(caveat: I'm not a number-theorist or Langlands-programme-er, and I don't expect to understand all the answers to this question, but I figured they might be useful to someone besides me).

I've been making videos of symmetries of Klein's $j(\tau)$:

in an attempt to have a better visual feel of the symmetries of $\mathbf{SL}(2,\mathbb{Z})$.

I know that $\mathbf{SL}(2,\mathbb{Z})$ and $\mathrm{Gal}(\mathbb{\bar{Q}},\mathbb{Q})$ are related -- for instance this paper by Ribet: http://math.berkeley.edu/~ribet/Articles/motives.pdf but that's outside of my scope of knowledge, and that $\mathrm{Gal}(\mathbb{\bar{Q}},\mathbb{Q})$ is something which no one quite understands yet, so:

Is $\mathbf{SL}(2,\mathbb{Z})$ a subgroup of $\mathrm{Gal}(\mathbb{\bar{Q}},\mathbb{Q})$? and if so,

What properties does their quotient $\mathrm{Gal}(\mathbb{\bar{Q}},\mathbb{Q})/\mathbf{SL}(2,\mathbb{Z})$ have?