In this paper we classify finite non-abelian groups having a unique non-abelian proper subgroup.

We characterize finite groups with exactly two non-abelian proper subgroups. When G is nilpotent, we show that G is either the direct product of a minimal non-abelian p -group and a cyclic q -group or a 2 -group. When G is non-nilpotent supersolvable group, we obtain the presentation of G . Finally, when G is non-supersolvable, we show that G is a semi direct product of a p -group and a cyclic group.