We　explore　the　numerical　approximation　of　the　stochastic　Burgers　equation　driven　by　fractional　Brownian　motion　with　Hurst　index　H　∈　(1/4,　1/2)　and　H　∈　(1/2,　1),　respectively.　The　spatial　and　temporal　regularity　properties　for　the　solution　are　obtained.　The　given　problem　is　discretized　in　time　with　the　implicit　Euler　scheme　and　in　space　with　the　standard　finite　element　method.　We　obtain　the　strong　convergence　of　semidiscrete　and　fully　discrete　schemes,　performing　the　error　estimates　on　a　subset　Ωk,　h　of　the　sample　space　Ω　with　the　Gronwall　argument　being　used　to　overcome　the　difficulties,　caused　by　the　subtle　interplay　of　the　nonlinear　convection　term.　Numerical　examples　confirm　our　theoretical　findings.　©　2024　the　Author(s),　licensee　AIMS　Press.　This　is　an　open　access　article　distributed　under　the　terms　of　the　Creative　Commons　Attribution　License　(http://creativecommons.org/licenses/by/4.0)