D. J. Benson, J. F. Carlson, and J. Rickard [1997, Fund. Math.153, 59–80] classified the tensor-closed thick subcategories of finite-dimensional representations of finite groups over algebraically closed fields. In this paper, we remove the algebraically closed hypothesis by applying some Galois theory. Our methods apply more generally to finite-dimensional cocommutative Hopf algebras over a field. Thus they allow us to drop the algebraically closed hypothesis in the classification of thick subcategories of modules over finite-dimensional sub-Hopf algebras of the Steenrod algebra as well.