Should a sub-game perfect Nash Equilibrium always be a part of the Nash Equilibrium of a game?

I’m solving a 3 person game and the Subgame Perfect Nash Equilibrium I get is not a part of the Nash Equilibrium. Isn’t a SPNE supposed to be a refinement over the Nash? I’ve seen some Nash Equilibria not being a part of the SPNE (i.e., some NE being redundant), but I wonder if a SPNE is supposed to just cut down the NE into one SPNE or if it can be something completely different.

The SPNE has a mixed strategy for 2 players and a pure strategy for one player.

Here is the game for anyone interested.