The history of the Wizarding World is long and rife with conflict. Much of the conflict stems from blood purity. Over the centuries, categories have been developed, which mark blood purity. The categories are Pure-bloods - wixen with four magical grandparents - Muggle-borns - wixen with two non-magical parents - and Half-blood, who are the wixen with any parentage in-between the two.
I have taken it upon myself to list the families with Pure blood. They are as follows:
Abbott, Avery, Black, Bulstrode, Burke, Carrow, Crouch, Fawley, Flint, Gaunt, Greengrass, Lestrange, Longbottom, Macmillan, Malfoy, Nott, Ollivander, Parkinson, Prewett, Rosier, Rowle, Selwyn, Shacklebolt, Shafiq, Slughorn, Travers, Weasley, and Yaxley.
Many of these families have claimed seats in the House of Lords on the Wizengamot. There are only twenty-four Houses represented on the Wizengamot, which include Houses Ravenclaw, Slytherin, Gryffindor, Hufflepuff, Black, Nott, Potter, Weasley, Bulstrode, Brown, Carrow, Cornfoot, Crabbe, Flint, Greengrass, Harper, Higgs, Lestrange, Longbottom, Lovegood, Malfoy, Prince, Shacklebolt and Pucey.
One might notice that several of the Houses on the Wizengamot are not listed among the Sacred Twenty-Eight and vice versa. The reason for this is that - simply put - these Houses are not of Pure blood. The seats on the Wizengamot are hereditary and - unfortunately - have nothing to do with blood purity.
As an example, House Potter was considered Pure-blooded until very recently; a few years ago, they showed their true colors, and fought for Muggles, as well as marrying one. It is quite clear that House Potter is not one of the many Houses that values the gifts they have been given. It must be said, though, that it was not a Muggle-born who married into the family - that would be disgraceful. After all, everyone knows that Muggle-borns steal magic from deserving Pure-bloods, and that they are uncouth beasts who don't understand our culture.
House Potter's allegiances are not surprising, as they have a history of fighting for the Muggles. Ralston Potter is the one who trapped wixen into a pocket world so that we wouldn't harm the Muggles.
It is for reasons such as these that some of the Houses on the Wizengamot do not appear within the Sacred Twenty-Eight, even if they are old enough to be in good standing with the real Pure-bloods.