I was hunting an example of a non-trivial finite group in which
1) All non-trivial normal subgroup are non-abelian.
2) There exists a nontrivial subnormal abelian subgroup.
Is there any hope to find this out?
A subgroup $H$ of a given group $G$ is a subnormal subgroup of $G$ if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at $G$.