I want to show that the following two composition series are not the same:
$D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle s \right \rangle \triangleright (1)$
$D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle r^{2}s \right \rangle \triangleright (1)$
I got these series from its lattice.
If I show this, then I can conclude they are distinct series for $D_{8}$. But how would I show this? I am not very familiar with dihedral groups!