An R-subalgebra of a bounded, distributive lattice L is a sublattice that includes 0 and 1, and is closed under existing relative complements. The collection \(S_ R(L)\) of all R-subalgebras of L is a lattice under inclusion. It is shown that \(S_ R(L)\) is dually isomorphic to the lattice of all congruence relations (that is, kernels of morphisms) of the Priestley dual of L. This duality is then used to characterize those L such that \(S_ R(L)\) is semimodular, modular, distributive, and Boolean. For example, it is shown that \(S_ R(L)\) is distributive iff \(S_ R(L)\) is Boolean iff L is a chain or a four element Boolean algebra.