When I look for a classification of the normal subgroups of $SL(2,Z)$, I found that people tend to only study $PSL(2,Z)=SL(2,Z)/\{1,-1\}$'s subgroups or normal subgroups, as far as I know Morris Newman studied $PSL(2,Z)$'s normal subgroups classification systematically . So now the question is how do we get normal subgroups of $SL(2,Z)$ from normal subgroups of $PSL(2,Z)$ (I don't know if this problem is mathematically trivial.)
All I know so far is the following very easy fact: There is a one-to-one correspondence between the subgroups of $PSL(2,Z)$ and the subgroups of $SL(2,Z)$ that contain −1. (I also know that from the point of view of computational group theory, it is perfectly possible for a computer to generate the normal subgroups of $SL(2,Z)$ or $PSL(2,Z)$, but I do not have a good understanding.)