The relativistic SU(6) supermultiplets which can be constructed from two quarks and two antiquarks, three quarks, or four quarks and one antiquark are examined as possible candiates for the classification of meson and baryon resonances. The multiplets can be labeled either by the SU(12) formalism or by the U(6)×U(6) group of Dashen and Gell-Mann; the two groups lead to equivalent classifications. Those static SU(6) multiplets within a given SU(12) multiplet which represent physical particles (i.e., satisfy Bargmann-Wigner equations) are just the states having the maximal eigenvalue of γ0 in the SU(12) multiplet. These constitute a representation of the Dashen-Gell-Mann U(6)×U(6) subgroup of U(12) which commutes with γ0. Applications to specific resonances are discussed.