My question is the following: what is the subgroup of $O(d)$ that preserves a given collection of linear subspaces $V_1, \ldots, V_n\subset \mathbb{R}^d$?
For a single subspace $V\subset \mathbb{R}^d$, the answer is $O(\dim V)\times O(\dim V^\perp)$, consisting of rotations that act within $V$ or within its orthogonal complement.
A first guess might be that the general answer consists of rotations that act within all possible intersections of the subspaces and their orthogonal complements: \begin{equation} \prod_{(i_1, \ldots, i_n)\in \{0, 1\}^n} O(\dim(V_1^{i_1}\cap\cdots \cap V_n^{i_n})), \end{equation} where $V_i^0\equiv V_i$, $V_i^1\equiv V_i^\perp$, and the group $O(m)$ is understood to be trivial unless $m$ is positive. However, the actual answer might be larger, and in particular, the above guess might be missing some discrete subgroups of $O(d)$. How close is this guess to being correct? In particular, is its dimension (as a group) correct?