Non-normal subgroups 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 following two conjugate subgroups:

$$\! H_1 = \{ e, x \}, H_2 = \{e, a^4x \}$$

The two subgroups are conjugate by any element not centralizing either of them. Specifically, we can choose any of the elements $$a,a^3,a^5,a^7,ax,a^3x,a^5x,a^7x$$ to conjugate either subgroup into the other.