Here is the full list of all 38 House of the Dying Sun achievements.
Assassinate Traitor Admiral Bergautur.
Assassinate Lord Commodore Naybard.
Assassinate Lord Commodore Ganuun.
Assassinate Lord Commodore Seigfast.
Assassinate Commodore Freyv.
Survive for at least 10 battles in Challenge Mode.
Survive for at least 5 battles in Challenge Mode.
Complete at least 1 battle in Challenge Mode
Assassinate Commodore Elduette.
To avenge the Emperor.
Earn a second wingman.
Earn a frigate for your fleet.
Earn a new weapon.
Add a destroyer to your fleet.
Complete a bonus objective.
Destroy a vessel while inside its shields.
Complete a mission on Harbinger.
Complete a mission on Warlord or higher.
Earn a wingman.
Purchase an upgrade.
Complete a mission on Invader or higher.
Destroy an escape pod.
Complete host uplink training.
Purchase all upgrades.
Run over a traitor lord's escape pod.
So I could rest.
To free the galaxy.
Because I love to kill.
Because this isn't real.
Because I am not human.
Assassinate Commodore Ausgauth.
Assassinate Traitor Admiral Anrune.
Assassinate Traitor Admiral Alfmundur.
Assassinate Prince Aum.
Assassinate Princess Admiral Du Moch.
Assassinate Kingsclone A1.
Assassinate Kingsclone B1.
Assassinate The Traitor King.