Adam Thomas, 22, and Claudia Patatas, 38, were found guilty after a trial of being members of the extreme right-wing organisation National Action, which was banned in 2016.
Thomas and Patatas were jailed at Birmingham Crown Court for six years and six months, and five years, respectively.
The sentencing judge said both had “a long history of violent racist beliefs”.
Their close friend, Darren Fletcher, who admitted National Action membership before trial, was also jailed for five years for the same offence.
In all, six people were sentenced on Tuesday, for being members of what Judge Melbourne Inman QC described as a group with “horrific aims”.
He said: “Its aims and objectives are the overthrow of democracy in this country by serious violence and murder, and the imposition of a Nazi-style state which would eradicate whole sections of society by such violence and mass murder.”
Last week, the court heard the prosecution claim that Fletcher had taught his daughter to give a Nazi salute, and that he sent a message to Patatas saying “finally got her to do it”.
The jury was told that Thomas and Patatas gave their child the middle name “Adolf”, which Thomas said was in “admiration” of Hitler, and the couple had Swastika scatter cushions in their home.
Photographs recovered from their address also showed Thomas cradling his newborn son while wearing the hooded white robes of a Ku Klux Klansman.
In conversation with another National Action member, Patatas said “all Jews must be put to death”, while Thomas had once told his partner he found “all non-whites intolerable”.
Former Amazon security guard Thomas and Patatas, a wedding photographer originally from Portugal who also wanted to “bring back concentration camps”, were found guilty after a seven-week trial.
Thomas, a twice-failed Army applicant, was also convicted on a majority verdict of having a terrorist manual, namely the Anarchist’s Cookbook, which jurors heard contained instructions on making “viable” bombs.
The couple, of Waltham Gardens, Banbury, Oxfordshire, held hands and wept as they were sentenced.
Daniel Bogunovic, 27, of Crown Hills Rise, Leicester, a leading member in National Action’s Midlands chapter, was also jailed.
He was convicted of membership of National Action after standing trial with Patatas and Thomas, and sentenced to six years and four months.
He was described by prosecutors as a “committed National Action leader, propagandist and strategist”, within the group’s Midlands cell.
Two other men, cyber security worker and National Action Midlands cell “banker” Joel Wilmore, 24, and van driver Nathan Pryke, 26, described as the group’s “security enforcer”, were also sent to prison.
Fletcher, 28, of Kitchen Lane, Wednesfield, Wolverhampton, Wilmore, 24, of Bramhall Road, Stockport, Greater Manchester, and Pryke, 26, of Dartford Road, March, Cambridge, all admitted membership of the banned group prior to the trial.
Pryke was handed a jail term of five years, five months, and Wilmore was sentenced to five years, 10 months.
Opening the case, Barnaby Jameson QC, prosecuting, said all six defendants had been members post-ban and taken part in the organisation’s chat groups, which were staging posts for comments of “virulent racism, particularly from Thomas, Patatas and Fletcher”.
He added: “Leaders Pryke, Wilmore and Bogunovic were more circumspect in their views but on occasion the true depth of their racial hatred leeched out.”