  We classify the linearly reductive finite subgroup schemes $G$ of
$SL_2=SL(V)$ over an algebraically closed field $k$ of positive characteristic,
up to conjugation. As a corollary, we prove that such $G$ is in one-to-one
correspondence with an isomorphism class of two-dimensional $F$-rational
Gorenstein complete local rings with the coefficient field $k$ by the
correspondence $G\mapsto ((\mathop{\mathrm{Sym}} V)^G)\,\hat{~}$.
