Since I happen to know the OP is number-theoretically inclined, let me add the following remark:

For "most" finite simple groups $G$ it is indeed the case that $G = \langle x, y \rangle$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a quotient of the free product $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z} \cong \operatorname{PSL}_2(\mathbb{Z}) = \Gamma(1)$.

This has the following geometric consequence: there is some subgroup $\Gamma_G \subset \Gamma(1)$ such that $X_G = \Gamma_G \backslash \overline{\mathcal{H}}$ is a modular curve and $X_G \rightarrow X(1) \cong \mathbb{P}^1$ is a $G$-Galois branched covering. By taking $G$ to be something else than $\operatorname{PSL}_2(\mathbb{Z}/p\mathbb{Z})$ one sees that $\Gamma(1)$ admits many **non-congruence subgroups**. For instance, it is well-known (**added**: I should have said "a well-known theorem of J.G. Thompson") that one can take $G$ to be the Fischer-Griess Monster.

I don't want to make precise what I mean by "most". Note that there are infinitely many finite simple groups with order prime to $3$ (although one has to look fairly far down the list of all finite simple groups to see them: **Suzuki groups**), so I definitely do not mean "all but finitely many".