Non-cyclic subgroup of order 4 in non-dihedral group

Question

A group $G$ has sixteen elements:

$$\{e, r, r^2, \dots , r^7, s, rs, r^2s, \dots , r^7s\},$$

where $r$ and $s$ satisfy the relations $r^8 = e, s^2 = e, sr = r^3s$. (Note that $G$ is not a dihedral group.)

a) Make a list of the cyclic subgroups of $G$. Make sure to state the elements of each cyclic subgroup, and do not list any subgroup more than once.

b) The group $G$ has two subgroups of order $4$ which are not cyclic. State the elements of each of these subgroups.

I have managed to answer part a) and for part b) I have found the klein group, $\{e, r^4, s, r^4s\}$ but I can't find the second group of order $4$, could someone please help with this? Thanks for your answers!

Answer 1

Let $G=\{e,r,r^2,...,r^7,s,rs,...,r^7s\}$
and condition $sr=r^3s \Rightarrow$ $sr^n=r^3sr^{n-1}=...=r^{3n}s$
we can calculate all element order
$|r| = |r^3| = |r^5| = |r^7| = 8$
$|r^2| = |r^6| = |rs| = |r^3s| = |r^5s| = |r^7s| = 4$
$|r^4| = |s| = |r^2s| = |r^4s| = |r^6s| = 2$
and $|e| = 1$
all non-cyclic order-4 group element is in $\{e, r^4, s, r^2s, r^4s, r^6s\}$
simple calculate we can found other non-cyclic order-4 group is
$\{e, r^2s, r^4, r^6s\}$

Answer 2

Hint: Find all elements of order two. Take two of them, $a,b$. Then if $ab=ba$, then $\{ e, a, b, ab\}$ is one such subgroup.

Answer 3

Consider

$$\{e, r^2s, r^4, r^6s\}.$$