Let X be a finite set and let T(X) denote the full transformation semigroup on X. For a fixed nonempty subset Y of X, we define S(X,Y) as the subsemigroup of T(X) given by S(X,Y) = {α ∈ T(X) : Yα ⊆ Y}. In this paper, we identify all maximal subsemigroups of S(X,Y ) when Y is a proper subset of X. Additionally, we show that within this semigroup, the maximal subsemigroups and the maximal regular subsemigroups coincide if and only if |Y | = 1.