Definition. Let $G < SO(n)$ be the rotation group of some compact subset of $\mathbb{R}^n$ We call $G$ chiral if there exists a compact set $K$ such that its symmetry group equals $G$.
Question: classification of chiral subgroups of $SO(n)$.
Statement. For $n \leq 3$ chiral $\Leftrightarrow$ discrete.
Proof. (non-trivial for n = 3) $\Leftarrow$ Any discrete subgroup is a symmetry group of some compact (place a small compact with no symmetries far from 0 and act on it with your group). $\Rightarrow$ If a subgroup is not discrete, then it necessarily contains all rotations around some line. Then the compact set with each point contains a circle around this line. Then the compact has all mirror symmetries with respect to the planes passing through this line
P.S. I used to think that this is true in all dimensions and put this hypothesis in the title of the question. Later I realized (which was also pointed out to me in the comments) that there are non-discrete chiral subgroups for $n = 4$, after which I edited my question. I think I understood a lot about the classification of chiral subgroups among one-dimensional ones, but I still need to check some things.