The suspect, identified only as Abdel J.S., was detained in the western city of Wuppertal, the federal prosecutor's office said in a statement.
He is accused of being a member of a foreign terrorist organisation and of involvement in the "war crimes" of killing, maiming and torturing in Iraq, they said.
He allegedly joined IS in Iraq in 2014, when the jihadists were rising to prominence by capturing swathes of the country as well as Syria.
Over several months that year, he participated in "draconian" public punishments in the Iraqi town of Al-Qaim, prosecutors said.
In one case, he supervised the execution of a prisoner, giving the signal for it to go ahead by firing a pistol, they said.
In another, he oversaw a thief having his hand amputated in public.
"IS imposed the punishments without the victims having access to a properly appointed court," prosecutors said.
He is also suspected, along with other IS fighters, of abusing a prisoner in custody to extort information, they said.
The suspect is being held in custody. The prosecutor's office did not say how he came to Germany or what he was doing there.
After enduring sustained offensives, IS's self-proclaimed "caliphate" eventually collapsed in Iraq in 2017 and Syria two years later but sleeper cells still carry out attacks in both countries.
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