On a truncated icosahedron / buckyball / Telstar-style soccer ball, consider two adjacent hexagons and the two pentagons that are adjacent to both. These four faces can be removed, rotated by a right angle, and reattached, causing only a small change to the overall shape. Most fullerenes have at least one such patch.
If I ever get around to making more printable models of fullerenes, I would omit those that can be changed, by the above twist, into one of higher symmetry. I have a pretty good idea of how I’d go about listing the fullerenes and finding their siblings; but I do not have a grip on distinguishing symmetry groups of the same order – e.g., that of a regular tetrahedron versus that of a hexagonal prism – and a subgroup of one may not be a subgroup of the other.
So I got out An Atlas of Fullerenes in the hope of understanding how they did it – and happened to open to a chapter I had not looked at before, which covers the Stone-Wales transformation (for so it is named) and lists which fullerenes change with which, up to C50 (15 hexagons).
The 812 smallest fullerenes are thus cut to 72 in 47 families. The biggest of these families has six remaining members, four with C2v symmetry (one axis of twofold rotation, and a reflection plane containing that axis) and two with C3 symmetry (chiral with one threefold axis). Their symmetry numbers are 4 and 3 respectively, but as C3 is not a subgroup of C2v I keep them all.
Surprisingly the ten families of C50 include two with no nontrivial symmetry at all.