There are more known inherited genetic factors to type 2, but it is also more susceptible to environment. There is a thing called 'heritability' which is worked out by concordance between monozygotic (identical) and dizygotic (fraternal) twins, looking at if there is a difference in levels of incidence - identical twins have (almost) identical DNA, whereas fraternal twins are no more similar in their DNA than any other pair of siblings. Concordance is higher for type 2, but it is also much more dependent on the environment. If you have a risk allele for type 1 - e.g. HLA-B-DR3 or DR4 - you could get it no matter your environment, but if you have a risk allele for type 2, you can be more at risk, but you can decrease that risk depending on environment. I hope that makes sense! Although actually, T2 shows only about 10% more heritability than T1 when it comes down to it. Also, whilst there aren't many 'big' contributing alleles to T2, only lots of little ones that have been identified, anyway, the DR3 and DR4 HLA-B alleles contribute to a huge proportion of risk for T1.
I don't have the exact figures to hand, but whilst T2 shows very high heritability, when you break down the risk, for *most* people it works out at about 10% genetics and 90% environmental, although there are exceptions, so you see T2 families, where *everybody* has it, due to inheritance of some allele/alleles that haven't been identified yet.
Inheritance is a funny old thing to study - the best way is identical twins separated at birth, but even that isn't perfect, and it also isn't very ethical. Next Generation Sequencing Technologies are making it easier though.
Sorry for the essay
Edited to add that anyone who is really interested in the inheritance of T2D should look up the 'Pima Indian Study'. Fascinating stuff, although the original paper is not what you would call particularly readable.