I was recently at AT&T visiting for a few weeks, and I was lucky enough to catch a talk by Amit Chakrabarti on lower bounds for multi-player pointer jumping. A complexity class that figured prominently in his talk was the class ACC0, which consists of constant depth, unbounded fanin, poly sized circuits with AND, OR, NOT and MOD m gates, for all m.
Suppose we make our life simple by fixing m to be a single prime, like 3. It turns out that the corresponding class AC0[m] can be contained strictly within NC1: this arises from results in the 80s by Razborov and Smolensky. Now suppose that instead of picking a prime integer, we choose a number like 6, which is the product of distinct primes. We do not even know whether AC0 = NP or not !