I spent a helluva lot of time on this!
Burning Star: 11
I Bradley Prime
II Lyrea
III New Genesse
IV Gizadra
V Ormon
VI Hemlock
VII Prima Two
VIII Qazat
IX Alpha Majori
X Rigel
XI Orzock
Earth Coalition: 28 Plus Earth
I Carnak
II Taurus
III Huju
IV Atlas
V Betazed
VI Mentara
VII Alpha Major
VIII Alpha Minor
IX Derezat
X Sacnez
XI Ramazat
XII Aries
XIII Malfadoris
XIV Undara
XV Delta Upsilon Theta
XVI Vesuiv
XVII Cardassia
XVIII Wirtuy
XIX Lambda Lambda Lambda
XX Zoltan
XXI Eden
XXII Zamara
XXIII Zeta Iota Theta
XXIV Vorva
XXV Beta Majoris
XXVI Pleadies
XXVII Burundi
Sol
Draconis Empire: Fourteen
I Jalltal
II Gamolan
III Fyzan
IV Chi Lambda Tau
V Cretanus
VI Srutu
VII Gemini
VIII Xtapa
IX Bzintus
X Ragnarock
XI Nidhogge
XII Grendel
XIII Fenris
XIV Orion
MacroHard Corporation: Thirteen
I Tyran
II Mu Omicron Nu Epsilon Upsilon
III Udran
IV Narcissus
V New Seattle
VI Voltar
VII Jukala
VIII Karndor
IX Alpha Omega
X Lanknok
XI Qwerty
XII Delta Gamma
XIII Cintak
Darkfire Empire: Twenty
I Omega Prime
II Gamma Hyrdonz
III Anogdt
IV Quartz
V Opiuchi Junction
VI Darkfire
VII Dark Junction
VIII Untsvy
IX Pfi Chi Si Rho
X Havrgry
XI Pi Gamma Upsilon
XII Jobu
XIII Pi Rho
XIV Hyonfdelk
XV Troback’s System
XVI Smaltz
XVII Thirsk
XVIII Flander
XIX Kurlerbach
XX Sabark’s World
Independent Systems: 9
I Stalz
II New Russia
III Telume
IV Sparta
V New Berlin
VI Nova Terra
VII Zeta Upsillon Chi
VIII Polaski’s World
IX Benton
NOT PICTURED
Mechanical Worlds Of Sector Zero Empire: Two Systems
(If I forgot a system I specifically mentioned in the RP, just let me know and I'll replace one of the gibberish ones. Also, assume that there are more than one planets in most systems.)