However, nilpotent normal subgroups do not in general form a complete lattice, as a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent, though it will be normal.
The concept of nilpotent normal subgroups and their relationship to lattices is complex.  A subgroup generated by an infinite collection of nilpotent normal subgroups may not be nilpotent, but it will still be a normal subgroup.