The subgroup correspondence theorem says that for a group $G$ and a normal subgroup $N\subset G$, the canonical projection $\pi\colon G\rightarrow G/N$ establishes a bijection between the subsets of subgroups of $G$ containing $N$ and the subgroups of $G/N$ (mapping a subgroup $U$ of $G$ containing $N$ to $\pi(U)$ and a subgroup $V$ of $G/N$ to $\pi^{-1}(V)$), and that this bijection identifies normal subgroups on both sides.
Now I'm searching for simple, yet pedagogically valuable examples to illustrate this, on the level of an introductory course on algebra. In particular, in view of the addendum for normal subgroups, the groups involved should be non-abelian.
A first example I found is to use the symmetric group $S_4$ as $G$ and let $N:=V_4$ be the Klein four group, so that $G/N\simeq S_3$. Then, one may identify $A_3\lhd S_3$ with $A_4\lhd S_4$ and the three subgoups of order $2$ in $S_3$ with the three dihedral groups $D_4$ in $S_4$.
However, I would really like to know about other and possibly even shorter examples! So my question is: What are your favorite examples for this? Many thanks in advance!