M16 in holomorph of Z8

Definition
The group $$G$$, defined as the holomorph of Z8, has the following presentation (with $$e$$ denoting the identity element):

$$\! G := \langle a,x,y \mid a^8 = x^2 = y^2 = e, xax^{-1} = a^{-1}, yay^{-1} = a^5 \rangle$$

The subgroup $$H$$ of interest is:

$$H = \langle a, y \rangle$$

It is a subgroup of order 16, and is isomorphic to M16.