They and two others – Ryan Ackroyd, 25, and a 17-year-old boy – deny other similar hacking charges.
Lulzsec claimed responsibility for attacking the UK’s Serious Organised Crime Agency (Soca) website.
The group is also believed to have hacked computers operated by the NHS, News International, Sony, Nintendo, film studio 20th Century Fox, US public broadcaster PBS, and US computer security organisations HBGary, Black & Berg and Infraguard.
A trial date has been set for April 2013.
Mr Cleary also faces charges in the US, where he stands accused of breaking into a number of websites, including that of the US X Factor, in order to deface them and steal personal details.
Arrested in June 2011, Mr Cleary was released on bail but later jailed after breaching his conditions.
He had been banned from accessing the internet, but is said to have contacted hacker Hector Xavier Monsegur, known as “Sabu”, who had been working as an FBI informant following his own arrest.