game of thrones - Under which kingdom are the Iron Islands?

According to the Game of Thrones Wiki, there seem to be 7 kingdoms but 8 governors. It seems to me that this is the correspondence:

1. Kingdom of the North -- The Lord Paramount of the North and Warden of the North

? -- The Lord of the Iron Islands

2. 
   
   

   Kingdom of the Mountain and the Vale -- The Lord Paramount of the Vale and Warden of the East

   
   


3. 
   

   Kingdom of the Isles and Rivers -- The Lord Paramount of the Riverlands

   
   


4. 
   

   Kingdom of the Rock -- The Lord Paramount of the Westerlands and Warden of the West

   
   


5. 
   

   Kingdom of the Stormlands -- The Lord Paramount of the Stormlands

   
   
   


6. 
   

   Kingdom of the Reach -- The Lord Paramount of the Reach and Warden of the South

   
   


7. 
   

   Principality of Dorne -- The Prince of Dorne

   
   

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My understanding based on the rest of the webpage, Why do they still refer to the "Seven Kingdoms" in Game of Thrones? and What are the different provinces mentioned in a Game of Thrones? is that Kingdom #3 was split into 2 regions and hence there are 8 regions of Westeros that are subject to the rule of the Iron Throne.

Is that right?

Answer

The Iron Islands were a part of the Kingdom of Isles and Rivers. Before the Targeryen invasion, House Hoare ruled over the Iron Islands and the Riverlands.

House Hoare was the Royal House of the Iron Islands. Harwyn Hoare conquered the Riverlands and House Hoare controlled the Riverlands until Harwyn's grandson Black Harren was killed during the Targeryen Invasion.

Yes, you are correct that there are 8 regions due to the fact that the Kingdom of Isles and Rivers was split into two regions - The Iron Islands and The Riverlands.