For　one-dimensional　nonlinear　lattices　with　momentum　conserving　interparticle　interactions,　intensive　studies　have　suggested　that　the　heat　conductivity　kappa　diverges　with　the　system　size　L　as　kappa　similar　to　L-alpha　and　the　value　of　alpha　is　universal.　But　in　the　Fermi-Pasta-Ulam-beta　lattices　with　nearest-neighbor　(NN)　and　next-nearest-neighbor　(NNN)　coupling,　we　find　that　alpha　strongly　depends　on　gamma,　the　ratio　of　the　NNN　coupling　to　the　NN　coupling.　The　correlation　between　the　gamma-dependent　heat　conduction　behavior　and　the　in-band　discrete　breathers　is　also　analyzed.