Families of hyperfinite subfactors with the same standard invariant and prescribed fundamental group

We construct irreducible hyperfinite subfactors of index 6 with a prescribed fundamental group from a large family containing all countable and many uncountable subgroups of R_+. We also prove that there are unclassifiably many irreducible hyperfinite group-type subfactors of index 6 that all have the same standard invariant. More precisely, we associate such a subfactor to every ergodic measure preserving automorphism of the interval [0,1] and prove that the resulting subfactors are isomorphic if and only if the automorphisms are conjugate.