I'm interested in conformal maps of the three-torus $\mathbb T^3$, or (I think relatedly) of $\mathbb R^2 \times S^1$. (Of course if you allow the diameters of $\mathbb T^3$ to be different, then, $\mathbb R^2 \times S^1$ can be thought of as a particular limit.)
In the case of the three-torus, I can see that there are three different subgroups of translations, as well as some reflections. I can't think of any other conformal transformations. In the case of $\mathbb R^2 \times S^1$, we also get to add the group $SO(2)$ of rotations of the $\mathbb R^2$.
Are there any others? Assuming not, how can one show that no others exist?