Finite groups with three nonabelian subgroups

Abstract

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.