Whoever put this in was really trying to pinch pennies. It is an 8-space panel that, except for the shorter bus bars, could just as easily be a 12-space panel.
In fact, it looks like Square D used the exact same box and front panel, and most of the other parts (except bus bars 8 vs 12) for:
- 8 space subpanel (no main breaker)
- 12 space subpanel (no main breaker)
- 12 space main panel (main breaker)
You might be able to take the guts of this 12 space Square D Homeline panel and replace the corresponding parts of your panel. That would cost you $ 65 in "parts" (the entire subpanel). You might find that you can get just the guts from a Square D distributor (not Home Depot/Lowes/etc. but rather an electrical supply house) and if they will sell it you (i.e., if it is available and if they will sell it to you knowing full well you are a DIY and not an electrician) then you might save a few $.
But once you are doing that, you should seriously consider putting in a bigger panel as that really won't cost much more. Currently at Home Depot (not necessarily the best choice, but easy to find pricing online) a 20 space is only $ 10 more! Or a 30 space with several "free" breakers for $ 129 total. That's just looking at the same brand/type so that you can use all of your existing breakers.
But the easiest solution is half size breakers. You can replace the 3 20A single breakers with 2 x 20A half-size breaker pairs (so 4 20A circuits total, including one new circuit) and have two regular spaces available for your new 240V circuit. Home Depot has the 20A breakers:

for about $ 16 each.