yes
a person is permitted to have a dog.
but it causes all kinds of very serious Halachic problems.
the dog is muktzah, you cant lead it on a leash on Shabbos. (there are some lenient opinions here.) you cant pet it or hold it on Shabbos.
you cant feed it cholov v basur, most commercial dog foods are assur.
there are other problems i cant recall.
in addition, although i personally like dogs, in Yiddish tradition, all throughout our history, in Tanach, and Gemorah, dogs are considered disgusting. All of our Bubbies and Zadies from two generations ago back to Moshe Rabbeinu, would gag at the thought of a dog in the house, (though they were used for working purposes)