SQLITE: select rows where a certian column is contained in a given string

I have a table which has a column named "directory" which contains strings like:

c:\mydir1\mysubdir1\
c:\mydir2
j:\myotherdir
...

I would like to do something like

SELECT FROM mytable WHERE directory is contained within 'c:\mydir2\something\'

This query should give me as a result:

c:\mydir2