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A subgroup was defined by O. Ore to be quasinormal in a group if it permuted with all subgroups of the group, and he proved  that such a subgroup is subnormal (= subinvariant = accessible) in a finite group. Finite groups in which all subgroups are quasinormal were classified by K. Iwasawa , and more recently N. Ito and J. Szép  and the author  proved that a quasi-normal subgroup is an extension of a normal subgroup by a nilpotent group. Similar results were obtained by O. Kegel doi:10.1017/s0027763000012034 fatcat:z6d5cvqkyjeh3lbslntgq27ukq