  By using the quantum extremal island formula, we perform a simple calculation
of the generalized entanglement entropy of Hawking radiation from the two
dimensional Liouville black hole. No reasonable island was found when
extremizing the generalized entropy. We explain qualitatively the reason why
the page curve cannot be reproduced in the present model. This suggests that
the islands may not necessarily save the information paradox for the Liouville
black holes.
