$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \O(n) $ of signed permutations has order $ n!2^n $. It is equal to the Weyl group of type $ C_n $, denoted $ W(C_n) $, see here for a list of properties.
Question:
Is it true that $ W(C_n) $ is a maximal proper closed subgroup of $ \O(n) $ if and only if $ n \neq 2,4 $?
Note that $ \O(2) $ and $ \O(4) $ are famously exceptional, for example they are not simple.
Equivalently we can define $ W_n $ to be the determinant $ 1 $ subgroup of $ W(C_n) $ and ask if $ W_n $ is a maximal proper closed subgroup of $ \SO(n) $ if and only if $ n \neq 2,4 $.
Context:
$ W_3 \cong S_4 $ is isomorphic to the symmetric group on 4 letters. It is a maximal proper closed subgroup of $ \SO(3) $ sometimes called the octahedral group https://en.wikipedia.org/wiki/Octahedral_symmetry.
It is also true that there are no closed subgroups of $ \SO(5) $ containing $ W_5 $ and none in $ \SO(6) $ containing $ W_6 $, for details on these claims see Maximal Closed Subgroups of $ SO_5(\mathbb{R}) $.
$ W_2 \cong C_4 $ is the cyclic group of order 4. It is not maximal. The infinitely many cyclic subgroups of order $ 4m $ all contain $ W_2 $.
And for $ W_4 $ we have a chain of strict containments $$ W_4 \subsetneq \Lift(W_3 \times W_3) \subsetneq \SO(4) $$ where $\Lift$ denotes the Lift through the double cover $ \SO(4) \to \SO(3) \times \SO(3) $. Observe that $ W_4 $ has order $ (4!)(2^3)=192 $ while $ \Lift(W_3 \times W_3) $ has order $ (24)(24)(2)=1152 $.
