*SU(3)* *is* a subgroup of *SU(6)*: its generators span the 3×3 block of the 6×6 generators of the latter. 

Yes, the eightfold way trivially extends to a notional 35-way of all six flavors, which is, nevertheless, largely useless.

You may, of course, build classification tables of such hadrons in a 5-dimensional space (rank of *SU(6)*),  but with little logical benefit. People did build 3d pictures of the first 4 quarks' *SU(4)* when charm was discovered,

[![charmed states][1]][1]

However, the whole point of the 8-fold way was  that the lightest 3 quarks are lighter than the scale of QCD, ~200 MeV , which binds them together, so their masses could serve as small perturbations to a robust underlying pattern, explicitly broken (corrected) by *small SU(3)* violations. 

For more flavors, the violations are, evidently, huge, and so your proposal might be a fool's errand. 

Perversely, the 3 heaviest quarks, *c,b,t,* form a separate *SU(3)* of their own, as they are so heavy that you may transcribe across their differences, and treat the much lighter (scale-separated)  QCD "glue" as invariant around them: "brown mud", in Bjorken's words, for example.


  [1]: https://i.sstatic.net/cqUPM.gif