The group $(\mathbb{Z}/2\mathbb{Z}) \wr S_{n}$ is finite maximal for $n$ sufficiently large (probably $n>72$ will do, by a theorem of M.Collins on a sharp form of Jordan's theorem on finite complex linear groups, and this is likely more generous than the true bound). I mean here the group of $n \times n$ monomial matrices with all  non-zero entries $\pm 1$.

To roughly sketch out the argument for large enough $n$: think of the hyperoctahedral group as a complex linear group of degree $n$. If it is contained in larger finite subgroup X of ${\rm O}_{n}(\mathbb{R}),$ the (viewed as complex linear group), $X$ has an Abelian normal subgroup $A$ with $[X:A] \leq (n+1)!,$ and the only way any index greater than $n!$ can be attained is if $X/A$ has a composition factor $A_{n+1}.$ But if $X/A$ has a composition factor $A_{n+1},$ then $X$ must be (irreducible and) primitive as complex linear group, in which case $A$ must consist of scalars. But since $X$ is really a subgroup of ${\mathbb{O}}_{n}(\mathbb{R})$, we have $|Z(X)| \leq 2$ and $|X| \leq (2n+2)n!,$ contrary to $|X| > 2^{n}n!.$ I should add that if $[X:A] \leq n!,$ we get $|A| \leq 2^{n},$ which is also a contradiction.