Summary: The knowledge of fuzzy sets and systems has become a considerable aspect to apply in various mathematical systems. In this paper, we apply knowledge of fuzzy sets to group structures. We consider fuzzy subgroups of finite abelian groups, denoted by \(G=\mathbb{Z}_{p^n}+\mathbb{Z}_{q^m}\), where \(\mathbb{Z}\) are the integers, \(p\) and \(q\) are distinct primes and \(m,n\) are natural numbers. The fuzzy subgroups are classified using the notion of equivalence classes. In essence the equivalence relations of fuzzy subsets \(X\) are extended to equivalence relations of fuzzy subgroups of a group \(G\). We then use the notion of flags and keychains as tools to enumerate fuzzy subgroups of \(G\). In this way, we characterize the properties of the fuzzy subgroups of \(G\). Finally, we use maximal chains to construct a fuzzy subgroups-lattice diagram for these groups of \(G\).