Beta-1,3-galactosyltransferase that transfers galactose from UDP-galactose to substrates with a terminal beta-N-acetylglucosamine (beta-GlcNAc) residue. Can also utilize substrates with a terminal galactose residue, albeit with lower efficiency. Involved in the biosynthesis of the carbohydrate moieties of glycolipids and glycoproteins. an N-acetyl-beta-D-glucosaminyl derivative + UDP-alpha-D-galactose = a beta-D-galactosyl-(1->3)-N-acetyl-beta-D-glucosaminyl derivative + H(+) + UDP a beta-D-GlcNAc-(1->3)-beta-D-Gal-(1->4)-beta-D-Glc-(1<->1)-Cer(d18:1(4E)) + UDP-alpha-D-galactose = a beta-D-Gal-(1->3)-beta-D-GlcNAc-(1->3)-beta-D-Gal-(1->4)-beta-D-Glc-(1<->1')-Cer(d18:1(4E)) + H(+) + UDP a neolactoside IV(3)-beta-GlcNAc-nLc4Cer(d18:1(4E)) + UDP-alpha-D-galactose = a neolactoside IV(3)-beta-[Gal-beta-(1->3)-GlcNAc]-nLc4Cer(d18:1(4E)) + H(+) + UDP Protein modification; protein glycosylation. Detected in brain and heart. Belongs to the glycosyltransferase 31 family. b3GalT2