  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 n and 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
Gal(F_{p^n}/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.
