Generated Subgroup connected in topological group?

Question

Let $G$ be a non-abelian, uncountable topological Hausdorff group (not necessarily connected) and $x \in G$. Then of course $\overline{\langle x \rangle}$ is a closed subgroup of $G$. My question is: Under which circumstances (if there are any?) can we say that $\overline{\langle x \rangle}$ is connected in $G$? Thanks for any help in advance - sorry if I may have missed to add additional information!