Plot hole: Hogan has his men play romantic music over the radio. Earlier they were afraid of the radio in the observation post being detected. Elsewhere in the series, they try to avoid sending too long to avoid being homed in on - which is correct. Now, if they were to play a whole record at once, wouldn't the homing devices pick up on the transmitter at the camp?
Factual error: Klink and Hogan sit behind each other in the P-51 they try to steal. The P-51 is a single-seat airplane, the only twin-seat P-51 are trainers. A trainer would not be at the flight line with the regular airplanes, and if a trainer would actually scramble with the others, it would at least arouse suspicion.
Continuity mistake: After Shultz has been discredited, Klink is taking away the "badges of rank". When Klink goes to crush the monocle Shultz was wearing, he has his monocle on. After the shot cuts to a closeup of Shultz's monocle being crushed, and widens out, you see Klink's monocle is suddenly gone. It remains missing until Klink goes to the hat rack. The shot cuts from Klink, to Hogan and Shultz, then back, and the famous monocle is suddenly back, both without Klink stopping to take it off, or put it back on.
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Factual error: Hogan calls the radio detection truck "radar" when he orders the SS guard to switch it off. From other episodes, we know that Hogan knows what radar is, and back then, the difference between radio homing equipment and radar was even clearer to people than it is today, because radio homing was an established technology, while radar was brand new, and most people were not even aware it existed.
Factual error: Baker picks up a lot of static in his radio, then suddenly signs off and says "Sounds like the radio detection unit picked up our signal." Unlike radar, radio signal homing relies entirely on measuring the signals emitted by the transmitter that is tracked. It works by comparing the strength of the signal arriving at each component of an array of directional antennae. The process is completely passive and does not cause any alteration of the signal measurable at either receiver or transmitter at all, and certainly not any audible interference or humming.