I would like to know all finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$.
I am aware that any automorphism of $\mathbb{P}^1$ is given by a Möbius transformation $$ z\mapsto\frac{az+b}{cz+d} $$ and thus there is an identification $$ \operatorname{Aut}(\mathbb{P}^1)\cong \operatorname{PSL}(2,\mathbb{C})\cong \operatorname{SO}(3, \mathbb{C}). $$ I thought this solved the question, but what I know is the classification of finite subgroups of real orthogonal group $\operatorname{SO}(3, \mathbb{R})$.