The sphere centred on $(1,1,1)/2$ and radius $\sqrt{131}/2\approx 5.723$ contains $24+48+48=120$ points with integer coordinates, thanks to $$2\{\pm x,\pm y,\pm z\}-(1,1,1)=\{9,5,5\},\{9,7,1\},\{11,3,1\}.$$