I've had two Daewoo DP51s and a DH-40 (the .40 version of the same design), and now have an LH-9. The LH-9 comes with one mag, I think, and it's flush fit. (I got mine used.) The longer ones are probably S&W Model 59 or S&W 59xx mags. They all stick out about an extra 1/4". (I've also got a S&W 20 rounder somewhere that will work. Awkward to use, though -- and too heavy.)
Mags are hard to come by for any of these guns, and the full-size models will use S&W Model 59/59XX mags. I also have a S&W 40XX mag that seems to cycle rounds properly in the LH9, but I've never tried it at the range.
If the LH9CN is a compact model, the flush-fit mags may be from a S&W 469 or 669 or similar S&W Third Gen. "compact".
Getting mags for the DH-40 (the .40 version of the DP-51) was always a problem. I did get a 59-series mag to work in the DH-40 by lowering the top front of the mag (with a Dremel) so that the follower (which sets lower when using .40 ammo in a 9mm mag) would allow the .40 round to feed.
I never tried the S&W 40XX mags in the DH-40 -- didn't have any S&W 4006s or 4043s then -- but did have some old S&W 59 mags. I suspect the S&W 40XX mags could be made to work with a little or no effort.