  In the late 19th century, Klein inaugurated a program for describing the
finite subgroups of $PGL_2(k)$ by treating the case in which the field $k$ is
the complex numbers. Fifteen years later Dickson extended Klein's arguments to
deal with finite fields. In the past century, major contributions were made by
Serre, Suzuki, and Beauville, among others. We complete this program by giving
a classification of the finite $p$-irregular subgroups of $PGL_2(k)$, up to
conjugation, for an arbitrary field $k$ of positive characteristic $p$.
