Well it depends on the style of societal breakdown. In the zombie apocalypse shows, the human population is less than halved, and the sorts of daily things we take for granted (electricity, clean water, gasoline, etc.) become hard to come by, along with police, fire, and medical services becoming lost or completely disorganized, with governments completely falling apart at every level.

But I could imagine a scenario where the AI-owning capital class becomes indistinguishable from the government, and the government retains control of the military. Regular disenfranchised folks usually don't stand much of a chance against their own country's military. It would be super super bloody, but it's a toss-up as to who would win. In reality, though, in the end, everyone would lose.

How big is the government/supporters compared to the population in any dictatorship? Tiny.