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

Let G be a finite group and 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 s(G) >= 6 whenever p(lambda) = 2(3); s(G) >= (p + 1)(lambda - 1) + 2 whenever p(lambda) not equal 2(3). In this paper, we generalize the results of Aivazidis and Muller on all finite non-cyclic nilpotent groups. Lower bounds on s(G) of non-cyclic nilpotent groups G are established.