We present lattice calculations of the low-lying spectrum of $^{12}$C using a simple nucleon-nucleon interaction that is independent of spin and isospin and therefore invariant under Wigners SU(4) symmetry. We find strong signals for all excited states up to $sim 15$~MeV above the ground state, and explore the structure of each state using a large variety of $alpha$ cluster and harmonic oscillator trial states, projected onto given irreducible representations of the cubic group. We are able to verify earlier findings for the $alpha$ clustering in the Hoyle state and the second $2^+$ state of $^{12}$C. The success of these calculations to describe the full low-lying energy spectrum using spin-independent interactions suggest that either the spin-orbit interactions are somewhat weak in the $^{12}$C system, or the effects of $alpha$ clustering are diminishing their influence. This is in agreement with previous findings from {it ab initio} shell model calculations.
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