We figure out that the ground state of a multiple-spin exchange model applicable to thin films of solid {3}He possesses an octahedral spin nematic order. In the presence of a magnetic field, it is deformed into an antiferroquadrupolar order in the perpendicular spin plane, in which lattice Z{3} rotational symmetry is also broken. Furthermore, this system shows a narrow magnetization plateau at half, m/m{sat}=1/2, which resembles recent magnetization measurements [H. Nema et al., Phys. Rev. Lett. 102, 075301 (2009)].