No Labels, the centrist group that could field a third-party presidential bid, has asked the Justice Department to investigate what it calls unlawful intimidation by groups that oppose it.

The group filed a complaint on Jan. 11, accusing a number of political figures and other critics of engaging in voter suppression and violating federal law, including the Racketeer Influenced and Corrupt Organizations Act, or RICO, which is often used to combat organized crime.

Leaders of New Labels who described the complaint during a news conference on Thursday pointed largely to previously reported details of efforts to oppose the group, as well incendiary statements that some of its critics had made on political podcasts.

The group compared the efforts of its opponents to those of the Ku Klux Klan in the 1950s and ’60s and the fictional mob boss Tony Soprano. A montage of clips shown by the group included Rick Wilson, a founder of the anti-Trump Republican group the Lincoln Project, saying last spring that the group had to “be burned to the ground,” using an expletive — although the clip had been cut off before Mr. Wilson adds the word “politically.” (After being asked about the shortened clip, the group uploaded a version of the video with the full statement.)

Other critics featured in the montage were Jonathan V. Last of The Bulwark, a conservative news outlet, and Matt Bennett of Third Way, a centrist Democratic group.

Former Gov. Pat McCrory of North Carolina, a national co-chairman of No Labels, said that opponents of the group were “using intimidation to keep people off the ballot,” and attacking “the rights of the American people and our democracy.”

In a statement on Thursday, the Lincoln Project accused No Labels of trying to “weaponize the D.O.J.” to “attack their opponents for protected political speech.” The Bulwark and Third Way did not immediately respond to messages seeking comment.

The complaint filed by No Labels was unusual for a group involved in campaign politics. Attacks and pressure campaigns against potential candidates, donors and supporters of a party are common in politics, particularly in high-profile presidential races, and are considered widely permissible by courts under First Amendment speech protections.

The group’s claims that its opponents are meaningfully infringing on voting rights are also complicated by the fact that a number of third-party presidential candidates have already entered the race, including Robert F. Kennedy Jr., the prominent environmental lawyer turned anti-vaccine activist.

In contrast, No Labels has yet to commit to entering the presidential race, and it remains unclear who would run under the group’s ballot line if it did. Senator Joe Manchin III, a conservative Democrat from West Virginia, who has been touring New Hampshire, is seen as a top potential candidate.

The group has also fought to block candidates from using its ballot line to run for offices other than president and vice president, in part to avoid disclosing its donors.

Ryan Clancy, the group’s chief strategist, said on Thursday that the group was “not trying to stand in the way of either party nominating whoever they want.” But Mr. Clancy and other leaders of the group said they would decide to enter the race only when it became clear that the two major parties would nominate President Biden and former President Donald J. Trump.

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