$x^3+y^3+z^3=(x+y+z)^3-3(x+y)(y+z)(x+z)\Longrightarrow x^3+y^3-z^3=(x+y-z)^3-3(x+y)(y-z)(x-z)$ $\Longrightarrow 414=6^3-3(x+y)(y-z)(x-z)\Longrightarrow 66=(x+y)(y-z)(z-x)$ Substituting $z=x+y-6$ into $66=(x+y)(y-z)(z-x) $ $\Longrightarrow 66=(x+y)(y-x-y+6)(x+y-6-x) \Longrightarrow 66=(x+y)(6-x)(y-6)=2*3*11 $ $\Longrightarrow (x,y,z)=(3,8,5), (8,3,5), (3,-5,-8), (-5,8,-3),(-5,3,-8), (8,-5,-3)$