Let G be a group. If every nontrivial subgroup of G has a proper supplement, then G is called an aS-group. We study some properties of aS-groups. For instance, it is shown that a nilpotent group G is an aS-group if and only if G is a subdirect product of cyclic groups of prime orders. We prove that if G is an aS-group which satisfies the descending chain condition on subgroups, then G is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an aS-group. Finally, it is shown that if G is an aS-group and |G| not equal pq, p, where p and q are primes, then G has a triple factorization.