Finite 2-groups with index of every cyclic subgroup in its normal closure no greater than 4

In this paper, we study the finite 2-group G such that |G〈a〉:〈a〉|⩽22 for every a∈G. We prove that G8⩽Z(G), G4 is abelian and cl(G)⩽5. We also get that exp(G′)⩽23.