Checking whether $\langle s,r^2\rangle$ is normal in $D_8$

Question

I'm preparing for a midterm and one of the suggested practice questions was Chapter 3.1 #32 in Dummit and Foote (p.88):

32. Which subgroups of $D_8$ are normal? (You can use the lattice.)

The textbook's advice on how to check if $N$ is a normal subgroup includes:

• If you have a set of generators of $N$, you can check all conjugates of the generators lie in $N$.
• If you have a set of generators of $G$, you can check that the generators for $G$ normalize $N$.

I have some questions below about technical details in my following attempt. I want to clarify my understanding of the textbook's advice, so going through this analysis is useful for me even if it's not the recommended method (which I would like to hear in addition if there is one.)

Using the lattice of $D_8$, I'm examining each subgroup one by one.

Consider $\langle s,r^2\rangle$.

First question: When checking that all conjugates of $s$ and $r^2$ live in $\langle s,r^2\rangle$, does it suffice to check the generators of $G$?

If so, we verify that

• $rsr^-1 \in \langle s,r^2\rangle$
• $sss^-1 \in \langle s,r^2\rangle$

Or, do we need to actually conjugate by every single element in $D_8$, including for example the following element?

• $r^3sr^{-3} \in \langle s,r^2\rangle$

If there are other approaches that could be useful, I'd be glad to hear them. But note that since I'm preparing for an exam, hopefully other approaches can be generalized for other groups and not just work for $D_8$.

(I also noticed that the center of $D_8$ is $\langle r\rangle$, and that the center is a normal subgroup, but I don't know how to exploit this fact further for this question.)

Thank you!