Finite p-groups with many minimal nonabelian subgroups

In Theorem 2.1 we characterize finite p-groups G such that each nonabelian subgroup H of G which possesses an abelian maximal subgroup is minimal nonabelian. In Theorem 3.1 the problem 2331 of Y. Berkovich (stated in Y. Berkovich and Z. Janko, in preparation [4]) about finite p-groups with “many” minimal nonabelian subgroups is solved.