We characterize finite groups with exactly two nonabelian proper subgroups. When $G$ is nilpotent, we show that $G$ is either the direct product of a minimal nonabelian $p$-group and a cyclic $q$-group or a $2$-group. When $G$ is nonnilpotent supersolvable group, we obtain the presentation of $G$. Finally, when $G$ is nonsupersolvable, we show that $G$ is a semidirect product of a $p$-group and a cyclic group.
Finite groups, minimal nonabelian groups, minimal nonnilpotent groups, critical groups
TAERI, BIJAN and BEYG, FATEMEH TAYANLOO
"Finite groups with three non-abelian subgroups,"
Turkish Journal of Mathematics: Vol. 45:
6, Article 3.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss6/3