How can I create such subchapters for Google on my website:

Is this HTML or something server specific?
And how to make this on my own homepage?
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They’re called Sitelinks, and you don’t. Google does this automatically, but only if they feel it’s going to be useful for users – typically, only on high-traffic sites.
Via the Google Webmaster Tools, you can prevent certain pages from showing up in these suggestions, but you can’t enable/add any.