Let $K$ be a function field of genus $g$ with a finite constant field ${\mathbb{F}}_q$. Choose a place $\infty$ of $K$ of degree $\delta$ and let ${\mathbb{C}}$ be the arithmetic Dedekind domain consisting of all elements of $K$ that are integral outside $\infty$. An explicit formula is given (in terms of $q$, $g$ and $\delta$) for the minimum index of a non-congruence subgroup in SL$_2({\mathcal{C}})$. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL$_2({\mathcal{C}})$. The minimum index of a normal non-congruence subgroup is also determined.