Currently, the Indonesian SAR team is still looking for AirAsia QZ 8501 losing contact somewhere in near the Belitung regions. QZ 8501, PK-AXC is registered in Indonesia. Here's the aircraft detail according to airfleets.net:

Status : Written off
Last Registration : PK-AXC

Last Airline Indonesia Airasia     Country : Indonesia
    Date : 2005 - 
    Codes QZ AWQ 
    Callsign : Wagon Air

    Web site : http://www.airasia.com 

 General information & flightlog
Serial number3648Type320-216First flight date25/09/2008Test registrationF-WWBZSeat configurationW10 Y180 
Flights recorded
QZ8501SUB->SIN28/12/14QZ8501SUB->SIN28/12/14QZ663SIN->SRG10/11/14See details - Add a flight Engines2 x CFMI CFM56-5B6/3

Here is the plane's picture from planepictures, taken in Changi in February, 2014. More pictures availabe here.

According to ICAO regulations, Indonesia as the state of registry is responsible in monitoring and issuing the certificate of airworthiness and in ensuring airworthiness regardless of where it is operated.