For trivial tree-processing problems, a recursive solution is generally easiest. This might not be the best way if your tree is truly gigantic, but it should work in most cases. This is untested but should get you on the right track:

my $dbh = ... # set up database

sub display_groups { 
    my ( $parent, $level ) = @_;
    
    # get all children of this parent
    my $sth = $dbh->prepare("SELECT id, gname
                             FROM groups WHERE subgroup=?
                             ORDER BY gname");
    $sth->execute( $parent );

    # display each child and its descendants 
    while( my $row = $sth->fetchrow_hashref ) { 
        print " " x $level;  # indent
        print $row->{gname}, "\n";
        display_groups( $row->{id}, $level + 1 );
    }
}

# start with children of group 0
display_groups( 0, 0 );
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Update: Added ORDER BY clause.