If you want to go really concrete, take any connected finite $3$-valent graph in the $2$-sphere with disks as components of the complement.
This yields a conjugacy class of genus $0$ finite index torsion free subgroups of $\Gamma(1)=PSL(2,\mathbb{Z})$. In fact the index is $6n$ if the graph has $2n$ vertices (hence $3n$ edges and $n-2$ faces).
But there are clearly infinitely many such graphs (up to oriented homeomorphism), whereas there are only finitely many congruence subgroups of genus $0$ in $\Gamma(1)$ (about $33$ conjugacy classes of torsion free ones, with maximum index $60$, according to this table).