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