A countable group is $$C^*$$ -simple if its reduced $$C^*$$ -algebra is simple. It is well-known that $$C^*$$ -simplicity implies that the amenable radical of the group must be trivial. We show that the converse does not hold by constructing explicit counter-examples. We additionally prove that every countable group embeds into a countable group with trivial amenable radical and that is not $$C^*$$ -simple.