German Interior Minister Nancy Faeser called for police to act vigorously if crimes are committed during political demonstrations.
Faeser was speaking after a rally by more than 1,000 Islamists in Hamburg on Saturday, which passed off peacefully, but where allegations of anti-Islamic policies and an anti-Islam media campaign in Germany were heard.
Some carried posters that read "Caliphate is the solution" or "Germany - a dictatorship of values."
"Seeing an Islamist demonstration of this kind on our streets is difficult to bear. It's a good thing that the Hamburg police counteracted crime with a large presence," she said in remarks to the Tagesspiegel newspaper to be published on Monday.
The red line at which Germany's protection of the right of assembly and freedom of speech ended had to be clear, she said.
"No terrorism propaganda for Hamas, no hate speech directed at Jews. If crimes like this occur, there has to be immediate and forceful intervention at demonstrations," Faeser told the Berlin daily.
According to the Hamburg authorities, the people registering the demonstration in advance, as required under German law, are close to the Muslim Interaktiv group, which has been classified by the intelligence services as extremist.
The security authorities were monitoring Islamist organizations, Faeser said. Following German bans on the Palestinian organization Hamas and the Samidoun group, which supports Palestinian prisoners, other groups were being monitored, she said.
"Other groups that raise emotions, radicalize and recruit new Islamists are also being watched by our security authorities," she said. This included the group at the Hamburg demonstration.