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).

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). ADDED: Indeed, since $\Gamma(1)\simeq Z/2*Z/3$, there is a transitive action of $\Gamma(1)$ on oriented edges of the graph, flipping orientation for $Z/2$, and rotating around source vertex for $Z/3$. The conjugacy class of stabilizers subgroups determines the graph.

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).