The officer has SS lightning insignia on both sides of his collar. The lightnings should be on his right and rank insignia on the left.
The German officer is addressed as Oberleutnant. The corresponding rank in SS-Waffen would be Obersturmbannführer.
SS troopers carry their swastika armbands all the time. In reality this did not belong to their everyday uniform. The officer even has the armband on his shirt after removing his jacket.
Just after 52:50, a US soldier removes both dogtags from his fellow soldier that just died. A soldier will never remove both dogtags, because one dogtag is always supposed to remain with the dead soldier (that is also why there are two tags). Another mistake is that the dogtags do not look like the dogtags that are used by the USA in WW2: real ones from WW2 have a notch at the bottom of each dogtag.
At 39:03, when the black suited SS officer is kneeled over the just burned body in the hall, you see a drawing of a Snoopy figure on the wall in the back. Snoopy was first introduced to the public in 1950, so it can not be drawn on a wall in WW2.
When the prisoner is released it is obvious there is no keyhole. The actor just rubs keys on the door for a second.
At 39:00, two SS officers enter a hall where another officer lies burned on the ground. You can see that the hall has airconditioning ducts below the roof. Airconditioning was not invented yet in the 2nd world war.
At 1:00:08 - when the US soldier "opens" the cell door - you can see that the cell door has no lock. The soldier seems to just very shortly touch the door with a key.
At 2:07 - when they have some video shots of a concentration camp with barbed wire - you can clearly see that all the barbed wire is completely rusted away. Of course this was not the case in the camps during WW2.
The movie poster shows planes and a tank. The movie itself however does not have a single tank or plane in it.
Success of Operation Overlord relied on utmost secrecy and disinformation. All Americans sent in this film behind German lines seem to be aware of all details about the coming invasion a few days before. That would be risky if they were caught by Germans.
During the sad scene when the American soldier finds his friend dead, the would-be dead guy clearly blinks.