Using perl rename and perl's File::Basename module (which is included with perl). I've included some minimal safety checking to make sure it actually succeeds creating the directory:
$ rename -n 'use File::Basename;
my ($bn, undef, $ext) = fileparse($_, qr/\.part.*/);
mkdir $bn;
die "Error creating directory \"$bn\".\n" unless -d $bn;
s:$_:$bn/$bn$ext:' *.rar
rename(LQ0gfKQej7GKG44Cn0sSAC.part01.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part01.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part02.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part02.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part03.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part03.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part04.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part04.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part05.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part05.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part06.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part06.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part07.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part07.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part08.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part08.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part09.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part09.rar)
rename(LQ0gfKQej7GKG44Cn0sSAC.part10.rar, LQ0gfKQej7GKG44Cn0sSAC/LQ0gfKQej7GKG44Cn0sSAC.part10.rar)
rename(no9e1hvqlVEbYHs9YU3.part1.rar, no9e1hvqlVEbYHs9YU3/no9e1hvqlVEbYHs9YU3.part1.rar)
rename(no9e1hvqlVEbYHs9YU3.part2.rar, no9e1hvqlVEbYHs9YU3/no9e1hvqlVEbYHs9YU3.part2.rar)
rename(NwgdNaort1EqT0ch.part1.rar, NwgdNaort1EqT0ch/NwgdNaort1EqT0ch.part1.rar)
rename(NwgdNaort1EqT0ch.part2.rar, NwgdNaort1EqT0ch/NwgdNaort1EqT0ch.part2.rar)
rename(NwgdNaort1EqT0ch.part3.rar, NwgdNaort1EqT0ch/NwgdNaort1EqT0ch.part3.rar)
rename(NwgdNaort1EqT0ch.part4.rar, NwgdNaort1EqT0ch/NwgdNaort1EqT0ch.part4.rar)
rename(VEwMzBPH91J.part1.rar, VEwMzBPH91J/VEwMzBPH91J.part1.rar)
rename(VEwMzBPH91J.part2.rar, VEwMzBPH91J/VEwMzBPH91J.part2.rar)
The -n makes this a dry-run. Remove the -n (or replace it with -v for verbose output) when you're sure it's going to do what you want.
Note: this will work whether the .rar files are in the current directory or if you have to specify the full path. The sub-directories will be created in the current directory. It wouldn't be hard to change the script to create the subdirs in a different directory if needed.
As usual, don't confuse perl rename with the rename in util-linux. Check which version you have by running rename -V. If it mentions perl or File::Rename, it's the perl version.
xargsdoesn't use the shell to execute the command, so wildcards have no special meaning to it.