The concept of ρG-strong semilattice of semigroups is introduced. By using this concept, we study Green’s relation H on a completely regular semigroup S. Necessary and sufficient conditions for S/H to be a regular band or a right quasi-normal band are obtained. Important results of Petrich and Reilly on regular cryptic semigroups are generalized and enriched. In particular, characterization theorems of regular cryptogroups and normal cryptogroups are obtained.