I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$.


The smallest I could find has even entries $a\_{3,1}$ and $a\_{3,2}$,
along the bottom row. I could not figure out whether there are
subgroups of index $2$ or $3$.


A search found lots of information about ${\rm SL}(2,\mathbb{Z})$,
but not about ${\rm SL}(3,\mathbb{Z})$.



Small-index subgroups of SL(3,Z)
To fill in the comments, there are basically two serious issues involved.


1) You want to know that every subgroup of finite index in $\mathrm{SL}(3,\mathbb{Z})$ contains some *congruence kernel*: the kernel of the natural reduction homomorphism induced by $\mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z}$. In other words, the original group satisfies the *congruence subgroup property* (as HW comments). This is already quite nontrivial to prove and evolved from work of Bass-Lazard-Serre [here](http://www.ams.org/journals/bull/1964-70-03/S0002-9904-1964-11107-1/) followed by more definitive work by Bass-Milnor-Serre [here](http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1967__33__59_0).
(There is some exposition in the last part of my 1980 Springer Lecture Notes 789.)


2) Now you have to pin down the maximal subgroups of a typical finite group $\mathrm{SL}(3,n)$ and pull these back to the big group in order to sort out which give you subgroups of minimal index there. Maximal subgroups of the finite groups in question have been extensively studied, especially over finite fields, but in this case some trial-and-error leads pretty quickly to the number 7, as indicated by Aakumadula. This results from the first possibility $n=2$, where you get the simple nonabelian group $\mathrm{SL}(3,2)$ of order $ 8 \cdot 3 \cdot 7$, from the general formula for a prime $p$ given by $p^3 (p^2-1)(p^3-1)$. Here there is a maximal parabolic subgroup of index 7. 


The search for maximal subgroups is itself a major project, for which I don't have all relevant references at hand. But over a field some basic insight comes from algebraic groups, where you see that a maximal (closed) proper subgroup will be either parabolic or reductive. I'm not sure in the special case at hand how hard it is to pin down the maximal subgroups of the various finite groups directly.


ADDED: As pointed out in comments, it's possible here to rule out subgroups of index 2 or 3 by combining the known commutator group result with a study of actions of the big group on left cosets of a hypothetical subgroup. But a systematic study (especially for other Chevalley-type groups) probably requires something like the program I've sketched above. Given the congruence subgroup property, the study of subgroups of finite index then reduces quickly to the study of finite groups of the same type over finite rings and then prime fields. In your case, this helps to provide a conceptual answer to such questions as: *Why does $\mathrm{SL}(3, \mathbb{Z})$ have a subgroup of index 7?* 


There is a vast literature by now on maximal subgroups of known finite simple groups, by people including Aschbacher, Kleidman, Liebeck, Scott, Seitz, .... An older brief survey by Kleidman-Liebeck, with an impressive bibliography, appeared in *Geom. Dedicata* 25 (1988). For your specific group there are very old results, but the general approach is more unified even though intricate. 


