A subgroup of $SL_2(\mathbb{R})$ is called arithmtic if it is communserable with $SL_2(\mathbb{Z})$.

An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for some $N\in \mathbb{N}$.

Question: What are concrete examples of subgroups of $SL_2(\mathbb{R})$, which are arithmetic, but not congruence?

I have heard the Belyi theorem produces some examples, but I have never seen a concrete one.

Further Question: Can such things exist in higher rank Lie groups (real rank $\geq 2$)?

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.

An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for some $N\in \mathbb{N}$.

Question: What are concrete examples of subgroups of $SL_2(\mathbb{R})$, which are arithmetic, but not congruence?

I have heard the Belyi theorem produces some examples, but I have never seen a concrete one.

Further Question: Can such things exist in higher rank Lie groups (real rank $\geq 2$)?