  We initiate the study of subgroups $H$ of the general linear group
$GL_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th
exterior power of an elementary group $\bigwedge^mE_n(R)$. Each such group $H$
corresponds to a uniquely defined level $(A_0,\dots,A_{m-1})$, where
$A_0,\dots,A_{m-1}$ are ideals of $R$ with certain relations. In the crucial
case of the exterior squares, we state the subgroup lattice to be standard. In
other words, for $\bigwedge^2E_n(R)$ all intermediate subgroups $H$ are
parametrized by a single ideal of the ring $R$. Moreover, we characterize
$\bigwedge^mGL_n(R)$ as the stabilizer of a system of invariant forms. This
result is classically known for algebraically closed fields, here we prove the
corresponding group scheme to be smooth over $\mathbb{Z}$. So the last result
holds over arbitrary commutative rings.
