Let $n \geq 3$. Let $A_0$ denote the $n \times n$ symmetric matrix with $1$'s on the antidiagonal and $0$'s everywhere else. We can define the associated special orthogonal group $$ \mathrm{SO}(A_0, \mathbb{F}_p) := \{ g \in \mathrm{SL}(n, \mathbb{F}_p) : g^t A_0 g = A_0 \}. $$ We can suppose that $p \neq 2$.

This group appears in *The average size of the $2$-Selmer group of Jacobians of hyperelliptic curves...* by Bhargava and Gross (arxiv link) and several follow-up works. This group acts on the group of $n \times n$ symmetric matrices by conjugation, and Bhargava and Gross study sizes of certain orbits under this action.

I'm wondering what is known about the general structure of subgroups of $\mathrm{SO}(A_0, \mathbb{F}_p)$? In particular, do we understand the subgroups of small index?

Morally, subgroups of small index could potentially be large stabilizer groups for the action, corresponding to small orbit sizes. I'm actually looking for methods to show that there aren't too many small orbits "except for expected ones", and I'm wondering if abstract group theory might be enough.

As a closely related tiny question, I note that I think that of the subgroup structure of $\mathrm{SO}(A_0, \mathbb{F}_p)$ as being essentially the same as the subgroup structure of $\mathrm{SO}(\mathbb{F}_p)$ --- is that right?