What is one of the main problems with Nietzsche's idea of the "ubermensch," or superman?

Frederich Nietzche's conception of a "superman," one who is literally "beyond human," grew out of his previous pronouncement that "God is dead," by which he meant that human civilization had outgrown the need for a mythologically-based value system (Christianity), and that the values of Christianity and other religions had ceased to be useful and even relevant in the age of science and reason. Although Nietzsche himself never advocated for the notion of racial or ethnic supremacy, many historians believe that his philosophy laid the groundwork for fascism and the rise of totalitarian dictatorships.

Nietzsche's idealized "ubermensch" would, according to Nietzsche, be so advanced and wise that he would create his own system of values and ethics to govern himself and the world. According to Nietzche, this new race of advanced humans, specifically men, would rely on women solely for reproduction, so that those women could give birth to more supermen. In this way, the philosophy of the superman justified relegating women to second class citizenship, and Hitler (among others) later used this philosophy to justify misogyny and sexism.

Most disturbing of all, the notion of a "race of superior men" who would evolve from other ordinary humans, helped justify all sorts of racism, eugenics, and eventually, genocide. If God was dead and the moral components of Christianity became obsolete, as Hitler and other despots argued, then the supermen, often self-appointed as such, could justify any actions they took based on the notion that they were God-like humans, to whom morality and rule of law did not apply. 

Ultimately, Nietzsche's conception of the ubermensch, though revolutionary and shocking at the time, was not problematic in and of itself. Nietzsche could not have fathomed how his idea would have been used. Yet it did become very dangerous in the hands of those who lacked a nuanced understanding of what Nietzsche was discussing, or simply didn't care. Nietzsche was ultimately pondering how to improve the world and create a new morality, which later philosophers like Bertrand Russell would term "humanistic." In the wrong hands, however, the concept of the ubermensch became twisted, and it was used to  advance the murderous agendas of numerous dictators during the 20th century.