The present paper numerically investigates the finite deformation (i.e. non-breakup) and breakup of a double-core compound droplet in an external flow confined in an axisymmetric channel. The compound droplet is axisymmetric around the vertical center axis of the channel, and its two inner droplets are located symmetrically around the horizontal center plane of the channel. The problem is solved by an axisymmetric front-tracking method that models the interface by connected elements. Many parameters (such as the Capillary number Ca, the Reynolds number Re, the interfacial tension ratio sigma(21) of the inner to outer interfaces, the channel size ratio C-21, the ratio R-21 of the inner to outer droplet radii, the inner droplet location and the outer droplet size R-1) are varied to reveal their influences on the dynamic behavior of the compound droplet. Numerical results show that the compound droplet exhibits two distinct modes - finite deformation or breakup - depending on the flow condition. In the finite deformation mode, the inner droplet first moves away from and then moves back to the center. It stays there for the rest of time, resulting in the non-breakup of the outer droplet. However, in the breakup mode, after moving back to the center, the inner droplet again moves away while the outer droplet is continuously stretched and eventually performs necking and breakup at its ends. The transition between these two modes occurs when Ca varies in the range of 0.004-0.1, C-21 varies in the range of 0.5-1.4 or the ratio of the outer droplet radius to the channel size varies in the range of 03-0.9. Changing Re (from 0.01 to 1.0), sigma(21) (from 0.25 to 4.0), R-21 (from 0.3 to 0.45) or the inner droplet location does not make any transition. The effect of the kinematic viscosity ratios on the transition is also studied. In addition, a phase diagram in terms of Ca and the droplet size, on which the breakup and non-breakup modes are recognized, is proposed. (C) 2019 Elsevier Ltd. All rights reserved.