  Quantum wells formed by layers of HgTe between Hg$_{1-x}$Cd$_x$Te barriers
lead to two-dimensional (2D) topological insulators, as predicted by the BHZ
model. Here, we theoretically and experimentally investigate the
characteristics of triple HgTe quantum wells. We describe such heterostructure
with a three dimensional $8\times 8$ Kane model, and use its eigenstates to
derive an effective 2D Hamiltonian for the system. From these we obtain a phase
diagram as a function of the well and barrier widths and we identify the
different topological phases composed by zero, one, two, and three sets of edge
states hybridized along the quantum wells. The phase transitions are
characterized by a change of the spin Chern numbers and their corresponding
band inversions. Complementary, transport measurements are experimentally
investigated on a sample close to the transition line between the phases with
one and two sets of edges states. Accordingly, for this sample we predict a
gapless spectrum with low energy bulk conduction bands given by one parabolic
and one Dirac band, and with edge states immersed in the bulk valance bands.
Consequently, we show that under these conditions, local and non-local
transport measurements are inconclusive to characterize a sole edge state
conductivity due to bulk conductivity. On the other hand, Shubnikov-de Haas
(SdH) oscillations show an excellent agreement with our theory. Particularly,
we show that the measured SdH oscillation frequencies agrees with our model and
show clear signatures of the coexistence of a parabolic and Dirac bands.
