Finite nonabelian \(2\)-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent \(4\). (Q2390493)

The finite groups satisfying the property as given in the title of this paper, are classified here. It completes earlier investigations as proved in [\textit{Z. Janko}, J. Algebra 315, No. 2, 801-808 (2007; Zbl 1127.20018)], for instance. In addition, a lot of results are proved here, to wit among others: 1. Let \(G\) be a nonabelian 2-group all of whose minimal nonabelian subgroups are non-metacyclic and have exponent 4. Then \(G\) must be of exponent 4. 2. All finite non-Abelian finite 2-groups of exponent \(2^e\) (\(e\geq 3\)) are classified which do not have any minimal nonabelian subgroup of exponent \(2^e\); it solves problem No. 1475 of \textit{Y. Berkovich, Z. Janko}, Groups of prime power order. Vol. 1, Vol. 2. [de Gruyter Expositions in Mathematics 46, 47 (2008; Zbl 1168.20001, Zbl 1168.20002)].