  The theory of fuzzy sets has a wide range of applications, one of which is
that of fuzzy groups . The fuzzy sets were introduced by Zadeh. Even though,
the story of fuzzy logic started much earlier, it was specially designed
mathematically to represent uncertainty and vagueness. It was also, to provide
formalized tools for dealing with the imprecision intrinsic to many problems. A
group is said to be nilpotent if it has a normal series of a finite length n.
By this notion, every finite p-group is nilpotent. Nilpotent structures such as
the p-groups, have normal series of finite length. Any finite p-group has many
normal subgroups and consequently, the phenomenon of large number of
non-isomorphic subgroups of a given order. This makes it an ideal object for
combinatorial and cohomological investigations. Cartesian product (otherwise
known as the product set) plays vital roles in the course of synthesizing the
abstract groups. Previous studies have determined the number of distinct fuzzy
subgroups of various finite p-groups including those of square-free order.
However, not much work has been done on the fuzzy subgroup classification for
the nilpotent groups formed from the Cartesian products of p-groups through
their computations. This work is therefore designed to classify the nilpotent
groups formed from the Cartesian products of p-groups through their
computations. In this paper, the Cartesian products of p-groups were taken to
obtain nilpotent groups. the explicit formulae is given for the number of
distinct fuzzy subgroups of the Cartesian product of the dihedral group of
order eight with a cyclic group of order of an n power of two for, which n is
not less than three
