The arrests followed raids, some by heavily-armed special units, which hit 13 locations in six German states.
The four prime suspects planned to spark “a civil-war-like situation… via as yet undefined attacks on politicians, asylum seekers and people of Muslim faith,” federal prosecutors said in a statement.
A further eight suspects were alleged to have agreed to “financially support the group, provide it with weapons or take part in future attacks”.
The twelve included a police officer previously suspended over suspicions he had links to the far-right, a source at the interior ministry in North-Rhine Westphalia state told AFP, though it was not immediately clear if he was one of the prime suspects.
From its founding in September 2019, the group's ultimate aim was “to shake the state and social order in Germany and in the end to overturn it,” investigators believe.
In order to plan their attacks, the group allegedly held regular meetings which were coordinated and organised by two of the main suspects, named only as Werner S. and Tony E.
The suspects, all of whom are German citizens, also communicated using messenger apps.
Investigators launched Friday's raids to determine whether the suspects already had weapons or other supplies that could be used in an attack.
The twelve men are set to appear before a court on Friday or Saturday to hear whether they will be imprisoned on remand.
Far-right in spotlight
German authorities have turned increased attention to the country's underground extreme right scene since the murder of conservative local politician Walter Lübcke last June and an October attack on a synagogue in eastern city Halle.
Suspects arrested in both cases have ties to the extreme right.
According to Spiegel magazine, police discovered several weapons in Friday's raids, including one self-made “slam gun” similar to the one used in the Halle attack.
Interior minister Horst Seehofer announced in December 600 new posts across the federal police and domestic security services to track far-right extremist threats, citing a growing danger.
At the time, federal police said they had identified 48 people on the extreme right as “dangerous” individuals who could carry out an attack.
Reacting to reports of the arrests on Friday, a spokesman for the Federal Interior Ministry said that measures to protect religious institutions would be reviewed by local authorities.
A spokesman for Chancellor Angela Merkel, meanwhile, said that threats to attack Islamic institutions in Germany amounted to “abominable behaviour”.
“We as the federal government feel an obligation to ensure that anyone in Germany can practice their religion within the bounds of our legal order,” said Steffen Seibert at a government press conference.
By Kit Holden