Invariant rings of all finite subgroups of GL(1,C)

Question

I am trying to determine the invariant rings of all finite subgroups $\Gamma$ of $\text{GL}(1,\mathbb{C})=\mathbb{C}^\ast$. I know that since $\Gamma$ is finite, the invariant ring is finitely generated, there is $1$ algebraically independent invariant and the invariant ring has an algebra consisting of at most $1+|\Gamma|$ invariants of degree bounded above by $|\Gamma|$. I am not sure how to classify all the invariant rings however as finite groups of $\mathbb{C}^\ast$ come in various forms (I'm thinking of roots of unity here for example). Any help/hints would be appreciated.