Let me add to this nice answer of Ian Agol, that the isometry group is always finite, and contains at most $84(g-1)$ elements, if we count orientation-preserving (conformal) isometries, where $g\geq 2$ is the genus (Hurwitz, Math Ann, 41 (1893)). For an exposition in English, see the book by Tsuji, Potential theory in modern function theory.
EDIT. Let me also mention the beautiful book, The Eightfold Way (S. Levy, editor) which contains several surveys on the subject, in particular Hurwitz grous and surfaces, by Murray Macbeath, where he describes history and motivation for this theorem.