Finite Group Databasebeta

← #7624554

Subgroups — #7624554

Order 512.

1389 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62417C211yesmaximal, characteristic, complemented
2#61431C211yesmaximal, characteristic, complemented
3#33963C212yesmaximal, complemented
4#33928C212yesmaximal, complemented
5#33900C212yesmaximal, complemented
6#33889C214yesmaximal, complemented
7#33840C212yesmaximal, complemented
8#33839C214yesmaximal, complemented
9#33820C211yesmaximal, characteristic, complemented
10#33817C212yesmaximal, complemented
11#33778C212yesmaximal, complemented
12#33771C211yesmaximal, characteristic, complemented
13#33667C211yesmaximal, characteristic, complemented
14#33660C212yesmaximal, complemented
15#33582C211yesmaximal, characteristic, complemented
16#33580C212yesmaximal, complemented
17#33505C211yesmaximal, characteristic, complemented
18#3508—21no—
19#3481—21no—
20C2×#509C2212yes—
21C2×#509C2214yescomplemented
22#3396—21no—
23#3396C2212yescomplemented
24#3396C2211yescharacteristic, complemented
25#3380C2211yescharacteristic, complemented
26#3361—21no—
27#3339C2212yes—
28#3338C2211yescharacteristic, complemented
29#3297—22no—
30#3295—21no—
31#3287—21no—
32#3287C2212yes—
33#3280—22no—
34#3280C2212yes—
35#3280C2214yescomplemented
36D4×D8C2211yescharacteristic, complemented
37#3208—21no—
38#3207—22no—
39#3170C2211yescharacteristic, complemented
40#3169C2211yescharacteristic
41#3169C2212yescomplemented
42#3168C2212yescomplemented
43#3147C2212yescomplemented
44C2×#493C2211yescharacteristic, complemented
45#3106—21no—
46#3105—22no—
47C8×#41C2211yescharacteristic
48C4×#135C2211yescharacteristic
49C2×C4×D8C2212yescomplemented
50#2249C2214yescomplemented
51#2246C2212yes—
52#2242C2214yescomplemented
53#2240C2212yescomplemented
54#2237C2212yescomplemented
55#2232C2214yescomplemented
56#2228C2214yescomplemented
57#2227C2212yescomplemented
58#2225C2211yescharacteristic, complemented
59#2224C2212yescomplemented
60#2224C2211yescharacteristic, complemented
61#2220C2214yescomplemented
62#2220C2214yescomplemented
63#2218C2214yescomplemented
64#2210C2214yescomplemented
65#2210C2214yescomplemented
66#2210C2214yescomplemented
67#2209C2211yescharacteristic, complemented
68#2208C2212yescomplemented
69#2208C2211yescharacteristic, complemented
70#2207C2212yescomplemented
71#2207C2212yescomplemented
72#2207C2212yescomplemented
73#2206C2212yes—
74#2206C2212yes—
75#2206C2212yescomplemented
76#2206C2212yescomplemented
77#2189C2212yes—
78#2189C2214yescomplemented
79#2186C2214yescomplemented
80#2184C2214yescomplemented
81#2184C2214yescomplemented
82Q8×C16C2212yescomplemented
83C4×Q32C2212yescomplemented
84C4×SD32C2212yescomplemented
85C4×D16C2214yescomplemented
86C4×D16C2214yescomplemented
87#2169C2214yescomplemented
88#2169C2212yescomplemented
89#2162C2212yescomplemented
90#2162C2211yescharacteristic, complemented
91#2150C2212yescomplemented
92#2150C2211yescharacteristic, complemented
93#2140C2212yescomplemented
94#2139C2214yescomplemented
95C2×#135—22no—
96C2×#135C2311yescharacteristic
97C2×#135C2314yes—
98C22×D8—21no—
99C22×D8C2312yes—
100C22×D8C2312yes—
101#559—21no—
102#559—22no—
103#559—41no—
104#554—21no—
105#554—21no—
106#553—22no—
107#553—41no—
108#551—42no—
109#550—22no—
110#550—41no—
111#546—22no—
112#546—24no—
113#546—41no—
114#546—42no—
115D42—21no—
116#540—42no—
117#535—41no—
118#533—21no—
119#532—21no—
120#532—22no—
121#532—22no—
122C2×#127—21no—
123#529—22no—
124#529—41no—
125#525—42no—
126#518—41no—
127C4×#41—21no—
128C4×#41D411yescharacteristic
129C2×C4×D4—22no—
130C2×C4×D4—41no—
131#509—44no—
132#509—48no—
133C2×Q32C2312yes—
134C2×SD32C2312yes—
135C2×SD32C2314yes—
136C2×SD32C2314yes—
137C2×D16C2314yes—
138C2×D16C2314yes—
139C2×D16C2314yes—
140C2×M32C2312yes—
141C2×M32C2314yes—
142#500C2311yescharacteristic
143#500C2311yescharacteristic
144#499C2312yes—
145#495—22no—
146#495C2314yes—
147#495C2312yes—
148#494C2311yescharacteristic
149#493—21no—
150#493—21no—
151#493C2312yes—
152#493C2311yescharacteristic
153#493C2312yes—
154#492—22no—
155#488—22no—
156#487—21no—
157#486—21no—
158#486—22no—
159#484—22no—
160#482—22no—
161#481—22no—
162#478—22no—
163#478—22no—
164#477—21no—
165#476—21no—
166#467—21no—
167#467C2312yes—
168#467C2312yes—
169#466—21no—
170#466—22no—
171#466C2312yes—
172#466C2312yes—
173#466C2314yes—
174#465—22no—
175#463—22no—
176#463—22no—
177#463—24no—
178#463—24no—
179#460—21no—
180#460—22no—
181#460—22no—
182#459—21no—
183#450—22no—
184#450—24no—
185#449—22no—
186#449—24no—
187#449—24no—
188#447—22no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—21no—
192C4×Q16C2311yescharacteristic
193C4×Q16C2312yes—
194C4×Q16C2311yescharacteristic
195C4×SD16—21no—
196C4×SD16—22no—
197C4×SD16—21no—
198C4×D8—21no—
199C4×D8—22no—
200C4×D8—22no—
201C4×D8—22no—
202C4×D8C2312yes—
203C4×D8C2312yes—
204C4×D8C2311yescharacteristic
205C4×D8C2314yes—
206C4×D8C2312yes—
207C4×D8C2311yescharacteristic
208C8×D4D412yes—
209C8×D4D411yescharacteristic
210C8×D4C2311yescharacteristic
211C8×D4C2312yes—
212#431D412yes—
213#431D411yescharacteristic
214#427C2311yescharacteristic
215C2×#107C2312yes—
216#416—21no—
217C2×#102—22no—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#367C2312yes—
221#366C2312yes—
222#366C2314yes—
223C4⋊C16C2312yes—
224C4⋊C16C2311yescharacteristic
225C4⋊C16C2312yes—
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2312yes—
229#358C2314yes—
230#358C2312yes—
231#358C2312yes—
232#357C2314yes—
233#357C2314yes—
234#357C2314yes—
235#357C2312yes—
236#357C2314yes—
237#357C2314yes—
238C22⋊C16C2312yes—
239C22⋊C16C2314yes—
240C4×C16C2314yes—
241C4×C16C2312yes—
242C2×#41—21no—
243C2×#41—24no—
244C2×#41—41no—
245C2×#41—42no—
246C2×#41D811yescharacteristic
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—41no—
250C22×D4—41no—
251C22×D4—41no—
252C23×C4—42no—
253#135—22no—
254#135—41no—
255#135—44no—
256#135—44no—
257#135—84no—
258C2×Q16—21no—
259C2×Q16C2412yes—
260C2×Q16C2411yescharacteristic
261C2×SD16—21no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—22no—
265C2×D8—24no—
266C2×D8—22no—
267C2×D8—22no—
268C2×D8—22no—
269C2×D8—22no—
270C2×D8—21no—
271C2×D8—42no—
272C2×D8—42no—
273C2×D8—42no—
274C2×D8—82no—
275C2×D8C2414yes—
276C2×D8C2412yes—
277C2×D8C2411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285#127—22no—
286#127—22no—
287#127—21no—
288#127—42no—
289#127—41no—
290#127—41no—
291#126—42no—
292#126—42no—
293#125—22no—
294#125—41no—
295#125—41no—
296#124—22no—
297#124—24no—
298#124—41no—
299#124—42no—
300#124—42no—
301#124—41no—
302#123—42no—
303#123—42no—
304#123—42no—
305#123—44no—
306#123—44no—
307#123—42no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—41no—
311C22⋊Q8—41no—
312C22⋊Q8—42no—
313C22⋊Q8—42no—
314#121—22no—
315#121—24no—
316#121—22no—
317#121—42no—
318#121—42no—
319#121—42no—
320#121—42no—
321#121—41no—
322#121—41no—
323#121—42no—
324#121—44no—
325#121—44no—
326#120—42no—
327#120—44no—
328#120—42no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—22no—
332C4×Q8—41no—
333C4×Q8D811yescharacteristic
334C4×Q8C2×D411yescharacteristic
335C4×D4—21no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—21no—
340C4×D4—24no—
341C4×D4—41no—
342C4×D4—42no—
343C4×D4—41no—
344C4×D4—41no—
345C4×D4—41no—
346C4×D4—42no—
347C4×D4—42no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—41no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4D811yescharacteristic
356C4×D4D812yes—
357C4×D4C2×D411yescharacteristic
358C4×D4C2×D412yes—
359#117—21no—
360#117—41no—
361#117—41no—
362#117—42no—
363#117D811yescharacteristic
364#117D812yes—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—41no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—44no—
370C2×C42—41no—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373Q32—42no—
374SD32—24no—
375SD32—44no—
376SD32—44no—
377D16—28no—
378D16—44no—
379D16—44no—
380M32—42no—
381M32—44no—
382C2×C16C2412yes—
383C2×C16C2414yes—
384C2×C16C2412yes—
385#107—22no—
386#107—21no—
387#107—21no—
388#107C2411yescharacteristic
389#107C2412yes—
390#107C2411yescharacteristic
391#107C2412yes—
392#106—21no—
393#106—22no—
394C4⋊C8C2×D411yescharacteristic
395C4⋊C8C2×D412yes—
396C4⋊C8C2×D411yescharacteristic
397C4⋊C8C2×D412yes—
398#103—21no—
399#103—21no—
400#103—22no—
401#103—22no—
402#103—22no—
403#102—21no—
404#102—22no—
405#102—22no—
406#102—22no—
407#102—21no—
408#102—24no—
409#102—22no—
410#102—24no—
411#102—22no—
412C22⋊C8C2×D412yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C4×C8C2×D412yes—
416C4×C8C2×D411yescharacteristic
417C4×C8C2×D412yes—
418C4×C8C2×D411yescharacteristic
419C4×C8C2412yes—
420C4×C8C2411yescharacteristic
421C4×C8C2411yescharacteristic
422C24—42no—
423C24—42no—
424#41—42no—
425#41—42no—
426#41—81no—
427#41—84no—
428#41—84no—
429#41—162no—
430#41—164no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8C2×D811yescharacteristic
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—44no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—41no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—84no—
457C2×D4—84no—
458C2×D4—162no—
459C2×D4C2×D811yescharacteristic
460C2×D4C2×D812yes—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—164no—
471C22×C4C2×D811yescharacteristic
472C22×C4C2×D812yes—
473C22×C4C2×D811yescharacteristic
474C22×C4C2×D812yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477Q16—22no—
478Q16—21no—
479Q16—21no—
480Q16—22no—
481Q16—42no—
482SD16—42no—
483SD16—82no—
484SD16—84no—
485SD16—84no—
486D8—24no—
487D8—22no—
488D8—21no—
489D8—22no—
490D8—24no—
491D8—44no—
492D8—44no—
493D8—81no—
494D8—82no—
495D8—82no—
496D8H(5,𝔽2)12yes—
497C2×C8—22no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—22no—
501C2×C8—41no—
502C2×C8—42no—
503C2×C8C22×D411yescharacteristic
504C2×C8C22×D412yes—
505C2×C8C22×D411yescharacteristic
506C2×C8C22×D412yes—
507C2×C8C22×D411yescharacteristic
508C2×C8C2511yescharacteristic
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4C2×D811yescharacteristic
526C4⋊C4C2×D811yescharacteristic
527C4⋊C4C2×D812yes—
528C4⋊C4C2×D812yes—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4C2×D812yes—
546C22⋊C4C2×D812yes—
547C22⋊C4C2×D812yes—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—42no—
552C42C2×D811yescharacteristic
553C42C2×D811yescharacteristic
554C42C2×D812yes—
555C42C2×D812yes—
556C42C22×D411yescharacteristic
557C42C22×D412yes—
558C42C22×D411yescharacteristic
559C16—24no—
560C16—22no—
561C16—22no—
562C23—41no—
563C23—42no—
564C23—44no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—162no—
569C23—162no—
570C23—162no—
571C23C22×D811yescharacteristic
572C23C22×D812yes—
573Q8—22no—
574Q8—42no—
575Q8—41no—
576Q8—41no—
577Q8—42no—
578Q8—82no—
579Q8—161no—
580D4—22no—
581D4—22no—
582D4—42no—
583D4—41no—
584D4—44no—
585D4—42no—
586D4—42no—
587D4—44no—
588D4—84no—
589D4—84no—
590D4—82no—
591D4—84no—
592D4—82no—
593D4—84no—
594D4—84no—
595D4—162no—
596D4—162no—
597D4—161no—
598D4—164no—
599D4—162no—
600D4—164no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4—162no—
620C2×C4—162no—
621C2×C4—164no—
622C2×C4—164no—
623C2×C4—162no—
624C2×C4C22×D811yescharacteristic
625C2×C4C22×D811yescharacteristic
626C2×C4C22×D812yes—
627C2×C4C22×D812yes—
628C2×C4C22×D812yes—
629C2×C4C22×D811yescharacteristic
630C2×C4C22×D811yescharacteristic
631C2×C4C22×D811yescharacteristic
632C2×C4C23×D411yescharacteristic
633C8—21no—
634C8—21no—
635C8—22no—
636C8—22no—
637C8—21no—
638C8C2×H(5,𝔽2)11yescharacteristic
639C8#58511yescharacteristic
640C22—21no—
641C22—22no—
642C22—22no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22—168no—
656C22—324no—
657C22C23×D811yescharacteristic
658C4—21no—
659C4—22no—
660C4—22no—
661C4—21no—
662C4—21no—
663C4—22no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4#341412yes—
671C4#341512yes—
672C4#348811yescharacteristic
673C4#358511yescharacteristic
674C2—42no—
675C2—41no—
676C2—82no—
677C2—162no—
678C2—164no—
679C2#6194711yescharacteristic
680C2C2×#341411yescharacteristic
681C2C2×#341511yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62417C2semidirect
2. #61431C2semidirect
3. #33963C2semidirect
4. #33928C2semidirect
5. #33900C2semidirect
6. #33889C2semidirect
7. #33840C2semidirect
8. #33839C2semidirect
9. #33820C2semidirect
10. #33817C2semidirect
11. #33778C2semidirect
12. #33771C2semidirect
13. #33667C2semidirect
14. #33660C2semidirect
15. #33582C2semidirect
16. #33580C2semidirect
17. #33505C2semidirect
21. C2×#509C22semidirect
23. #3396C22semidirect
24. #3396C22semidirect
25. #3380C22semidirect
28. #3338C22semidirect
35. #3280C22semidirect
36. D4×D8C22semidirect
39. #3170C22semidirect
41. #3169C22semidirect
42. #3168C22semidirect
43. #3147C22semidirect
44. C2×#493C22semidirect
49. C2×C4×D8C22semidirect
50. #2249C22semidirect
52. #2242C22semidirect
53. #2240C22semidirect
54. #2237C22semidirect
55. #2232C22semidirect
56. #2228C22semidirect
57. #2227C22semidirect
58. #2225C22semidirect
59. #2224C22semidirect
60. #2224C22semidirect
61. #2220C22semidirect
62. #2220C22semidirect
63. #2218C22semidirect
64. #2210C22semidirect
65. #2210C22semidirect
66. #2210C22semidirect
67. #2209C22semidirect
68. #2208C22semidirect
69. #2208C22semidirect
70. #2207C22semidirect
71. #2207C22semidirect
72. #2207C22semidirect
75. #2206C22semidirect
76. #2206C22semidirect
78. #2189C22semidirect
79. #2186C22semidirect
80. #2184C22semidirect
81. #2184C22semidirect
82. Q8×C16C22semidirect
83. C4×Q32C22semidirect
84. C4×SD32C22semidirect
85. C4×D16C22semidirect
86. C4×D16C22semidirect
87. #2169C22semidirect
88. #2169C22semidirect
89. #2162C22semidirect
90. #2162C22semidirect
91. #2150C22semidirect
92. #2150C22semidirect
93. #2140C22semidirect
94. #2139C22semidirect

Maximal subgroups

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

subgroupindex
#624172
#614312
#339632
#339282
#339002
#338892
#338402
#338392
#338202
#338172
#337782
#337712
#336672
#336602
#335822
#335802
#335052