  We study the phases and phase transition lines of the finite temperature G(2)
Higgs model. Our work is based on an efficient local hybrid Monte-Carlo
algorithm which allows for accurate measurements of expectation values,
histograms and susceptibilities. On smaller lattices we calculate the phase
diagram in terms of the inverse gauge coupling $\beta$ and the hopping
parameter $\kappa$. For $\kappa\to 0$ the model reduces to G(2) gluodynamics
and for $\kappa\to\infty$ to SU(3) gluodynamics. In both limits the system
shows a first order confinement-deconfinement transition. We show that the
first order transitions at asymptotic values of the hopping parameter are
almost joined by a line of first order transitions. A careful analysis reveals
that there exists a small gap in the line where the first order transitions
turn into continuous transitions or a cross-over region. For $\beta\to\infty$
the gauge degrees of freedom are frozen and one finds a nonlinear O(7) sigma
model which exhibits a second order transition from a massive O(7)-symmetric to
a massless O(6)-symmetric phase. The corresponding second order line for large
$\beta$ remains second order for intermediate $\beta$ until it comes close to
the gap between the two first order lines. Besides this second order line and
the first order confinement-deconfinement transitions we find a line of
monopole-driven bulk transitions which do not interfer with the
confinement-deconfinment transitions.
