what you got there is a TRIPLE action agitator…..its spring loaded, to move up and down, as well as in the normal spiral motion...
you have to use a really wide flat blade screwdriver, or if I am correct, a 3/4 drive socket extension to unscrew that plastic piece....which will give you access to the bolt underneath, which will have threads on both sides....sometimes it will release from the bolt, other times the bolt will come with it....
sometimes you can maneuver needle nose pliers down inside there against the corners to unscrew...little tricks we come across
there were several designs of this type.....
in any case, the dogs are the same through out all the designs, clean it up inside, and apply a bit of Vaseline to all the parts, then reassemble....
any issues, let us know....