England were at one of their lowest points on Tuesday after Iceland's battlers dumped them out of Euro 2016, coach Roy Hodgson quit, and the abject failure added to the national sense of post-Brexit turmoil.

Ragnar Sigurdsson and Kolbeinn Sigthorsson claimed their place in sporting history with the goals on Monday that secured a 2-1 win in Nice for the smallest nation ever to play in a major tournament.

Wayne Rooney gave a full-strength England the perfect start with a fourth minute penalty, but Hodgson's men rapidly ran out of ideas and energy against a nation ranked 34th in the world.

"Clueless. Utterly clueless," said The Times newspaper.

Goalkeeper Joe Hart, who let the second Iceland goal squirm under his fingers, said the humiliation had left English football "in a low place".

"We will get a lot of flak and we deserve it," said the Manchester City stopper.

A shellshocked Hodgson read out a statement announcing his resignation straight after the game -- the performance was "obviously not acceptable" and "now is the time for someone else to oversee the progress of this young, hungry and extremely talented group of players", he said.

The 68-year-old departed without answering a single question, but his critics filled the vacuum after England crashed out prematurely again just two years after exiting the World Cup in Brazil at the group stage under his tutelage.

Many asked how Jamie Vardy, whose 24 goals helped Leicester win the Premier League last season, Tottenham's Harry Kane, who netted 25 league goals, and Liverpool's often lethal Daniel Sturridge had failed to find the net when it mattered most.

Former England striker Alan Shearer said it was "the worst performance I have seen from an England team".

Another ex-international forward, Gary Lineker, tweeted that England, with a population of 53 million and the world's wealthiest league, had been beaten "by a country with more volcanoes than professional footballers".

England U-21 coach Gareth Southgate was the early favourite to replace Hodgson.