Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Let Nil be the class of nilpotent groups and G be a group. We call G a meta-Nil-Hamiltonian group if any of its non-Nil subgroups is normal. Also, we call G a para-Nil-Hamiltonian group if G is a non-Nil group and every non-normal subgroup of G is either a Nil-group or a minimal non-Nil group. In this paper we investigate the class of finitely generated meta-Nil-Hamiltonian and para-Nil-Hamiltonian groups.