Can a normal subgroup of a finite nonabelian group be nonabelian?

Question

We know that if a group is Abelian, then all its subgroups are normal. Also, if a group is nonabelian, it can contain a subgroup which is Abelian. Eg: The Dihedral group of order 2n, $D_{2n}$ is nonabelian for $n\geq 3$ but has $C_n$ sitting inside it. Now, my question is: "If a finite group is nonabelian, can it have a normal subgroup which is nonabelian?" In other words, can a normal subgroup of a nonabelian finite group be nonabelian? Thanks.

Answer 1

Yes, the most trivial example would be to consider the nonabelian group itself (which is a normal subgroup). If you are looking for a proper normal subgroup, then consider $A_4$ as a normal subgroup of $S_4$.