Let G be a group. If for every proper normal subgroup N and element x of G with N 〈 x 〉≠ G , N 〈 x 〉 is an FC -group, but G is not an FC -group, then we call G an NFC - group . In the present paper we consider the NFC -groups. We prove that every non-perfect NFC -group with non-trivial finite images is a minimal non- FC -group. Also we show that if G is a non-perfect NFC -group having no nontrivial proper subgroup of finite index, then G is a minimal non- FC -group under the condition “every Sylow p -subgroup is an FC -group for all primes p ”. In the perfect case, we show that there exist locally nilpotent perfect NFC - p -groups which are not minimal non- FC -groups and also that McLain groups M ( ℚ , G F ( p ) ) for any prime p contain such groups. We give a characterization for torsion-free case. We also consider the p -groups such that the normalizer of every element of order p is an FC -subgroup.