I noticed something particularly weird. While dictators can be socially progressive or conservative (though most of the terrible ones in history are usually conservative), all of their regimes are usually Socialist or Communist.

This is incredibly incorrect.

Taking both the two classic fascist dictators, their regimes were not socialistic in most meanings of the word employed by political scientists. In the case of Germany, there was little restriction of the ownership of private property and investment (excepting German Jews, of course) and much of the "socialism" of National Socialism took the form of highly cosmetic and anemic government programs. Some of this took the form of state-subsidized consumer goods which promised Germans things like radios and cars. The various Volksprodukte lines all failed- with the exception of radio- but it should also be noted that these state production lines coexisted with private enterprise producing the same product. A German could subscribe to the saver program for the KdF-wagen, but could also purchase and Opel or Mercedes Benz automobile as well. Both Italy and Germany had state-sponsored leisure programs and many fascist ideologues claimed that their political system was a third-way between communism and the extreme free-market capitalism exemplified by the US. But this argument though was a European cultural conservative argument. Where fascism departed from prior examples of Italian or German conservatism was that it was much more at ease with mass politics and mass mobilization than its predecessors on the right. Much of the state's intervention in the economy had much more to do with issues of either power or corralling private industry into rearmament drives, which some of the more sensible industrial chiefs thought (often quite belatedly) was ill-advised.

Although a number of American public intellectuals in the Cold War maintained that free markets were inimical to unfree political systems. Yet there were a large number of dictators in the post-1945 period who presided over what could only be termed capitalist systems. Augusto Pinochet of Chile was perhaps the most famous of these, employing a group of American-trained economic advisors colloquially known as the "Chicago Boys" who helped structure Chile's economy into one amenable to free trade agreements and investments. The Chicago Boys were to a degree successful, but they also operated in the context of a very violent and repressive dictatorship that was infamous for its human rights violations. The Chicago Boys also had an East Asian predecessor in the form of the Indonesian Berkeley Mafia who operated under Suharto. These US-trained economists helped undo the communist-inspired state control over the economy. As with Chile, the dictatorship that these men helped strengthen was one that was infamous for its many human rights abuses.

The Berkeley Mafia and the Chicago Boys were two examples of numerous dictatorships that found free markets and investments to be quite useful to this illiberal state order. This did not mean that these post-1945 dictatorships did not intervene in the economy or have some degree of state control. The various Brazilian military dictatorships between 1964 and 1985 had tendencies to favor programs of state-nationalization and heavy control. But this was not a system of mass state-control of the economy like that seen in the Eastern bloc; there was a decided attempt to court foreign investment. Emílio Garrastazu Médici in particular sought to bring in foreign economic investment and cooperation with free-market entities as part of a larger program to modernize Brazil. Spain under Franco also saw a move away from fascist-inspired autarky and creating an economic environment more amenable to free-market investments and private enterprises in the 1950s and 60s.

One of the common threads in these dictatorships is that many of these illiberal regimes recognized that economic development would strengthen the state. Rather than going the route of various Eastern bloc countries, these men recognized that it was much easier to work within the global free-market system than to resist it. South Korea's Park dictatorship exemplified this with significant state sponsorship aimed to achieving high levels of growth and to transform South Korea's economy. While this level of state control and interference might seem out of place in the free-market, this is not uncommon around the word and has parallels in Japan's MITI or various European consortiums such as Airbus which has a rather tangled relationship with various European governments.

These are just a few examples of how the free market is largely indifferent to dictatorships. A good many dictatorships in the twentieth century allowed private business and courted foreign investments as a means to strengthen the state.