S-semipermutable and Weakly S-permutable Subgroups

S-semipermutable subgroup and weakly 8-permutable subgroup are two different generalizations of 8-permutable subgroup. In this paper, we investigate the influence of s-semipermutable and weakly 8-permutable subgroups on the structure of finite groups. We give some conditions of p-nilpotency and supersolvability under assumption that some primary subgroups (for example, maximal subgroups or minimal subgroups of Sylow subgroups) are either s-semipermutable or weakly 8-permutable. Meanwhile, some results are extended by using formation theory.