For a maximal subgroup M of a finite group G, a θ -pair is any pair of subgroups (CD) of G such that (i) DjG, D>C, (ii) (M, C=G, <M,D> = M and (iii) C/D has no proper normal subgroup of G/D. A natural partial ordering is defined on the family of 0 -pairs. We study the further properties of the maximal 0 -pairs of M and obtain several results on 0 -pairs which imply G to be n -solvable, 7t - supersolvable and π - nilpotent