The modular group is commonly described as the group of linear fractional transforms $z \mapsto \displaystyle \frac{az+b}{cz+d}$ with $a,b,c,d$ integers and $ad-bc = 1$. Of course, a great deal is known about the subgroups of the modular group of finite index. But what about just subgroups of finite size?
We know that the modular group is generated by the maps $S: z \mapsto \displaystyle\frac{-1}{z}$ and $T: z \mapsto 1+z$, with $S$ of order 2 and $ST$ of order 3, so these generate groups of size 2 and 3 respectively. But has any work been done on enumerating all possible groups of finite orders?
I would also be interested in knowing about finite subgroups of the extended modular group, which is defined as similar to the modular group but with determinants $ad-bc = \pm 1$.