This is probably a delta. It is what I would call a manual plug... that is if you want the tub to fill, you push down the plug, when you want it to drain you pull it up. It isn't part of the overflow.
This one has a removal knob on top, but once we get that off, we can't get the rest of the stopper/plug thing out.
The tub barely drains right now. We've run a drain auger down the overflow - got about 4' in there, and didn't encounter any gunk. Seemed to get stuck a couple times, but figured it was bends in the pipe. So now we figure the clog is right at the drain, but we can't get the stopper out..