Call a positive integer n triadic if n can be written as a sum n $=$ x $+$ y $+$ z$,$ where x $<$ y $<$ z$,$ x $|$ y$,$ and y $|$ z$.$ For example, $9=1+2+6,$ so $9$ is triadic. Find, with proof, the largest non$-$triadic number.

The non triadic is not $8.$ What else should it be or what should I do$.$ Since x y and z can be any number that fits the condition. I know that all the odd numbers are triadic. The non triadic I have so far are $1,2,3,4,5,6,8,12,24$ how can I find the largest.