Following theoretical calculations, particle accelerator experiments in the 1970s confirmed that quarks are responsible for forming the nuclear particles - protons and neutrons.
Six 'flavours' of quark have now been identified. (ref.) Suggesting the obvious question, are quarks themselves formed from even more elementary particles? At present, the consensus is that they're not. If they can be broken down, experiments underway at CERN's Large Hadron Collider may show their finer structure. Until then, they appear to be fundamental - in the same way that electrons and photons appear to be.
Sub-quark particles, named as 'Preons' in 1974, have been suggested as possible building blocks for quarks. Some researchers maintain that they could help provide theoretical support for experimental findings. As yet, no experiment has confirmed (or refuted) their existence. See : Wikipedia
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