D8 in 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 subgroups $$H_1$$ and $$H_2$$ of interest are:

$$\!H_1 = \{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}, \qquad H_2 = \{ e, a^2, a^4, a^6, ax, a^3x, a^5x, a^7x \}$$

Both subgroups are normal but are related by the outer automorphism class of $$a \mapsto a, x \mapsto ax$$.

Cosets
Both subgroups has index two and are normal subgroup (See index two implies normal), so left cosets coincide with right cosets.

The cosets of $$H_1$$ are:

$$\! H_1 = \{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}, G \setminus H_1 = \{ a, a^3, a^5, a^7, ax, a^3x, a^5x, a^7x \}$$

The cosets of $$H_2$$ are:

$$H_2 = \{ e, a^2, a^4, a^6, ax, a^3x, a^5x, a^7x \}, G \setminus H_2 = \{ a, a^3, a^5, a^7, x,a^2x,a^4x,a^6x \}$$