Des a magnetic field flows though the inside of a magnet

Magnetic field lines are defined to have the direction that a small compass points when placed at a location. (a) If small compasses are used to map the magnetic field around a bar magnet, they will point in the directions shown: away from the north pole of the magnet, toward the south pole of the magnet.

Explanation:

However, the magnetic field lines do not just end at the tip of the magnet. They go right through it, so that inside the magnet the magnetic field points from the south pole to the north pole. Thus the magnetic field lines form a closed loop and do not have any ends.

Explanation:

So it's probably YES.