Let $K$ be a field and let $k$ be a subfield of it. Subgroups $H$ in $\mathrm{GL}(n,K)$ are considered which contain all diagonal matrices with nonzero elements in the subfield $k$. It is said that $H$ is rich in transvections if for any pair of indices $i\ne j$ $H$ contains a transvection with a nonzero element in the position $(i,j)$. In the work a description is given of all intermediate subgroups $H$ rich in transvections under the condition that $n\ge3$, $(K:k)\ge3$. A similar question is solved also for the special linear group. Bibl. 5 titles.