structure of subrepresentations of (infinite) sums of irr. representations

Question

Let $G$ be a (locally compact) group and $ ( \pi_1 , V_{\pi_1} ) , ( \pi_2 , V_{\pi_2} ) , \ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum representation $$\big( \pi_1 \oplus \ldots \oplus \pi_n , V_{\pi_1} \oplus \ldots \oplus V_{\pi_n} \big) $$ every subrepresentation $(\eta , U)$ is $G$-isomorphic to a direct sum of the representations showing up in the direct sum, so $$(\eta , U) \cong \left( \bigoplus_{k=1}^l \pi_{j_k} , \bigoplus_{k=1}^\mathcal{l} V_{\pi_{j_k}} \right) ,$$where $1 \leq j_k \leq n$ and $l \leq n$.
Now the question is, when we start with the $\textit{infinite}$ direct sum of representations, do we have a similar statement? Namely, is every subrepresentation of $$\left( \bigoplus_{k=1}^\infty \pi_{j_k} , \widehat{\bigoplus_{k=1}^\infty} V_{\pi_{j_k}} \right)$$a direct sum of the $\pi_i$'s? I strongly suspect this to be false, but I can't find a counterexample.