Factual error: When the SS guard is calling Sergeant Shultz after seeing the burning fuse, he addresses Shultz as "Sergeant", rather than the proper German word for Shultz's rank, "Feldwebel."
Factual error: The guard outside Klink's office and the two Gestapo soldiers guarding the gold truck have MP38/40 and a Thompson submachine gun, but carry rifle ammunition pouches instead of the long ammo pouches that carried the 30-round magazines used by the MP38/40.
Factual error: The American women the SS brings into camp claim to be entertainers having performed for troops. The series is allegedly set in 1942. Before June 6th, 1944 there were no allied troop concentrations in central Europe, certainly not in Germany, and very certainly none of a size and security rating the USO (or probably rather its predecessor organization, since the USO was founded in 1941 and would not have been fully operational yet) would send a troupe of female entertainers to.
Factual error: Hogan mentions a store that offers "thirty percent off on T-shirts." T-shirts were mostly unknown in Germany during the 3rd Reich, and didn't see widespread use before the 1960s. Before that, the traditional Unterhemd, known in America as tank top or wife beater, was worn almost exclusively.
Factual error: When the three captured heroes stand before the Gestapo officer, he sweeps three sets of US identification tags into his hand. The names and uniforms suggest that the three captured personnel are from three different armies. Identification tags differ greatly between armies, all wearing US-style with their usual uniforms they would be worse off than wearing none at all. The Geneva convention would allow for them to be shot on the spot as spies under these circumstances since they initiated combat (blew something up) wearing false uniforms. (00:03:20)
Factual error: When the men are lined up to take the strongboxes of gold to switch with bricks, each man in line easily hefts and tosses them to the next man. Gold bricks weigh over 27.5 pounds (12.4kg) apiece. If there 10 bricks per box (going by the size of the boxes), then that comes to 275 pounds per box, plus the weight of the box, which these guys are tossing around without a problem.