Finite automorphism groups of $\mathbb{P}^1$

I would like to know all finite groups of $\operatorname{Aut}(\mathbb{P}^1)$.

I am aware of that any automorphism of $\mathbb{P}^1$ is given by 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 groups of real orthogonal group $\operatorname{SO}(3, \mathbb{R})$.

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I think the third groupe of your line of isomorphism is in fact $SO(3,\mathbb{R})$, not $SO(3,\mathbb{C})$ and hence your problem is solved (you can check that $SO(3,\mathbb{C})$ doesn't have the good dimension as a real manifold if you're not covinced...). –  Simon Henry Oct 13 '12 at 11:22