In the Austrian Empire, what group (s) wanted to establish an autonomous state?

Remember, at the time, it was the Austro-Hungarian Empire. The Empire (unlike any of the other major states in Europe) was a patchwork of over a dozen major ethnic groups. Nationalism tends to organize along ethnic boundaries (that is, nations tend to form around a large concentration of one ethnic group). Thus, with a very large number of different ethnic groups, the Empire had to worry about each group wanting to split from the Empire, and form its own nation. Indeed, after WW1, this is what happened to the Empire - it was split into about a 8 different countries (or, more accurately, portions of 8 countries included lands formerly part of the Austro-Hungarian Empire).