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We distinguish 2-maximal and strictly 2-maximal subgroups of a finite group and give examples of soluble and simple groups in which every 2-maximal subgroup is strictly 2-maximal. Let  M  be a maximal subgroup of a group  G . We prove that every maximal subgroup of  M  is strictly 2-maximal in  G  if  M  is normal in  G  or if  G  is  p -soluble andWe describe the structure of a finite group in which all 2-maximal subgroups are Hall subgroups. In particular, it has a Sylow tower and all its Sylow subgroups are elementary abelian.

On 2-maximal subgroups of finite groups