Infinite locally finite simple groups with many complemented subgroups

We prove that the following families of (infinite) groups have complemented subgroup lattice: alternating groups, finitary symmetric groups, Suzuki groups over an infinite locally finite field of characteristic 2, Ree groups over an infinite locally finite field of characteristic 3. We also show that if the Sylow primary subgroups of a locally finite simple group G have complemented subgroup lattice, then this is also the case for G.