Let Λ be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal ideal m⊂C contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, Λ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals σ such that G(σ)≤H≤N(σ), where N(σ) is the normalizer of the D-net subgroup σ. A transparent classification of subgroups GL(n, Λ) normalizable by diagonal matrices is thus obtained. Further, the factor group N(σ)/G(σ) is studied.