Groups with restricted non-permutable subgroups

A subgroup H of a group G is said to be permutable if HK = KH for every subgroup K of G and the group G is called metaquasihamiltonian if all subgroups of G are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group G is soluble with derived length at most 4 and contains a finite normal subgroup N such that all subgroups of the factor G/N are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian.