I suspect you have Phlox panniculata. You can dig the clump up in either autumn or spring, break it up into smaller clumps and replant. The root ball will be quite tough but easily split.

I agree. Cut off the dead heads and let the plant die down naturally. Then in the autumn split it all up if you want to.

I agree with the above comments but what I like about dividing in spring is I can see where the fresh young growth is.

This means I can discard the old woody parts and only plant on the parts with new growth.

But it I very much a better a case of what suits you best!

The replanted sections will be much better next year. You'll be amazed! This is one of the perennials that the 'old books' suggest is divided regularly. Improve the soil in the new planting holes with bonemeal and compost, and water well in dry spells.