Every mineral has it's own internal building scheme, a lattice on which it's atoms lie. Mathematically, this can be described by the space group of the mineral. From mathematical considerations it's known that there are exactly 230 symmetry groups which can occur as the space group of a mineral. Now my question is: Is each of the 230 space groups realized as the symmetry group of a mineral?
Now all of the 230 space groups are realized by a natural occurring mineral. I found this blog: crystalsymmetry.wordpress.com where space groups are listed with at least one chemical compound is listed for each space group. Minerals are listed for all of the space groups.
For example, Benitoite has fairly unusual crystal symmetry. I think it was first mineral found for P-6c2 space group.
188 P-6c2 Example – Benitoite, BaTi(Si3O9)
From a crystallographical point of view this compound (Benitoite) is especially interesting, being the only known member of the ditrigonalbipyramidal symmetry class.
Quoting a mineralogist friend of mine:
I think there are mineral representatives of all 230 known by now - some of the groups are much rarer than others, but there is nothing specifically prohibiting...
This shouldn't be too hard to figure out. There are databases that contain this information, webmineral is one of them. Here an example mineral called lewisite where you can see: Space Group: F d3m . Unfortunately, that website does not aggregate space group data. Maybe it's possible to contact the people who run the website to see if it's possible to add this feature.
Well, one way would be to look through The rock forming minerals and see how many are represented. I strongly suspect that not all would be, so from that perspective the answer would be No.
But from an inspection perspective.. For any given mineral structure, we can make changes by simply substituting various cations and anions as in pyroxene; for example, we can substitute Mg(2+) for Fe(2+). Or SiO4(4-) for AlO4(5-) - requiring other changes. So, assuming you had suitable equipment, you could find a naturally occurring crystal that was close to the desired symmetry group and tweak it with substitutions until it matched that group. Or more simply, adjust the pressure and temperature until the symmetry changed. There is a lot of phase space to explore.
So from that perspective, the answer would be probably yes, we just haven't synthesized them all yet.