Symbol for finite

I have never seen a notation for 'finite,' but what I do very often see is denoting something finite as simply being less than infinity. For example, $|A| < \infty$, or $[G:H] < \infty$.

Small thing I'd like to add: Of course something like $[G:H] < \infty$ isn't technically meaningful, but it certainly gets the point across and in my experience at least seems to be pretty standard.

How about using $$\not\infty$$

I'm guessing you mean the symbol $\infty$, for a non-specific non-finite cardinality. In this case, in the same way you would say $|X|=\infty$ to mean "the set $X$ has infinitely many elements", I would write $|X|<\infty$ to mean "the set $X$ has finitely many elements".