Some additions/corrections to what has been said so far:
Hamburg Airlines doesn´t exist any more, neither does European Air Service.
Condor has a legally separate subsidiary called Condor Berlin. Lufthansa CityLine and Lufthansa Cargo are legally separate airlines as well.
Rheintalflug is Austrian.
You´ll find a complete list of all 31 German airlines under
Follow Germany´s country prefix "D". DaimlerChrysler Aviation who operate an A319 from Stuttgart to Detroit should be listed under "Private".