Mr Finlay, of Maplewood Road, and Mr Byrne, of Drumcairn Park, both Tallaght, came before the three-judge court where they were both charged with membership of an unlawful organisation styling itself the Irish Republican Army, otherwise Oglaigh na hEireann, otherwise the IRA on June 5.
They were also charged with the unlawful possession of a 9x18mm calibre Makarov PM semi-automatic pistol and magazine and 10 rounds of 9x18mm BR calibre Sellier & Bellot ammunition at Glenshane Crescent, Tallaght, on the same date.
Det Rory Geelon told state solicitor Liam Mulholland that he formally arrested Mr Finlay yesterday outside Tallaght garda station.
The accused man replied: “No comment”.
Det Colin Barker gave evidence to the court of the arrest and caution of Mr Byrne, who appeared before the court wearing a navy hoodie and black trousers.
He said Mr Byrne made no reply to the charge after caution.
There was no application for bail on behalf of either of the men, but presiding judge Mr Justice Paul McDermott approved applications for legal aid and a second counsel for both accused.
Mr Justice McDermott, sitting with Ms Justice Iseult O’Malley and Judge Margaret Heneghan, remanded both men in custody to appear before the court again on June 16.