A clamp structure especially for use in applications in which space conditions do not permit any projecting parts such as plastically deformable ears, and in which outwardly extending hooks adapted to engage in apertures are used to mechanically interconnect the clamp structure; to enable tightening of the band about an object to the fastened, cold-deformed pressed-out embossments are provided in inner and outer band portions, which extend generally in the circumferential direction so that each pressed-out embossment remains integral at least on both sides thereof, where the tool-engaging surfaces thereof are able to withstand very large tightening forces in the circumferential direction which permit a secure tightening of the clamp structure.