Subgroup structure of semidihedral group:SD16

We are interested in the group $$G = SD_{16}$$, the semidihedral group:SD16, given by the presentation:

$$\langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle$$

where $$e$$ denotes the identity element.

The group has 16 elements:

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