The paper provides a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring R that contain the elementary block-diagonal (or subsystem) subgroup Ep(ν,R) corresponding to a unitary equivalence relation ν such that all selfconjugate equivalence classes of ν are of size at least 4 and all nonselfconjugate classes of ν are of size at least 5. Namely, given a subgroup H ≥ Ep(ν,R) of Sp(2n,R), it is shown that there exists a unique exact major form net of ideals (σ, Γ) over R such that Ep(σ, Γ) ≤ H ≤ N_Sp(2n,R)(Sp(σ, Γ)). Next, the normalizer N_Sp(2n,R)(Sp(σ, Γ)) is described in terms of congruences.