The　main　objective　of　this　paper　is　to　study　the　existence　of　a　finite　dimensional　global　attractor　for　the　three　dimensional　Navier-Stokes　equations　with　nonlinear　damping　for　r　＞　4.　Motivated　by　the　idea　of　[I],　even　though　we　can　obtain　the　existence　of　a　global　attractor　for　r　＞=　2　by　the　multi-valued　semiflow,　it　is　very　difficult　to　provide　any　information　about　its　fractal　dimension.　Therefore,　we　prove　the　existence　of　a　global　attractor　in　H　and　provide　the　upper　bound　of　its　fractal　dimension　by　the　methods　of　l-trajectories　in　this　paper.