It is known that when arriving in the Moscow region Wehrmacht soldiers felt cold, especially since they didn't have any winter clothes and later they didn't arrive since they were blocked on the way. Therefore they stole the clothes from the soviet citizens. In the movie the Nazi soldiers received their clothes and didn't steal anything.
The first scenes of the movie, set in October 1941 in Moscow the city is shown as calm and clean. It is unlikely to be like that since the Nazi attacks on Moscow already began on the 2nd October 1941.
It is a historical fact that Zoya Kosmodemyanskaya was assigned to organize and to burn houses in various towns around Moscow, in which Wehrmacht soldiers were hiding. In the movie it is not shown that she is burning anything.
Zoya's brother Shura was 16 when he was taken to the army. He is portrayed by an actor who is older that 30. According to the movie makers, they use a photo of a 22 year old Shura for their casting. in fact, Shura died heroically at the age of 19
The speech Kosmodemyanskaya is saying during her hanging, in which she is mentioning Stalin, is based on the edited quotes printed in the soviet Newspapers after her death. According to witnesses her speech was a very different one and didn't mention Stalin at all. Instead she was praising the Red Army and was talking about their revenge.
When Zoya is walking through the snow after her arrest during the night there is no steam coming out of her mouth, which should happen during winter.
When Zoya is checked to become an agent, the officer is talking German to her. His subtitled line "In your questionnaire it is said that you learned German in school" is translated very incorrectly into the real German, including "es ist gesagt" (for "It is said") "Das Sie studieren Deutsch"". The real line should have been "In deinem Fragebogen steht, dass du in der Schule deutsch gelernt hast". Furthermore the word "Du" (Singular "You") was translated as "Sie" (plural version of "You").