Finite non-cyclic nilpotent group whose number of subgroups is minimal

Let G be a finite group and $$\mathfrak {s}(G)$$ denote the number of subgroups of G. Aivazidis and Muller proved that if G is a non-cyclic p-group of order $$p^{\lambda }$$ , then $$\mathfrak {s}(G)\ge 6$$ whenever $$p^{\lambda }=2^3$$ ; $$s(G)\ge (p+1)(\lambda -1)+2$$ whenever $$p^{\lambda }\ne 2^3$$ . In this paper, we generalize the results of Aivazidis and Muller on all finite non-cyclic nilpotent groups. Lower bounds on $$\mathfrak {s}(G)$$ of non-cyclic nilpotent groups G are established.