Charterhouse was historically an extra-parochial area (outside the control of any parish). It became a civil parish within Middlesex in 1858. In 1889 it was transferred from Middlesex to the newly-created County of London and in 1900 became part of Finsbury Metropolitan Borough. Finsbury Metropolitan Borough was abolished in 1965, becoming part of the London Borough of Islington.
For more information, see the EN Wikipedia article Charterhouse Square. This article explains the former history of Charterhouse Square from the time it was a monastery.