Do common seahorses belong to any domain group

I couldn't find any online and was about to give up. But I just had to be sure. If you don't know what I mean by domain, I meant like these:

Domain
Kingdom
Phylum
etc.

So common seahorses or Hippocampus kuda belong to a domain group?

According to the three-domain system of Carl Woese, introduced in 1990, the Tree of Life consists of three domains: Archaea, Bacteria and Eukarya. Respectively, single cells with no nucleus, single cells with nuclear membranes, and multicellular organisms. Seahorses, being multicellular, would belong to Domain Eukarya.

The common seahorse is classified as Hippocampus hudsonius of the genus Hippocampus in the family Syngnathidae.