quantum field theory - Representations of the Conformal Group in terms of the Poincare Group Reps

The Conformal group contains the Poincare group. Typically, if you take a representation of a group and then look at it as a representation of a subgroup, the representation will be reducible. I often hear that CFT's cannot have particles, and I have some understanding of this since $P_\mu P^\mu $ is not a Casimir of the conformal algebra. However, I would think reps of the conformal group should still be reducible representations of the Poincare group, and thus have some particle content.

Is it known how to decompose representations of the conformal group into reps of Poincare? Can we understand it as some sort of integral over masses that removes the scales of the theory? Are there any significant differences between reps of Poincare appearing in reps of the conformal group and the usual representations we are familiar with from QFT?

I'd appreciate any information or a reference that treats this thoroughly from a physics perspective.