As others have mentioned, there are many CFTs, but we can narrow down our list by looking at conditions that select for interesting automorphism groups. Perhaps the easiest is to consider holomorphic ($C_2$-cofinite) vertex operator algebras. By Zhu's theorem, these necessarily have central charge given by a nonnegative multiple of 8, so we can list them in some rough increasing order.

$c=0$: This is just the field of complex numbers, with trivial symmetry (as a ring over itself).

$c=8$: There is one example: the $E_8$ lattice algebra. Its automorphism group is the Lie group $E_8(\mathbf{C})$.

$c=16$: There are two cases, given by the lattice algebras for $E_8 \times E_8$ and $D_{16}^+$, and the automorphism groups are $E_8 \wr 2$ and $SO^+(32)$. This case and the previous case were proved by Dong and Mason.

$c=24$: In 1992, Schellekens conjectured that these are classified by their weight 1 Lie algebras, and gave 71 candidate types. The last candidate vertex operator algebra was finally constructed 6 weeks ago by Lam and Lin. Nontrivial Lie algebras yield positive-dimensional Lie groups of symmetries, so the only discrete case is where the Lie algebra is zero. The uniqueness question is still open, but if it is true, then the Monster is the only discrete automorphism group. If there is a counterexample to uniqueness of the moonshine module, then we still have a lot of control of its automorphisms due to genus zero constraints. That is, most of Borcherds's argument still works for such an object.

$c \geq 32$: The number of isomorphism types grows very quickly, just because they contain the vertex operator algebras attached to positive definite even unimodular lattices. For example, with $c=32$, King's mass formula gives over $10^9$ lattices, and over $10^7$ lattices with no roots. It is quite likely that by orbifolding some of the latter types, we get discrete symmetry groups, but it seems no one has bothered to work out any cases completely.

What we see from this list is that, from the lens of the category of holomorphic vertex operator algebras, the first nontrivial group is $E_8$, and the first nontrivial finite group is the Monster. At large central charge, we expect things to become boring from a symmetry standpoint, in much the same way that high rank lattices and high genus curves tend to become boring.