Subgraphs of Complete graphs

There's a fair amount of information at https://oeis.org/A000088: "Number of graphs on $n$ unlabeled nodes". The classic reference seems to be Harary and Palmer's book Graphical Enumeration.

As you've seen, $K_n$ has $\frac{n(n-1)}{2}=\binom{n}{2}$ edges. There are $2^\binom{n}{2}$ ways to select a subset of these edges. If "most" of the resulting subgraphs don't have much symmetry, then you'd expect this formula to overcount the number of non-isomorphic graphs by a factor of $n!$. It turns out that this is asymptotically true! But don't ask me to prove it. :)