In general, for dimensions $n>5$ we don't have  topological obstructions  to the existence of Einstein metrics (like as Hitchin-Thorpe inequality when $n=4$).

However, there exist   **compact homogeneous Riemannian manifolds** with **no $G$-invariant
Einstein metrics.** In this case, one works with cosets $M=G/K$ of  a compact Lie group $G\subset {\rm Isom}(M)$ and  the Einstein equation $Ric = c \cdot g$ is written with respect to  a $G$-invariant Riemannian metric $g$ on $M=G/K$.
For such a metric the Einstein equation reduces to a polynomial system and positive real solutions correspond to invariant Einstein metrics.

In particular, let $\frak{g}=\frak{k}\oplus\frak{m}$ be a reductive decomposition for $M=G/K$ (such an orthogonal decomposition always exists for a homogeneous Riemannian manifold $(M=G/K, g)$).  For simplicity, let us assume that the tangent space ${\frak{m}}\cong T_{o}G/K$ decomposes into $q$ pairwise inequivalent 
isotropy summands (i.e., irreducible $Ad(K)$-submodules)

$
\frak{m}= \frak{m}_1 \oplus \frak{m}_2 +\oplus ..... \oplus \frak{m}_q.
$

Then any $G$-invariant Riemannian metric is given by 

$
g = < , > = x_1\cdot (-B)|_{\frak{m}_1}+\cdots+x_s\cdot (-B)|_{\frak{m}_q} 
$

for some positive real numbers $x_1, x_2, ...  x_q$, where  $-B$ denotes the negative of the Killing form of $\frak{g}$ (restricted on $\frak{m}$).

Homogeneous Einstein metrics are positive real solutions of the system $\{r_1-r_2=0, ...., r_{q-1}-r_{q}=0\}$
where $r_{i}$ are the components of the invariant Ricci tensor on any isotropy summand. These are 
given by 

 \begin{equation}\label{ricc}
   r_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{i, j}\frac{x_{k}}{x_{i}x_{j}}[ijk]-\frac{1}{2d_{k}}\sum_{i, j}\frac{x_{j}}{x_{k}x_{i}}[kij], \qquad (k=1, \ldots, q).
 \end{equation}
Here $[ijk]$ are the **structure constants** of $G/K$.  Their computation is usually non-trivial.  For particular classes of homogeneous spaces (e.g. isotropy irreducible), they are related with the Casimir operator of $\frak{g}$.  They are non-zero only for  these triples $\frak{m}_i$, $\frak{m}_j$, $\frak{m}_k$, for which  

$B([\frak{m}_i, \frak{m}_j], \frak{m}_k)\neq 0$. 


 Notice that the study of homogeneous Einstein metrics becomes more complicated if some of the isotropy summands are equivalent each other.
Finally, it turns out that there are several homogeneous spaces with no invariant Einstein metrics. For example, we know by **M. Wang and W. Ziller**   that the **12-dimensional** space $SU(4)/SU(2)$ does not admit any homogeneous Einstein metric.

see: M. Wang and W. Ziller: Existence and non-existence of homogeneous Einstein metrics, 
  Invent.~Math.~84 (1986)  177--194.


 This space seems to be the lowest dimensional example of a compact homogeneous manifold with no invariant Einstein metrics.  
Other interesting examples of compact homogeneous spaces with no invariant Einstein metrics have been presented for instance  by

  J-S. Park and Y. Sakane:
   Invariant Einstein metrics on certain homogeneous spaces,
  Tokyo J. Math.  20   (1) (1997) 51--61.