Subgroup structure of nontrivial semidirect product of Z4 and Z4

This article describes the subgroup structure of nontrivial semidirect product of Z4 and Z4, given explicitly by the following presentation, where $$e$$ denotes the identity element:

$$\langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$$

The elements are:

$$\{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}$$

Table classifying subgroups up to automorphisms
In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.