The problem addressed is the classification up to conjugation of the finite subgroups of the (classical) Morava stabilizer group S_n and the extended Morava stabilizer group G_n(u) associated to a formal group law F of height n over the field F_p of p elements. A complete classification in S_n is provided for any positive integer n and prime p. Furthermore, we show that the classification in the extended group also depends on F and its associated unit u in the ring of p-adic integers. We provide a theoretical framework for the classification in G_n(u), we give necessary and sufficient conditions on n, p and u for the existence in G_n(u) of extensions of maximal finite subgroups of S_n by the Galois group of F_{p^n} over F_p, and whenever such extension exist we enumerate their conjugacy classes. We illustrate our methods by providing a complete and explicit classification in the case n=2.
