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For a group G, by δ(G) we denote the number of conjugate classes of the non-cyclic subgroups of G. In this paper, the groups G with δ(G)=3 are classified.

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set ${\cal M}_d(P)=\{P_1,\dots,P_d\}$ of maximal subgroups of P such that $\bigcap_{i=1}^d P_i=\Phi (P)$. It is shown that if every member ${\cal M}_d(P)$ of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

Let G be a finite group. By using the concept of strong θ-completions for maximal subgroups of G, we obtain in this paper some new conditions under which G is solvable or supersolvable.