What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup of $\Gamma$?

Attempt 1 at ``$\Lambda$ is very much not a subgroup of $\Gamma$'': $\Gamma$ has no subgroup commensurable to $\Lambda$.

Edit: the question is answered in the comments by YCor and Ian Agol. More examples are welcome. Specifically, I would be interested in $\Gamma$ that have many such $\Lambda$ (in particular, further $\Lambda$ that are neither QI to $\Gamma$ nor $\mathbb{Z}$).

Context: in comparing group theory to geometry one often looks at whether homomorphic embeddings are also quasi-isometric embeddings. This question is about the other way around. There are probably tons of examples, the problem is to verify that $\Lambda$ is very much not a subgroup of $\Gamma$. For example, any two points of the Cantor space $\partial F_2$ define a quasi-isometric embedding of $\mathbb{Z}$ into $F_2$ and only very few (countably many) of these are anywhere close to being homomorphic images, yet $\Gamma$ does have a subgroup isomorphic to $\mathbb{Z}$.