It's important to keep the 'flow'... of what Daniel wrote about the "little horn" in Daniel 7. Some like to pic-n-choose verses to cover while leaving others out which affect the interpretation...
Dan 7:2-9
2 Daniel spake and said, I saw in my vision by night, and, behold, the four winds of the heaven strove upon the great sea.
3 And four great beasts came up from the sea, diverse one from another.
4 The first was like a lion, and had eagle's wings: I beheld till the wings thereof were plucked, and it was lifted up from the earth, and made stand upon the feet as a man, and a man's heart was given to it.
That above 1st beast was the Babylon empire, under Nebuchadnezzar, king of Babylon.
5 And behold another beast, a second, like to a bear, and it raised up itself on one side, and it had three ribs in the mouth of it between the teeth of it: and they said thus unto it, Arise, devour much flesh.
The Medo-Persia empire came next, having conquered Babylon.
6 After this I beheld, and lo another, like a leopard, which had upon the back of it four wings of a fowl; the beast had also four heads; and dominion was given to it.
The 3rd one is represented by Alexander's Grecia, which conquered the Medo-Persia empire.
7 After this I saw in the night visions, and behold a fourth beast, dreadful and terrible, and strong exceedingly; and it had great iron teeth: it devoured and brake in pieces, and stamped the residue with the feet of it: and it was diverse from all the beasts that were before it; and it had ten horns.
That represented the pagan Roman empire of history. But there's a problem with that above, because it mentions that it had "ten horns". The ancient pagan Roman empire did NOT have "ten horns". And this is following the order of the pieces that make up the beast statue that Nebuchadnezzar was shown in his dream, per Daniel 2. So let's go back to Daniel 2 and see what's up.
Well, you are close, need to go over with history and match them up and then the little horn power becomes apperant.
Daniel 7:2-4
7:2 Daniel spake and said, I saw in my vision by night, and, behold, the four winds of the heaven strove upon the great sea.
7:3 And four great beasts came up from the sea, diverse one from another.
7:4 The first was like a lion, and had eagle's wings: I beheld till the wings thereof were plucked, and it was lifted up from the earth, and made stand upon the feet as a man, and a man's heart was given to it.
Most everyone agrees this is Babylon.
Daniel 7:5
7:5 And behold another beast, a second, like to a bear, and it raised up itself on one side, and it had three ribs in the mouth of it between the teeth of it: and they said thus unto it, Arise, devour much flesh.
Same agreement that this is Medi-Persia
Daniel 7:6
7:6 After this I beheld, and lo another, like a leopard, which had upon the back of it four wings of a fowl; the beast had also four heads; and dominion was given to it.
Pretty much agreement this is Greece
Daniel 7:7
7:7 After this I saw in the night visions, and behold a fourth beast, dreadful and terrible, and strong exceedingly; and it had great iron teeth: it devoured and brake in pieces, and stamped the residue with the feet of it: and it was diverse from all the beasts that were before it; and it had ten horns.
Here we see Pagan Rome with the ten horns, which are the nations which came up with it. We also get a description in Daniel 2..
Daniel 2:33 &41-43
33 His legs of iron, his feet part of iron and part of clay.
41 And whereas thou sawest the feet and toes, part of potters' clay, and part of iron, the kingdom shall be divided; but there shall be in it of the strength of the iron, forasmuch as thou sawest the iron mixed with miry clay.
42 And as the toes of the feet were part of iron, and part of clay, so the kingdom shall be partly strong, and partly broken.
43 And whereas thou sawest iron mixed with miry clay, they shall mingle themselves with the seed of men: but they shall not cleave one to another, even as iron is not mixed with clay.
We see how the nations of the empire progressed and become Western Civilization in Europe, and still have not been able to unite in the manner of the Roman Empire..
Daniel 7:20 &24
7:20 And of the ten horns that were in his head, and of the other which came up, and before whom three fell; even of that horn that had eyes, and a mouth that spake very great things, whose look was more stout than his fellows.
7:24 And the ten horns out of this kingdom are ten kings that shall arise: and another shall rise after them; and he shall be diverse from the first, and he shall subdue three kings.
So the ten toes mixed with iron and clay are of the fourth Beast who subdues three of the kings, and the Little Horn also comes out from the fourth Beast:
Daniel 7:8
7:8 I considered the horns, and, behold, there came up among them another little horn, before whom there were three of the first horns plucked up by the roots: and, behold, in this horn were eyes like the eyes of man, and a mouth speaking great things.
This is Papal Rome, we see much agreement in history..
"Out of the ruins of political Rome, arose the great moral Empire in the 'giant form' of the Roman Church." – A.C. Flick, "The Rise of the Medieval Church," 1900, p. 150.