Finite Group Databasebeta

← #7623378

Subgroups — #7623378

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60717C211yesmaximal, characteristic, complemented
2#60649C211yesmaximal, characteristic, complemented
3#33789C212yesmaximal, complemented
4#33788C212yesmaximal, complemented
5#33787C212yesmaximal, complemented
6#33786C212yesmaximal, complemented
7#33778C214yesmaximal, complemented
8#33777C214yesmaximal, complemented
9#33719C211yesmaximal, characteristic, complemented
10#33713C211yesmaximal, characteristic, complemented
11#33708C211yesmaximal, characteristic, complemented
12C2×#2242C211yesmaximal, characteristic, complemented
13#33590C212yesmaximal, complemented
14#33589C212yesmaximal, complemented
15#33527C212yesmaximal, complemented
16#33526C212yesmaximal, complemented
17#33420C211yesmaximal, characteristic, complemented
18C2×#578C2211yescharacteristic
19C2×#577—21no—
20C2×#576C2211yescharacteristic
21#3512—21no—
22#3500—21no—
23C2×#510C2212yescomplemented
24C2×#509C2212yes—
25C2×#508C2212yes—
26C2×#508C2212yescomplemented
27#3183—22no—
28#3177C2212yescomplemented
29#3176C2212yescomplemented
30#3174—22no—
31#3168C2214yescomplemented
32C2×#499C2211yescharacteristic, complemented
33#3156C2211yescharacteristic
34#3156C2212yes—
35#3155C2211yescharacteristic
36#3155C2212yes—
37#3154—22no—
38#3154—21no—
39#3153—21no—
40#3153—22no—
41#3147C2211yescharacteristic
42#3147C2212yes—
43#3147C2211yescharacteristic, complemented
44C2×#495C2211yescharacteristic
45C2×#495C2212yes—
46#3142—21no—
47#3061C2212yes—
48#3060—21no—
49#2952—21no—
50#2951C2212yescomplemented
51C4×#131C2211yescharacteristic
52#2251C2212yescomplemented
53#2250C2212yescomplemented
54#2249C2212yescomplemented
55#2249C2212yescomplemented
56#2249C2214yescomplemented
57#2248C2211yescharacteristic, complemented
58#2248C2211yescharacteristic, complemented
59#2247C2211yescharacteristic, complemented
60#2246C2211yescharacteristic, complemented
61#2245C2212yescomplemented
62#2244C2212yescomplemented
63#2242C2214yescomplemented
64#2242C2212yescomplemented
65#2242C2212yescomplemented
66#2242C2212yescomplemented
67#2242C2212yescomplemented
68#2211C2214yescomplemented
69#2211C2214yescomplemented
70#2210C2214yescomplemented
71#2210C2214yescomplemented
72#2209C2214yescomplemented
73#2208C2214yescomplemented
74#2207C2214yescomplemented
75#2206C2214yescomplemented
76#2189C2214yes—
77#2189C2214yescomplemented
78#2188C2214yescomplemented
79#2187C2214yes—
80#2186C2214yes—
81#2186C2214yescomplemented
82#2185C2214yescomplemented
83#2184C2214yes—
84#2164C2212yescomplemented
85#2164C2212yescomplemented
86#2162C2214yescomplemented
87#2151C2212yescomplemented
88#2139C2212yes—
89#2139C2212yescomplemented
90C2×#358C2212yescomplemented
91C2×#357C2212yes—
92#2113C2212yes—
93C4×M32C2211yescharacteristic, complemented
94C2×C4×C16C2211yescharacteristic, complemented
95C2×2−1+4—21no—
96C2×H(5,𝔽2)—21no—
97#578—44no—
98#577—48no—
99#576—44no—
100C2×#137—21no—
101C2×#137—22no—
102C2×#137—21no—
103C2×#137—22no—
104C2×AGL(1,ℤ/8)—21no—
105C2×AGL(1,ℤ/8)—22no—
106C2×AGL(1,ℤ/8)—21no—
107C2×AGL(1,ℤ/8)—22no—
108C2×#135—22no—
109C2×#135—21no—
110C2×#135—21no—
111C2×#135C2311yescharacteristic
112C2×#135C2311yescharacteristic
113C2×#135C2312yes—
114C2×#135C2312yes—
115C2×#135C2312yes—
116C22×Q16C2311yescharacteristic
117C22×Q16C2312yes—
118C22×SD16—22no—
119C22×SD16—21no—
120C22×D8C2311yescharacteristic
121C22×D8C2312yes—
122C2×#131D411yescharacteristic
123C2×#131C2311yescharacteristic
124#553—44no—
125#553—42no—
126#552—42no—
127#548—42no—
128#546—44no—
129#546—42no—
130#541—22no—
131#540—22no—
132#537—22no—
133#536—21no—
134#536—22no—
135#535—21no—
136#535—22no—
137#532—21no—
138#532—21no—
139#532—22no—
140#529—24no—
141C2×#125—21no—
142C2×#124—21no—
143C2×#124—22no—
144#520—22no—
145C4×#41D411yescharacteristic
146#510—44no—
147#509—44no—
148#508—44no—
149#508—44no—
150C2×Q32C2312yes—
151C2×Q32C2312yes—
152C2×Q32C2314yes—
153C2×SD32C2312yes—
154C2×SD32C2312yes—
155C2×SD32C2312yes—
156C2×SD32C2312yes—
157C2×SD32C2314yes—
158C2×SD32C2314yes—
159C2×D16C2312yes—
160C2×D16C2312yes—
161C2×D16C2314yes—
162C2×M32C2312yes—
163C22×C16C2312yes—
164#499—21no—
165#499—21no—
166#499C2312yes—
167#499C2312yes—
168#497—22no—
169#497—22no—
170#497—22no—
171#496—22no—
172#496—22no—
173#496—22no—
174#495—22no—
175#495—21no—
176#495—22no—
177#495—21no—
178#495—22no—
179#495—21no—
180#495—21no—
181#495C2312yes—
182#495C2311yescharacteristic
183#495C2312yes—
184#495C2312yes—
185#495C2312yes—
186#495C2311yescharacteristic
187#495C2312yes—
188#494C2312yes—
189#494C2311yescharacteristic
190#494C2311yescharacteristic
191#493C2312yes—
192#493C2311yescharacteristic
193#493C2311yescharacteristic
194#492—22no—
195#492—21no—
196#492—21no—
197#491—22no—
198#490—22no—
199#488—24no—
200#470—22no—
201#469—22no—
202#468—24no—
203#467C2314yes—
204#466C2314yes—
205#465—24no—
206#442—22no—
207#441—22no—
208#440—24no—
209C4×Q16C2314yes—
210C4×SD16—24no—
211C4×D8C2314yes—
212C2×#107C2312yes—
213C2×#106—21no—
214#417—22no—
215#417—22no—
216#405D412yes—
217#405D411yescharacteristic
218#405C2311yescharacteristic
219#405C2312yes—
220C4×M16D411yescharacteristic
221C4×M16D412yes—
222C2×C4×C8C2311yescharacteristic
223C2×C4×C8C2312yes—
224C4⋊C16C2314yes—
225C4⋊C16C2314yes—
226#358C2314yes—
227#358C2314yes—
228#358C2314yes—
229#358C2314yes—
230#357C2314yes—
231#357C2314yes—
232#357C2314yes—
233#357C2314yes—
234C22⋊C16C2314yes—
235C22⋊C16C2314yes—
236#346C2312yes—
237C4×C16C2312yes—
238C4×C16C2311yescharacteristic
239C4×C16C2311yescharacteristic
240C4×C16C2312yes—
2412−1+4—44no—
2422−1+4—44no—
243H(5,𝔽2)—44no—
244H(5,𝔽2)—44no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—21no—
248C2×#41—22no—
249C2×#41—21no—
250C2×#41—41no—
251C2×#41—41no—
252C2×#41—42no—
253C2×#41C2×D411yescharacteristic
254C22×Q8—22no—
255C22×Q8—21no—
256C22×Q8—41no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—42no—
260C22×D4—41no—
261#137—44no—
262#137—48no—
263#137—44no—
264#137—48no—
265AGL(1,ℤ/8)—44no—
266AGL(1,ℤ/8)—48no—
267AGL(1,ℤ/8)—44no—
268AGL(1,ℤ/8)—48no—
269#135—24no—
270#135—24no—
271#135—24no—
272#135—24no—
273#135—44no—
274#135—44no—
275#135—48no—
276#135—44no—
277#135—44no—
278C2×Q16—22no—
279C2×Q16—21no—
280C2×Q16—22no—
281C2×Q16—21no—
282C2×Q16—21no—
283C2×Q16—24no—
284C2×Q16—44no—
285C2×Q16C2412yes—
286C2×Q16C2411yescharacteristic
287C2×Q16C2411yescharacteristic
288C2×Q16C2412yes—
289C2×SD16—21no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—21no—
295C2×SD16—22no—
296C2×SD16—22no—
297C2×SD16—21no—
298C2×SD16—22no—
299C2×SD16—21no—
300C2×SD16—22no—
301C2×SD16—48no—
302C2×SD16—44no—
303C2×D8—22no—
304C2×D8—21no—
305C2×D8—22no—
306C2×D8—21no—
307C2×D8—21no—
308C2×D8—24no—
309C2×D8—44no—
310C2×D8C2412yes—
311C2×D8C2411yescharacteristic
312C2×D8C2411yescharacteristic
313C2×D8C2412yes—
314#131—24no—
315#131—24no—
316C2×M16C2×D411yescharacteristic
317C2×M16C2×D412yes—
318C2×M16C2×D411yescharacteristic
319C2×M16C2×D412yes—
320C22×C8C2×D412yes—
321C22×C8C2×D411yescharacteristic
322C22×C8C2412yes—
323C22×C8C2411yescharacteristic
324C4⋊Q8—22no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327#127—22no—
328#127—21no—
329#127—21no—
330#126—44no—
331#126—44no—
332#125—22no—
333#125—22no—
334#125—44no—
335#125—42no—
336#124—22no—
337#124—21no—
338#124—22no—
339#124—22no—
340#124—22no—
341#124—22no—
342#124—21no—
343#124—44no—
344#124—42no—
345#123—44no—
346#123—44no—
347#123—42no—
348#123—44no—
349C22⋊Q8—24no—
350C22⋊Q8—42no—
351C22⋊Q8—42no—
352C22⋊Q8—44no—
353#121—24no—
354#121—42no—
355#121—44no—
356#121—42no—
357#121—44no—
358#121—42no—
359#121—42no—
360#121—44no—
361#121—44no—
362#120—42no—
363#120—44no—
364C4×Q8—24no—
365C4×Q8D812yes—
366C4×D4—24no—
367C4×D4—44no—
368C4×D4—44no—
369C4×D4—44no—
370C4×D4—42no—
371C4×D4—42no—
372C4×D4D814yes—
373C4×D4D812yes—
374#117—42no—
375#117—44no—
376#117C2×D411yescharacteristic
377#117C2×D412yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—42no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—44no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385Q32—42no—
386Q32—42no—
387Q32—44no—
388SD32—42no—
389SD32—42no—
390SD32—42no—
391SD32—42no—
392SD32—44no—
393SD32—44no—
394D16—42no—
395D16—42no—
396D16—44no—
397M32—24no—
398C2×C16—24no—
399C2×C16C2412yes—
400C2×C16C2412yes—
401C2×C16C2414yes—
402#107—22no—
403#107C2414yes—
404#106—22no—
405#106—22no—
406#103—24no—
407#103—24no—
408#102—24no—
409#102—24no—
410#97C2×D412yes—
411#97C2×D412yes—
412#97C2×D412yes—
413C4×C8C2×D412yes—
414C4×C8C2×D411yescharacteristic
415C4×C8C2×D412yes—
416C4×C8C2×D411yescharacteristic
417C4×C8C2411yescharacteristic
418C4×C8C2411yescharacteristic
419C4×C8C2412yes—
420C24—42no—
421C24—41no—
422#41—22no—
423#41—22no—
424#41—44no—
425#41—44no—
426#41—44no—
427#41—44no—
428#41—42no—
429#41—42no—
430#41—44no—
431#41—42no—
432#41—42no—
433#41—44no—
434#41—88no—
435#41—84no—
436#41—84no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—84no—
448C2×Q8C2×D811yescharacteristic
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—41no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—88no—
469C2×D4—84no—
470C2×D4C2×D812yes—
471C2×D4C2×D811yescharacteristic
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—41no—
480C22×C4—44no—
481C22×C4C2×D812yes—
482C22×C4C2×D814yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485Q16—22no—
486Q16—22no—
487Q16—24no—
488Q16—22no—
489Q16—22no—
490Q16—44no—
491Q16—44no—
492Q16—42no—
493Q16—42no—
494Q16—42no—
495SD16—44no—
496SD16—44no—
497SD16—44no—
498SD16—44no—
499SD16—44no—
500SD16—42no—
501SD16—42no—
502SD16—42no—
503SD16—44no—
504SD16—44no—
505SD16—44no—
506SD16—42no—
507D8—22no—
508D8—22no—
509D8—24no—
510D8—22no—
511D8—22no—
512D8—44no—
513D8—44no—
514D8—42no—
515D8—42no—
516D8—42no—
517M16—22no—
518M16—22no—
519M16—24no—
520M16—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8AGL(1,ℤ/8)14yes—
524C2×C8C22×D412yes—
525C2×C8C22×D412yes—
526C2×C8C22×D411yescharacteristic
527C2×C8C22×D411yescharacteristic
528C2×C8C22×D411yescharacteristic
529C2×C8H(5,𝔽2)14yes—
530C2×C8C2511yescharacteristic
531C4⋊C4—24no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4C2×D812yes—
541C4⋊C4C2×D814yes—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4C2×D812yes—
554C22⋊C4C2×D814yes—
555C42—44no—
556C42C2×D814yes—
557C42C2×D812yes—
558C42C22×D411yescharacteristic
559C42C22×D412yes—
560C42C22×D411yescharacteristic
561C16—22no—
562C16—22no—
563C16—24no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—84no—
569C23—82no—
570C23—162no—
571C23—164no—
572C23C22×D812yes—
573C23C22×D811yescharacteristic
574Q8—22no—
575Q8—41no—
576Q8—42no—
577Q8—41no—
578Q8—42no—
579Q8—41no—
580Q8—42no—
581Q8—41no—
582Q8—42no—
583Q8—84no—
584Q8—82no—
585D4—24no—
586D4—22no—
587D4—42no—
588D4—41no—
589D4—41no—
590D4—42no—
591D4—41no—
592D4—42no—
593D4—41no—
594D4—42no—
595D4—84no—
596D4—84no—
597D4—84no—
598D4—82no—
599D4—84no—
600D4—82no—
601D4—82no—
602D4—164no—
603D4—162no—
604D4—162no—
605D4—164no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—84no—
622C2×C4—164no—
623C2×C4—162no—
624C2×C4#53512yes—
625C2×C4C22×D814yes—
626C2×C4C22×D812yes—
627C2×C4C22×D812yes—
628C2×C4C22×D812yes—
629C2×C4C22×D811yescharacteristic
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4C23×D411yescharacteristic
632C8—22no—
633C8—22no—
634C8#53512yes—
635C8C2×AGL(1,ℤ/8)12yes—
636C8C2×AGL(1,ℤ/8)12yes—
637C8C2×H(5,𝔽2)12yes—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—81no—
642C22—82no—
643C22—81no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22—82no—
650C22—322no—
651C22—324no—
652C22#320512yes—
653C22C23×D811yescharacteristic
654C4—22no—
655C4—22no—
656C4—24no—
657C4—22no—
658C4—81no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4#320512yes—
664C4C2×#53511yescharacteristic
665C4C22×AGL(1,ℤ/8)11yescharacteristic
666C2—22no—
667C2—42no—
668C2—161no—
669C2—162no—
670C2—162no—
671C2—161no—
672C2C2×#320511yescharacteristic
673C2#6263512yes—
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #60717C2semidirect
2. #60649C2semidirect
3. #33789C2semidirect
4. #33788C2semidirect
5. #33787C2semidirect
6. #33786C2semidirect
7. #33778C2semidirect
8. #33777C2semidirect
9. #33719C2semidirect
10. #33713C2semidirect
11. #33708C2semidirect
12. C2×#2242C2semidirect
13. #33590C2semidirect
14. #33589C2semidirect
15. #33527C2semidirect
16. #33526C2semidirect
17. #33420C2semidirect
23. C2×#510C22semidirect
26. C2×#508C22semidirect
28. #3177C22semidirect
29. #3176C22semidirect
31. #3168C22semidirect
32. C2×#499C22semidirect
43. #3147C22semidirect
50. #2951C22semidirect
52. #2251C22semidirect
53. #2250C22semidirect
54. #2249C22semidirect
55. #2249C22semidirect
56. #2249C22semidirect
57. #2248C22semidirect
58. #2248C22semidirect
59. #2247C22semidirect
60. #2246C22semidirect
61. #2245C22semidirect
62. #2244C22semidirect
63. #2242C22semidirect
64. #2242C22semidirect
65. #2242C22semidirect
66. #2242C22semidirect
67. #2242C22semidirect
68. #2211C22semidirect
69. #2211C22semidirect
70. #2210C22semidirect
71. #2210C22semidirect
72. #2209C22semidirect
73. #2208C22semidirect
74. #2207C22semidirect
75. #2206C22semidirect
77. #2189C22semidirect
78. #2188C22semidirect
81. #2186C22semidirect
82. #2185C22semidirect
84. #2164C22semidirect
85. #2164C22semidirect
86. #2162C22semidirect
87. #2151C22semidirect
89. #2139C22semidirect
90. C2×#358C22semidirect
93. C4×M32C22semidirect
94. C2×C4×C16C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#607172
#606492
#337892
#337882
#337872
#337862
#337782
#337772
#337192
#337132
#337082
C2×#22422
#335902
#335892
#335272
#335262
#334202