A suspended ellipsoidal particle inside a Poiseuille flow with Reynolds number up to 360 is studied numerically. The effects of tube diameter ( $D$ ), inertia of the particle and the flow, and the particle geometry (both prolate and oblate ellipsoids) are considered. When a prolate particle with $a/b=2$ is inside a wider tube (e.g. $D/A>1.9$ ), where $A=2a$ is the length of the major axis of the particle, the terminal stable state is tumbling. When the prolate particle is inside a narrower tube ( $1.0<D/A<1.9$ ), log-rolling or kayaking modes may appear. Which mode occurs depends on the competition between fluid and particle inertia. When the fluid inertia is dominant, the log-rolling mode appears, otherwise, the kayaking mode appears. Inclined and spiral modes may appear when $D/A<1$ and $D/A=1$ , respectively. For a prolate ellipsoid with $a/b=4$ , if $1<D/A<1.9$ , there is only the kayaking mode and the log-rolling mode is not observed. When an oblate particle is inside a wider tube (e.g. $D/A>3.5$ ), it may adopt the log-rolling mode. Inclined and intermediate modes are firstly identified in narrower tubes. The phase diagram of the modes is also provided. The modes in the phase diagrams were not found to be affected by the initial state of the particle based on limited observation.