Unhook the amp in the sub and see if the fuse blows.
Radio is probably hooked to that fuse which is fine, unless you add more power requirements, i.e. the subwoofer amp
Depending on that amps wattage you may need to install a direct 12v and ground with it's own fuse rated for that amp.
And then use current "Radio" 12v wire for a relay on the new direct 12v line this activates sub amp only when radio is on, so sub amp won't drain battery.
Look up the amps power rating for the correct fuse size, wire size and relay rating
Relays are designated by activation voltage and the amps it can safely pass, so a 12v 20A relay is activated by 12v and when contact close it cam safely pass 20amps
The relay itself, when closed, draws less than 1 amp