  We study the intersection lattice of the arrangement $\mathcal{A}^G$ of
subspaces fixed by subgroups of a finite linear group $G$. When $G$ is a
reflection group, this arrangement is precisely the hyperplane reflection
arrangement of $G$. We generalize the notion of finite reflection groups. We
say that a group $G$ is generated (resp. strictly generated) in codimension $k$
if it is generated by its elements that fix point-wise a subspace of
codimension at most $k$ (resp. precisely $k$).
  If $G$ is generated in codimension two, we show that the intersection lattice
of $\mathcal{A}^G$ is atomic. We prove that the alternating subgroup
$\mathsf{Alt}(W)$ of a reflection group $W$ is strictly generated in
codimension two, moreover, the subspace arrangement of $\mathsf{Alt}(W)$ is the
truncation at rank two of the reflection arrangement $\mathcal{A}^W$.
  Further, we compute the intersection lattice of all finite subgroups of
$GL_3(\mathbb{R})$, and moreover, we emphasize the groups that are "minimally
generated in real codimension two", i.e, groups that are strictly generated in
codimension two but have no real reflection representations. We also provide
several examples of groups generated in higher codimension.
