Yup, they do that. Others can chime in with specifics, and there are about a thousand threads on this topic.
GM addressed this in the 2008 model year with a change to the differential to add a brace. Some have swapped to this newer differential and fixed the problem.
Others claim a new differential mounting bushing made of a sitffer material (urethane) fixes a lot of it. DDMWorks sells one: DDMWorks Solstice Products
DDM sells lots of goodies for our cars
I've done neither to mine, and just enjoyed the car clunks and all. Then again, I ride a Sportster, too, so clunking drivelines are par for the course around my house...
Someday I may work on addressing it but for now there are other projects waiting.