Plot hole: In the cinema, Hogan and his female underground contact avoid attention from a passing plain clothes policeman by kissing. This is wrong for three reasons. Firstly, 3rd Reich Germany wasn't so liberal that open kissing in the cinema would have passed as normal, it would probably rather have been viewed as libertine and offensive. Secondly, the scene takes place with a speech of Hitler as the background, which can hardly be viewed as romantic, and thirdly, this is even more true for the Gestapo, who was tasked with ratting out political dissent. To a Gestapo officer, ignoring the Führer's speech and kissing instead would make them stand out doubly.
Plot hole: In several occasions throughout the story, e.g. S4E18, Hogan attends parties at Stalag 13 wearing an immaculate US dress uniform. Where did he get that? Dress uniforms are not part of the usual kit a bomber crew brings along with them on their missions.