This question already has an answer here:
Say I want to classify all groups of a given order. The abelian case is completely understood by the structure theorem for finitely generated abelian groups.
Assume our group is non-abelian, and we somehow managed to find a normal subgroup $N$ (e.g., by considering the core of some Sylow subgroup). It would be pretty nice if we could go on to construct all semidirect products of $N$ with groups of the remaining order – but $G$ need not be semidirectly reducible*.
The only counterexample that comes to mind is $0 \to C_2 \to C_4 \to C_2 \to 0$, which does not admit an appropriate section, i.e. a subgroup complementing our normal subgroup with trivial intersection.
What are counterexamples in the non-abelian case?