On my group's member page Subgroups is displayed, and we have no subgroups. I checked it with Member's List on and off with no change and turning the Directory on/off makes no change either. So it sounds like I'm seeing something different than you are.
1) I'm just logging off as an owner then logging back in as a member. Could it be there is something "left over" that is triggering Subgroups to display?
2) At one point this group did have a subgroup, which has been deleted. Could it be the system still thinks we have a subgroup, even though it was deleted?
In either case, it would seem to be a bug. :(