Oh, I think I need one of Don's daylily dividers! Another thing that works well for big clumps is 2 pitch forks inserted into the clump back-to-back, then gently pry the clump apart.
I have people come to my house and I dig daylilies for them to buy. I have them pick from the gardens, and I dig from the outer part of the clump. Since I don't want to dig up the whole clump and disturb it, I just usually insert a sharp knife and cut away a couple of fans, then ease put a shovel on the outside and ease the roots out of the soil, and out it comes with hardly a problem. The roots recover easily from being cut.