  A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic
if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots
,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$,
where $f(\theta)=0$. In a series of recent articles, complete classifications
of the Galois groups were given for irreducible polynomials \[{\mathcal
F}(x):=x^8+ax^4+b\in {\mathbb Z}[x]\] and \[{\mathcal
G}(x):=x^8+ax^6+bx^4+ax^2+1\in {\mathbb Z}[x], \quad a\ne 0.\] In this article,
for each Galois group $G$ arising in these classifications, we either construct
an infinite family of monogenic octic polynomials ${\mathcal F}(x)$ or
${\mathcal G}(x)$ having Galois group $G$, or we prove that at most a finite
such family exists. In the finite family situations, we determine all such
polynomials. Here, a ``family" means that no two polynomials in the family
generate isomorphic octic fields.
