In this paper, union of Fuzzy subsets, Fuzzy sub – Quadratic group, fuzzy sub - Pendant group and Fuzzy sub–N groups are discussed. Moreover, some properties and theorems based on these have been derived and derive the definitions of Union of Fuzzy sub – Quadratic group, definitions of Union of fuzzy sub- Pendant group and soon definitions of Union of Fuzzy sub–N groups, and derive the definitions of Fuzzy Quadratic group, definitions of fuzzy Pendant group and soon definitions of Fuzzy N group and definition of Quadratic group, definitions of Pendant group and soon definitions of N group.