The simplest way to do this (atleast for me) is with a hash. Now, the solution is a bit simplistic and it if you need to treat the data in any way before you print it it might not suffice, but it does give you the requested result.
I know I'm gonna be whacked on the head for not finding a better way of splitting the data.

#!/usr/bin/perl

my %hash;

while (<DATA>) {
        chomp;
        my ($name, $column2, $column3, $column4, $column5, $type) = sp
+lit/\s+/;
        $hash{$type} = $hash{$type} . "$name $column2 $column3 $column
+4 $column5\n";
}

foreach $key (sort keys %hash) {
        print "$key\n$hash{$key}\n";
}

__DATA__
name1 1 2 3 4 typex
name2 3 4 5 6 typey
name3 2 3 3 1 typex
[download]

which gives you the following output:

typex
name1 1 2 3 4
name3 2 3 3 1

typey
name2 3 4 5 6
[download]