 and index 4, so it has four cosets. Since it is a normal subgroup, the left cosets coincide with the right cosets:

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

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.