# Miller indices and FCC structure - non-existent planes

Below is a copy from Kittel's Introduction to Solid State Physics, 8th edition, page 42. It is a picture to illustrate that in FCC structures (loosely said and exactly cited) "no reflections can occur for which the indices are partly even and partly odd.". But how an FCC structure can have a peak named (400)? That translates to a plane that would be at $a/4$ where $a$ is the appropriate cubic cell constant and perpendicular to $x$ direction. There is no such plane in FCC structure.

I must admit that I am having huge trouble understanding the whole Miller indices business because if someone is referring to an $hkl$ reflection, it makes little sense because in order to see anything in the diffraction pattern, and the whole point of Bragg, is that there needs to be a constructive interference of two plane waves each scattered by different perpendicular planes. Another issue I have, if I look at plane that is perpendicular to $x$ axis and that intersects it at some point (other than 0), do I know that there is also a plane perpendicular to it at the origin (intersecting $x$ at 0)?

The answer to the question should answer the first paragraph, the second one is there only to illustrate items in crystallography that I do not understand.