http://www.proz.com/search/
Subdirección General has been translated in so many ways (as per ProZ glossary), a few examples which I’ve provided, but personally I would stay away from “Deputy” to describe an office/department.


Subdirección General de Comercialización Exterior de Turismo
General Section of Tourism Promotion
Subdirección General de Régimen Jurídico
Deputy general directorate of legal framework or system
Subdirección general de los archivos estatales
National Archives Bureau
Subdirección General de Productos Sanitarios
General Sub-directorate of health products
Subdirección General de Información Administrativa e Inspección General de...
Office of the Deputy Director of Administrative Information and General Services Inspection
Subdirección General de Régimen Jurídico y Coordinación Universitaria General Sub-Directorate of the University’s Legal Procedures and Coordination

http://administracion.gob.es/pagFront/espanaAdmon/directorio... (there is a translation here, but I caution, it is machine translated)
S. OF ADMINISTRATIVE RECORDS TO SUPPORT THE JUDICIAL ACTIVITY
Administration:
State administration
Ministry:
MINISTRY OF JUSTICE
S.GRAL. OF ADMINISTRATION OF JUSTICE
Address:
PLAZA DE JACINTO BENAVENT 3
28012 Madrid (Madrid)

So:
**Sub-Directorate General /General Sub-Office for Administrative Records for Judicial Activity Support (in Support of Judicial Activity).**

HTH!


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Note added at 4 days (2016-03-17 10:42:21 GMT) Post-grading
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Gracias Laura.