Abelian Covers and Non-Commuting Sets in a Non-Abelian p-Group Which its Central Quotient is Metacyclic

Finite groups with three nonabelian subgroups

“…If G is the group (2), then G is metacyclic. Since d(G) = 2, we have |G/Φ(G)| = 2 2 and so G has exactly three maximal subgroups.By virtue of[6, Theorem 2.2], the maximal subgroups of G are H = ⟨a 2 , b⟩, K = ⟨ba, a 2 ⟩ and L = ⟨b 2 , a⟩.Clearly H and K are non-abelian and L is abelian. Now to prove the converse of Theorem, suppose that G has exactly two non-abelian proper subgroups H and K , say.…”