  Let G be a finite group. A collection P={H1, ..., Hr} of subgroups of G,
where r > 1, is said a non-trivial partition of G if every non-identity element
of G belongs to one and only one Hi, for some 1 <=i<=r. We call a group G that
does not admit any non-trivial partition a partition-free group. In this paper,
we study a partition-free group G whose all proper non-cyclic subgroups admit
non-trivial partitions.
