So, let’s start with what we can say with any degree of certainty.
All of the critters that we know as mammals are grouped together within the class Mammalia; within this class sits an infraclass ( being Latin for “below”) called the Eutheria (or "true beasts"), which contains all the placental mammals (that is, all mammals except monotremes like the platypus and marsupials like the kangaroos).
Much has happened to the arrangement of the mammals in the last half-century and there are many groups -- of which deer are one -- for which the relationships are still not fully resolved.
That which follows is a summary of the situation to date; readers interested in more details of how we classify organisms are directed to the Taxonomy page.
While I can think of a few things to which that idea doesn’t apply, it certainly seems applicable to the task of classifying mammals.
hoofed animals) – whatever happened to this as a taxonomic unit?It is reasonably well established that the Eutheria can be broadly divided into four superorders: the Euarchontoglires (primates, rodents, hares and rabbits); the Xenarthra (anteaters and armadillos); the Afrotheria (elephants and manatees); and, of interest to us here, the Laurasiatheria, which holds the deer (along with various other critters including cows, bats and all the carnivores).The first major sticking point we encounter now is on the placement of the order Cetacea (the whales and dolphins).Nonetheless, the current consensus is that the ungulates don’t represent a genetic unit; that is to say that they’re not a group of mammals more closely related to one another than to other (non-ungulate) mammals.Rather, it seems that the ungulates form an evolutionary grade – in other words, they’re a group whose members have evolved similar adaptations (significantly, although not limited to, hooves in place of claws).