Q8 in central product of D8 and Z4

The group $$G$$ is central product of D8 and Z4:

$$G := \langle a,x,y \mid a^4 = x^2 = y^4 = e, xax^{-1} = a^{-1}, a^2 = y^2, ay = ya, xy = yx \rangle$$

The group is a central product of dihedral group:D8 and cyclic group:Z4, and is also a central product of the quaternion group and cyclic group:Z4. The group has 16 elements:

$$\! e, a, a^2, a^3, x, ax, a^2x, a^3x, y, ay ,a^2y, a^3y, xy, axy, a^2xy, a^3xy$$

We are interested in the subgroup:

$$H := \langle a, xy \rangle = \{ e, a, a^2, a^3, xy, axy, a^2xy, a^3xy \}$$

This group is isomorphic to the quaternion group, with $$e \mapsto 1, a^2 \mapsto -1$$.

Cosets
The subgroup has index two and index two implies normal, so it is a normal subgroup and thus its left cosets coincide with its right cosets. The two cosets are:

$$H = \{ e, a, a^2, a^3, xy, axy, a^2xy, a^3xy \}, G \setminus H = \{ x, ax, a^2x, a^3x, y, ay, a^2y, a^3y \}$$