The Harper-Hofstadter problem for a two-dimensional electron gas with Rashba spin-orbit coupling subject to periodic potential and perpendicular magnetic field is studied analytically and numerically. The butterfly-like energy spectrum, spinor wave functions as well as the spin density and average spin polarization are calculated for actual parameters of semiconductor structure. Our calculations show that in a two-dimensional electron gas subject to periodic potential and uniform magnetic field the effects of energy spectrum splitting caused by large spin-orbit Rashba coupling can be observed experimentally.