Finite Nilpotent Groups With Six Non-cyclic Subgroups

In this paper, we discuss the relationship between the structure of finite nilpotent groupand the conjugate class number of non-cyclic subgroups. Letδ(G) be the conjugate classnumber of non-cyclic subgroup of a group G. Obviously, the non-cyclic subgroups of Gare conjugated if and only if G is inner cyclic, we clearly know that ifδ(G)=1. And thesituation ofδ(G)=2,3,4,5have been completely classified by someone. Based the formerwork, this paper will continue discussing the question and will give the classification of afinite nilpotent group G whenδ(G)=6.