Number of Subgroups of Index m

[cyclic]

Homework Statement

The special linear group $SL(2, \mathbb{Z})$ over $\mathbb{Z}$ is the multiplicative group consisting of all $2 \times 2$ matrices with entries in $\mathbb{Z}$ and determinant $1$; that is,

$$
SL(2, \mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \mathbb{Z} \text{ and } ad - bc = 1 \right\}.
$$

Let $G$ be the quotient group $SL(2, \mathbb{Z}) / \{\pm I\}$, where $I$ is the $2 \times 2$ identity matrix. Find with proof the number of subgroups of $G$ of index $m$ for each $m \in \{2, 3, 4, 5, 6\}$.

Relevant Equations

Let $G$ be a group and $H$ be a subgroup of $G$. The index of $H$ in $G$, denoted $[G : H]$, is the number of distinct left cosets of $H$ in $G$. That is,
$$
[G : H] = \frac{|G|}{|H|},
$$
if $G$ is finite. For infinite groups, $[G : H]$ is the cardinality of the set of left cosets:
$$
[G : H] = |\{gH : g \in G\}|.
$$

Perhaps we can use congruence subgroups here? Or perhaps we can study SL(2,Z) using its action on the projective line over the integers modulo n? I'm pretty stumped and would appreciate any help.