We report on magnetotransport measurements on InSb nanowire quantum dots. The measurements show that the quantum levels of the InSb quantum dots have giant g factors, with absolute values up to approximately 70, the largest value ever reported for semiconductor quantum dots. We also observe that the values of these g factors are quantum level dependent and can differ strongly between different quantum levels. The presence of giant g factors indicates that considerable contributions from the orbital motion of electrons are preserved in the measured InSb nanowire quantum dots, while the level-to-level fluctuations arise from spin-orbit interaction. We have deduced a value of Delta(SO) = 280 mueV for the strength of spin-orbit interaction from an avoided level crossing between the ground state and first excited state of an InSb nanowire quantum dot with a fixed number of electrons.