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Abstract. Second-metacyclic finite 2-groups are finite 2-groups with some non-metacyclic maximal subgroup and with all second-maximal subgroups being metacyclic. According to a known result there are only four non-metacyclic finite 2-groups with all maximal subgroups being metacyclic. The groups pointed in the title should contain some of these groups as a subgroup of index 2. There are seventeen second-maximal finite 2-groups, four among them being of order 16, ten of order 32 and three of order 64.

Abstract. Nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic and possessing at least one nonabelian and at least one nonmetacyclic proper subgroup have been investigated and classified. Using the obtained result and two previously known results one gets the complete classification of all nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic.

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In this paper we investigate the title groups which we call isomaximal. We give the list of all isomaximal 2-groups with abelian maximal subgroups. Further, we prove some properties of isomaximal 2-groups with nonabelian maximal subgroups. After that, we investigate the structure of isomaximal groups of order less than 64. Finally, in Theorem 14. we show that the minimal nonmetacyclic group of order 32 possesses a unique isomaximal extension of order 64.