We study the unitary subgroupV*(F2G) in the group algebras F2Gof 2-groups of maximal class over the field F2of two elements. We show that there does not exist a normal complement toGinV*(F2G) if and only ifGis the dihedral group of order 2n(n≥ 5) or the semidihedral group of order 2n(n≥5). We also describeV*(F2G) when the subgroupGhas order 8 and 16 and in this case there exist a normal complement ofGinV*(F2G).