Sort By Image

How do you enable sorting by an image?

I am listing files in the grid which include word, excel, pdf, etc. I am displaying an icon for each file type and want the ability to sort similar
file types together.

The icon names are icon_word, icon_excel, icon_pdf, etc.

I thought this would be as simple as formatting the column as HTML but that doesn't seem to work. When I click the header, no sorting occurs at all.