Let $ G $ be a completely reducible algebraic group (group for which every finite dimensional algebraic representation is completely reducible). Let $ H $ be an algebraic subgroup of $ G $. Let $ V $ be a finite dimensional representation of $ G $. Since $ G $ is completely reducible there is a canonical direct sum decomposition of $ V $ into isotypic subspaces for $ G $ $$ V= \bigoplus_{\rho_i} V_{\rho_i} $$ where $ \rho_i $ are finite dimensional irreps of $ G $ and $ V_{\rho_i} $ denotes the sum of all subreps of $ V $ transforming in the irrep $ \rho_i $.
Since $ H $ is an algebraic subgroup of $ G $ then $ G $ completely reducible implies $ H $ completely reducible. Thus there is a canonical direct sum decomposition of $ V $ into isotypic subspaces for $ H $ $$ V= \bigoplus_{\tau_i} V_{\tau_i} $$ where $ \tau_i $ are finite dimensional irreps of $ G $ and $ V_{\tau_i} $ denotes the sum of all subreps of $ V $ transforming in the irrep $ \tau_i $.
Suppose that the direct sum decompositions of $ V $ exactly coincide for $ G $ and for $ H $. Prove that then every $ H $ subrep of $ V $ must also be a $ G $ subrep of $ V $. In other words, if the isotypic decomposition is the same as an orthogonal sum of subspaces then all the subreps are the same as vector subspaces.
For an interesting example like this where the isotypic decomposition for $ G $ coincides exactly with the isotypic decomposition for $ H $ consider $ G=GL(2,\mathbb{C}) $, $ H \cong SL(2,5) $ the binary icosahedral subgroup https://en.wikipedia.org/wiki/Binary_icosahedral_group and $ V= (\mathbb{C}^2)^{\otimes 5} $ the $ 5 $th tensor power of the natural representation of $ GL(2,\mathbb{C}) $.
The result is not true: consider the case where $V$ is irreducible for $G$ and $H$ is trivial. Then $H$ is isotypical for $H$, but no non-trivial sub-vector space of $V$ is a $G$-submodule (while it is of course an $H$-submodule).