Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the component group of the stabilizer $G_v$: $$ \Gamma= \pi_1(\mathcal O) = \pi_0(G_v) = G_v/G_v^{\mathrm o}. $$

Can one characterize the class of discrete groups $\Gamma$ that occur in this way (for some orbit in some representation of some $G$)? Failing that, what's the most complicated $\Gamma$ you've seen occur?

My (very possibly misguided) hunch is that e.g. $\mathbf{SL}(2,\mathbf Z)$, or the discrete Heisenberg group, don't occur "because I would have heard of it". On the other hand, it is clear that:

1. Every finite group $\Gamma$ occurs.

Indeed, using e.g. its regular representation, we can embed $\Gamma$ as a closed subgroup of $G =\mathbf{SU}(N)$ for some $N$; and then the Palais-Mostow theorem guarantees [e.g. Bourbaki, Lie Groups, IX.9.2, Cor. 2] that $\Gamma$ is the stabilizer of some vector in some $G$-module, q.e.d.

2. $\mathbf Z^n$ occurs.

Indeed, letting the additive group $G=\mathbf R^n$ act on $V=\mathbf C^n$ by $ \smash{\begin{pmatrix} e^{2\pi ig_1}\\ &\ddots\\ &&e^{2\pi ig_n} \end{pmatrix}} $ it is clear that $v={}^t(1,\dots,1)$ has stabilizer $\mathbf Z^n$.