Non-normal subgroups of dihedral group:D8

Definition
Suppose $$G$$ is the dihedral group of order eight (degree four) given by the presentation below, where $$e$$ denotes the identity element of $$G$$:

$$G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$$.

Then, we are interested in the following four subgroups:

$$A_0 := \langle x \rangle = \{ x, e \}, A_1 := \langle ax \rangle = \{ ax, e \}, A_2 := \langle a^2x \rangle = \{ a^2x, e \}, A_3 = \langle a^3x \rangle = \{ a^3x, e \}$$.

$$A_0$$ and $$A_2$$ are conjugate subgroups (via $$a$$, for instance). $$A_1$$ and $$A_3$$ are conjugate subgroups (via $$a$$, for instance). $$A_0$$ and $$A_1$$ are not conjugate but are related by an outer automorphism that fixes $$a$$ and sends $$x$$ to $$ax$$. Thus, all four subgroups are automorphic subgroups. These are the only non-normal subgroups of $$G$$ and they are all 2-subnormal subgroups.

Effect of subgroup operators
Specific values (in the second column) are for $$A_0 = \langle x \rangle$$.

Intermediate subgroups
We use $$A_0 = \langle x\rangle$$ here.

Invariance under automorphisms and endomorphisms
Suppose $$c_a$$ and $$c_x$$ denote conjugation by $$a$$ and $$x$$ respectively. Let $$\sigma$$ denote the automorphism that sends $$a$$ to $$a^3$$ and $$x$$ to $$ax$$. Then, $$\langle c_a, c_x\rangle$$ is the inner automorphism group and $$\langle c_a, c_x, \sigma \rangle$$ is the automorphism group.

The automorphism $$c_x$$ fixes $$A_0$$ and $$A_2$$ while interchanging $$A_1$$ and $$A_3$$. The automorphism $$c_a$$ interchanges $$A_0$$ and $$A_2$$ while also interchanging $$A_1$$ and $$A_3$$. The automorphism $$c_{ax} = c_a \circ c_x$$ fixes $$A_1$$ and $$A_3$$ while interchanging $$A_0$$ and $$A_2$$. The automorphism $$\sigma$$ interchanges $$A_0$$ and $$A_1$$ and also interchanges $$A_2$$ and $$A_3$$.