This paper studies the Galerkin finite element approximations of a class of stochastic fractional differential equations.The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained.The discretization in time is achieved via the piecewise constant,discontinuous Galerkin method and a Laplace transform convolution quadrature.We give strong convergence error estimates for both semidiscrete and fully discrete schemes.The proof is based on the error estimates for the corresponding deterministic problem.Finally,the numerical example is carried out to verify the theoretical results.