Conditional formatting referencing another sheet and returning multiple column items

What I'd like to do is pull info from one sheet into another.

The first sheet has a table, 5 columns wide and 250 rows (A5:D255). The fifth column has text from a 3-item pull down list. I would like to, on the second sheet, pull all the rows of info which have one of those three items labeling it. For instance, some have the label "Dog". I want to have a second sheet that pulls all the rows of info that have the label "Dog" from the first page.

The sample file shows how I want the second (and other) sheets to look like. I want to reference the first sheet, cell E2 as the index and if it says "Dog", then pull the info from columns A2:D2 into a sheet labeled Dog.

This file will be updated somewhat regularly, with new items added into the full list in the middle, so I'm trying to figure out an easier way to manage the other sheets that break down that list into categories than having to update those individually.

Right now I have the first page set up will the full list and the pull down menu items labeling each row, but I wasn't sure how to go about writing a formula for the other pages.