Cyclic maximal subgroup of dihedral group:D16

Definition
Here, $$G$$ is the dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation:

$$G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle$$

$$G$$ has 16 elements:

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

The subgroup $$H$$ of interest is the subgroup $$\langle a \rangle$$. It is cyclic of order 8 and is given by:

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

Cosets
The subgroup has index two and is hence normal (since index two implies normal). Its left cosets coincide with its right cosets, and there are two cosets:

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

Complements
There are eight possible permutable complements to $$H$$ in $$G$$, all of them automorphic to each other:

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