On finite groups with some Hall normally embedded subgroups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of HG, where HG is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called a PHN-group if its all minimal subgroups and cyclic subgroup of order 4 are Hall normally embedded in G. In this paper, we give the classification of minimal non-PHN-groups. Furthermore, we investigate the structure of finite group all of whose maximal subgroups of even order are PHN-groups.