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On finite groups close to completely factorisable groups

A subgroup H of a group G is called complemented in G if a subgroup K exists in G such that G = HK and H ∩ K = 1. A group is called completely factorisable if each subgroup of the group is complemented. Let D(G) be the subgroup of a group G generated by all subgroups of G which have no complements in G, Z(G) be the centre of the group G, and Φ(G) be the Frattini subgroup of the group G. If all subgroups of G are complemented in G, then we set D(G) = 1. Each cyclic subgroup of the Frattini subgroup Φ(G) of the group G has no complement in G, therefore Φ(G) ⊆ D(G). In the paper, we obtain a complete description of the structure of a finite group G such that D(G) ⊆ Z(G)Φ(G).