In the Standard Model, the Higgs is a fundamental particle, like the quarks, leptons (electron, muon, tau), and gauge bosons (photon, W, Z). It's not a combination of other particles, like protons and neutrons are combinations of quarks.
In our current understanding, it's not made up of anything. It just is. That's been thought before, of course, but at the moment it's where we're at. There is no evidence of any structure to it, nor to quarks, electrons and the other fundamental particles.
If I understood one of the conversations here correctly, one of the theories not yet ruled out by the LHC discovery of the Higgs implies that our current 'fundamental' particles are genuinely fundamental. There are other theories that say otherwise, and time and study will tell which (if any) is correct.
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