The former minister was pushing to stay in her education brief and was unwilling to accept a different job offered to her by the Prime Minister.
Health Secretary Jeremy Hunt also spent more than 90 minutes in Downing Street where he resisted May's attempts to persuade him to switch to Business, Energy and Industrial Strategy (BEIS) and allow Business Secretary Greg Clark to take on his role.
Karen Bradley moves from culture to replace Brokenshire.
Downing Street said that as a result of the reshuffle there are now more women attending Cabinet, more female ministers and more members of the Government from a black and minority ethnic background than before.
British Prime Minister Theresa May begins the new year with a reshuffle of her top team.
Work and pensions minister David Gauke is now justice secretary.
Justine Greening announced she was quitting the Government in protest at being moved from education to work and pensions. It was reported that Greening chose to go after refusing May's offer as Works and Pensions Secretary.
David Lidington replaced Mr Green, taking his other title as Cabinet Office Minister, while Brandon Lewis was installed to take over from Sir Patrick McLoughlin as Conservative Party chairman.
CONSERVATIVE PARTY STAFF Theresa May centre with the new Tory party chairman Brandon Lewis left and his deputy James Cleverly right
Mrs May's shake-up was sparked by former First Secretary of State Damian Green's resignation following allegations of pornography being found on a Commons computer. Green was sacked by the prime minister past year after making a misleading claiming about pornography found on his parliamentary computer in 2008. The Northern Ireland secretary, James Brokenshire, has resigned because he needs surgery for a lesion on his right lung. "Clearly my long-term health and my family are my priorities", he wrote.
Cabinet ministers are nearly over five times more likely to have gone to a fee-paying school compared to the general population.
And Sajid Javid stayed in post but in the newly-named role of Secretary of State for Housing, Communities and Local Government.
Boris Johnson will continue as foreign secretary despite a number of gaffes and his widely-criticised handling of the Nazanin Zaghari-Ratcliffe case, while Brexit Secretary David Davis, Chancellor Philip Hammond, and Home Secretary Amber Rudd are all set to hang on to their jobs.
The Tories also received derision after publishing a tweet congratulating Chris Grayling on becoming party chairman - then deleting it seconds later.
The Times later reported that there had been "internal pushback" against Grayling, with o ther reports suggesting that the Conservatives had mistakenly announced Grayling's appointment due to it being reported by the BBC.
Lewis also took on the role of Minister Without Portfolio.
A reshuffle that was intended as an assertion of the prime minister's authority has instead, in the cruellest way, exposed her lack of it, as middle-ranking ministers simply refused to budge from their posts.
This is why everyone wore white at the Critics' Choice Awards
The audience gave the 32-year-old actress a standing ovation as she received the award and humbly asked the crowd to sit. And Nicole made sure all eyes were on her when she arrived at the Critics' Choice Awards 2018 in Los Angeles last night.