JEANINMAINE...you actually may have some Japanese Iris or something similar...possibly even Peacock orchids. Daylilies can be divided or transplanted almost anytime but the perfect time for them is after they have bloomed and right before they start a resting season....
which is usually last of July until the first of August. (Per my daylily buddies bith of whom our daylily farms...combined farms of over 20,000 daylilies) You can take a spade or a sharp knife and break the clump of daylilies into smaller pieces...let the divisions dry for a few days before replanting to heal or cuts scab over.