According to the results of votes at Facebook's annual shareholder meeting, 68 percent of outside investors want the company to hire an independent chairman. The majority was up from 51 percent last year.

The proposals did not pass because of Zuckerberg's voting control of the stock, which means he can swat away shareholder demands. Facebook's voting rights are tilted heavily in favor of B-class shareholders, which consist exclusively of Zuckerberg and his small band of chums.

This means that the company's founder enjoys all the cash from being publicly held, but none of the discipline from shareholders.

Facebook's investors are now demanding an independent investigation into Zuckerberg's "outsized" power. 83.2 percent of outside shareholders also backed a proposal to scrap Facebook's dual-class share structure altogether.