Maximal subgroups in the Cremona group

We show that for any $n\ge 5$ there exist connected algebraic subgroups in the Cremona group $\mathrm{B}\mathrm{i}\mathrm{r}({\mathbb{P}}^n)$ that are not contained in any maximal connected algebraic subgroup. Our approach exploits the existence of stably rational, non-rational threefolds.