Let R be a commutative ring with 1, n a natural number, and let l = [n/2]. Suppose that 2 ∈ R^∗ and l ≥ 3. We describe the subgroups of the general linear group GL(n, R) that contain the elementary orthogonal group EO(n, R). The main result of the paper says that, for every intermediate subgroup H, there exists a largest ideal A ⊴ R such that EEO(n, R, A) = EO(n, R)E(n, R, A) ⊴ H. Another important result is an explicit calculation of the normalizer of the group EEO(n, R, A). If R = K is a field, similar results were obtained earlier by Dye, King, Shang Zhi Li, and Bashkirov. For overgroups of the even split elementary orthogonal group EO(2l, R) and the elementary symplectic group E_p(2l, R), analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3).