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The problem is that "types" is not a well-defined word in the contest of this problem. Do you mean morphology, lattice system, space group, or what?

There are more or less infinitely many possible morphologies (I'm pretty sure, though I wouldn't necessarily want to try to prove it, that it's a countable infinity).

There are 7 lattice systems: triclinic, monoclinic, orthorhombic, rhombohedral, tetragonal, hexagonal, and cubic.

There are 230 distinct space groups, and no I'm not going to list them. Get a graduate-level chemistry book on X-ray crystallography if you really want the details.

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