Finite Group Databasebeta

← #7624508

Subgroups — #7624508

Order 512.

1371 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62417C211yesmaximal, characteristic, complemented
2#61427C211yesmaximal, characteristic, complemented
3#33961C212yesmaximal, complemented
4#33928C212yesmaximal, complemented
5#33872C212yesmaximal, complemented
6#33863C214yesmaximal, complemented
7#33850C212yesmaximal, complemented
8#33846C214yesmaximal, complemented
9#33815C211yesmaximal, characteristic, complemented
10#33813C212yesmaximal, complemented
11#33778C212yesmaximal, complemented
12#33774C211yesmaximal, characteristic, complemented
13#33666C212yesmaximal, complemented
14#33661C211yesmaximal, characteristic, complemented
15#33579C212yesmaximal, complemented
16C2×#2206C211yesmaximal, characteristic, complemented
17#33502C211yesmaximal, characteristic, complemented
18#3508—21no—
19#3475—21no—
20C2×#508C2214yescomplemented
21C22×D16C2212yes—
22#3396—21no—
23#3396C2212yescomplemented
24#3393C2211yescharacteristic, complemented
25#3380C2211yescharacteristic, complemented
26#3360—21no—
27#3341C2211yescharacteristic, complemented
28#3336C2212yes—
29#3303—21no—
30#3296—22no—
31#3287—21no—
32#3287C2212yescomplemented
33#3283C2211yescharacteristic, complemented
34#3280—22no—
35#3280C2212yes—
36#3280C2214yes—
37#3208—21no—
38#3207—22no—
39#3170C2211yescharacteristic
40#3170C2211yescharacteristic
41#3169C2212yescomplemented
42#3168C2212yescomplemented
43#3147C2212yescomplemented
44C2×#493C2211yescharacteristic
45#3102—22no—
46#3101—21no—
47C8×#41C2211yescharacteristic
48C4×#135C2212yescomplemented
49C2×C4×D8C2211yescharacteristic
50#2249C2214yescomplemented
51#2248C2212yescomplemented
52#2242C2214yescomplemented
53#2240C2212yescomplemented
54#2235C2212yes—
55#2234C2214yescomplemented
56#2228C2214yescomplemented
57#2227C2212yescomplemented
58#2227C2212yescomplemented
59#2226C2211yescharacteristic, complemented
60#2226C2211yescharacteristic, complemented
61#2217C2214yescomplemented
62#2216C2214yes—
63#2215C2214yescomplemented
64#2211C2214yescomplemented
65#2210C2214yescomplemented
66#2210C2214yescomplemented
67#2209C2212yescomplemented
68#2209C2212yescomplemented
69#2208C2211yescharacteristic, complemented
70#2208C2211yescharacteristic, complemented
71#2207C2212yescomplemented
72#2207C2212yescomplemented
73#2206C2212yes—
74#2206C2212yes—
75#2206C2212yes—
76#2206C2212yes—
77#2189C2214yescomplemented
78#2187C2212yes—
79#2187C2214yes—
80#2186C2214yescomplemented
81#2186C2214yescomplemented
82Q8×C16C2212yescomplemented
83C4×SD32C2212yescomplemented
84C4×SD32C2214yescomplemented
85C4×SD32C2212yescomplemented
86C4×D16C2214yescomplemented
87D4×C16C2214yes—
88D4×C16C2212yescomplemented
89#2162C2211yescharacteristic
90#2162C2212yescomplemented
91C2×C4⋊C16C2211yescharacteristic
92C2×C4⋊C16C2212yescomplemented
93#2138C2214yescomplemented
94C2×#357C2212yes—
95C2×#135—22no—
96C2×#135C2312yes—
97C2×#135C2314yes—
98C22×D8—21no—
99C22×D8C2311yescharacteristic
100C22×D8C2312yes—
101#559—22no—
102#559—41no—
103#556—21no—
104#555—21no—
105#554—21no—
106#553—41no—
107#552—22no—
108#552—41no—
109#551—42no—
110#550—22no—
111#550—41no—
112#547—21no—
113#546—22no—
114#546—24no—
115#546—42no—
116#540—42no—
117#535—41no—
118#533—21no—
119#533—21no—
120#532—22no—
121#532—22no—
122C2×#127—21no—
123#529—22no—
124#529—42no—
125#525—41no—
126#518—41no—
127C4×#41—22no—
128C4×#41D411yescharacteristic
129C2×C4×D4—21no—
130C2×#117—41no—
131#508—48no—
132C2×Q32C2314yes—
133C2×SD32C2312yes—
134C2×SD32C2314yes—
135C2×SD32C2312yes—
136C2×SD32C2314yes—
137C2×D16—44no—
138C2×D16C2314yes—
139C2×D16C2314yes—
140C22×C16C2312yes—
141C22×C16C2314yes—
142#500C2311yescharacteristic
143#500C2311yescharacteristic
144#499C2312yes—
145#495—22no—
146#495C2312yes—
147#495C2314yes—
148#494C2311yescharacteristic
149#494C2311yescharacteristic
150#493—21no—
151#493—21no—
152#493C2312yes—
153#493C2312yes—
154#492—22no—
155#488—22no—
156#487—21no—
157#487—21no—
158#486—22no—
159#485—22no—
160#483—22no—
161#482—22no—
162#479—22no—
163#478—22no—
164#476—21no—
165#476—21no—
166#467—21no—
167#467—21no—
168#467C2312yes—
169#467C2312yes—
170#467C2312yes—
171#466—22no—
172#466C2312yes—
173#466C2314yes—
174#465—22no—
175#464—22no—
176#463—22no—
177#463—24no—
178#463—24no—
179#461—21no—
180#461—21no—
181#460—22no—
182#460—22no—
183#452—22no—
184#450—22no—
185#450—24no—
186#449—22no—
187#449—24no—
188#449—24no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—21no—
192C4×Q16—21no—
193C4×Q16C2312yes—
194C4×Q16C2312yes—
195C4×Q16C2312yes—
196C4×SD16—22no—
197C4×SD16—22no—
198C4×SD16—22no—
199C4×D8—21no—
200C4×D8—21no—
201C4×D8—22no—
202C4×D8C2311yescharacteristic
203C4×D8C2311yescharacteristic
204C4×D8C2312yes—
205C4×D8C2314yes—
206C4×D8C2311yescharacteristic
207C4×D8C2311yescharacteristic
208C8×D4D412yes—
209C8×D4D411yescharacteristic
210C8×D4C2312yes—
211C8×D4C2311yescharacteristic
212#431D411yescharacteristic
213#431D412yes—
214#427C2312yes—
215C2×#107C2311yescharacteristic
216#416—22no—
217C2×#102—21no—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#367C2312yes—
221#367C2312yes—
222#366C2314yes—
223C4⋊C16C2312yes—
224C4⋊C16C2312yes—
225C4⋊C16C2311yescharacteristic
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2312yes—
229#358C2314yes—
230#358C2314yes—
231#358C2312yes—
232#357C2314yes—
233#357C2314yes—
234#357C2314yes—
235#357C2312yes—
236#357C2314yes—
237#357C2312yes—
238C22⋊C16C2312yes—
239C22⋊C16C2314yes—
240C4×C16C2312yes—
241C4×C16C2314yes—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—42no—
245C2×#41D811yescharacteristic
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—41no—
249C22×D4—41no—
250C23×C4—41no—
251#135—24no—
252#135—42no—
253#135—44no—
254#135—44no—
255#135—84no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16C2412yes—
259C2×Q16C2412yes—
260C2×SD16—22no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—22no—
264C2×D8—22no—
265C2×D8—22no—
266C2×D8—21no—
267C2×D8—22no—
268C2×D8—22no—
269C2×D8—41no—
270C2×D8—42no—
271C2×D8—42no—
272C2×D8—82no—
273C2×D8C2414yes—
274C2×D8C2411yescharacteristic
275C2×D8C2411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2412yes—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284#127—22no—
285#127—22no—
286#127—42no—
287#127—41no—
288#127—41no—
289#126—42no—
290#126—42no—
291#126—42no—
292#125—22no—
293#125—42no—
294#125—41no—
295#125—41no—
296#124—22no—
297#124—24no—
298#124—42no—
299#124—42no—
300#124—42no—
301#123—42no—
302#123—42no—
303#123—42no—
304#123—44no—
305#123—44no—
306#123—42no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—41no—
311C22⋊Q8—41no—
312C22⋊Q8—41no—
313C22⋊Q8—41no—
314C22⋊Q8—42no—
315#121—22no—
316#121—24no—
317#121—41no—
318#121—41no—
319#121—42no—
320#121—42no—
321#121—42no—
322#121—42no—
323#121—44no—
324#121—44no—
325#120—42no—
326#120—44no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—41no—
331C4×Q8—41no—
332C4×Q8D811yescharacteristic
333C4×Q8C2×D411yescharacteristic
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—24no—
340C4×D4—42no—
341C4×D4—42no—
342C4×D4—42no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4—41no—
346C4×D4—41no—
347C4×D4—42no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4D812yes—
351C4×D4D811yescharacteristic
352C4×D4C2×D412yes—
353C4×D4C2×D411yescharacteristic
354#117—22no—
355#117—41no—
356#117—41no—
357#117—41no—
358#117—41no—
359#117—42no—
360#117—42no—
361#117—42no—
362#117D811yescharacteristic
363#117D812yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C22⋊C4—41no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—44no—
370C2×C42—42no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373Q32—44no—
374SD32—24no—
375SD32—42no—
376SD32—44no—
377SD32—44no—
378D16—28no—
379D16—44no—
380C2×C16—42no—
381C2×C16—44no—
382C2×C16C2412yes—
383C2×C16C2412yes—
384C2×C16C2414yes—
385#107—21no—
386#107—21no—
387#107—21no—
388#107C2412yes—
389#107C2411yescharacteristic
390#107C2411yescharacteristic
391#107C2412yes—
392#106—22no—
393#106—22no—
394C4⋊C8C2×D411yescharacteristic
395C4⋊C8C2×D412yes—
396C4⋊C8C2×D411yescharacteristic
397C4⋊C8C2×D412yes—
398#103—22no—
399#103—22no—
400#103—22no—
401#103—22no—
402#103—22no—
403#103—22no—
404#102—21no—
405#102—21no—
406#102—21no—
407#102—21no—
408#102—22no—
409#102—24no—
410#102—24no—
411#102—22no—
412C22⋊C8C2×D412yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C4×C8C2×D411yescharacteristic
416C4×C8C2×D412yes—
417C4×C8C2×D412yes—
418C4×C8C2×D411yescharacteristic
419C4×C8C2412yes—
420C4×C8C2411yescharacteristic
421C4×C8C2411yescharacteristic
422C24—41no—
423C24—42no—
424#41—24no—
425#41—44no—
426#41—82no—
427#41—84no—
428#41—84no—
429#41—164no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8C2×D811yescharacteristic
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—24no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—41no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—81no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4—84no—
453C2×D4—162no—
454C2×D4C2×D812yes—
455C2×D4C2×D811yescharacteristic
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—84no—
466C22×C4C2×D812yes—
467C22×C4C2×D811yescharacteristic
468C22×C4C2×D812yes—
469C22×C4C2×D811yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472Q16—22no—
473Q16—22no—
474Q16—22no—
475Q16—22no—
476Q16—42no—
477Q16—81no—
478SD16—44no—
479SD16—82no—
480SD16—84no—
481SD16—84no—
482D8—24no—
483D8—21no—
484D8—21no—
485D8—21no—
486D8—24no—
487D8—42no—
488D8—44no—
489D8—82no—
490D8—82no—
491D8H(5,𝔽2)12yes—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—22no—
496C2×C8—41no—
497C2×C8—42no—
498C2×C8C22×D411yescharacteristic
499C2×C8C22×D412yes—
500C2×C8C22×D411yescharacteristic
501C2×C8C22×D412yes—
502C2×C8C22×D411yescharacteristic
503C2×C8C2511yescharacteristic
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—41no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4C2×D811yescharacteristic
524C4⋊C4C2×D811yescharacteristic
525C4⋊C4C2×D812yes—
526C4⋊C4C2×D812yes—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4C2×D812yes—
544C22⋊C4C2×D812yes—
545C22⋊C4C2×D812yes—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42C2×D812yes—
551C42C2×D812yes—
552C42C2×D811yescharacteristic
553C42C2×D811yescharacteristic
554C42C22×D411yescharacteristic
555C42C22×D412yes—
556C42C22×D411yescharacteristic
557C16—24no—
558C16—22no—
559C16—22no—
560C23—41no—
561C23—41no—
562C23—44no—
563C23—82no—
564C23—82no—
565C23—84no—
566C23—161no—
567C23—162no—
568C23—162no—
569C23C22×D812yes—
570C23C22×D811yescharacteristic
571Q8—22no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—82no—
577Q8—161no—
578D4—24no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—44no—
583D4—41no—
584D4—41no—
585D4—44no—
586D4—82no—
587D4—84no—
588D4—84no—
589D4—84no—
590D4—84no—
591D4—162no—
592D4—162no—
593D4—162no—
594D4—164no—
595D4—164no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—162no—
616C2×C4—164no—
617C2×C4—164no—
618C2×C4#50814yescomplemented
619C2×C4C22×D812yes—
620C2×C4C22×D812yes—
621C2×C4C22×D811yescharacteristic
622C2×C4C22×D811yescharacteristic
623C2×C4C22×D812yes—
624C2×C4C22×D811yescharacteristic
625C2×C4C22×D811yescharacteristic
626C2×C4C22×D811yescharacteristic
627C2×C4C23×D411yescharacteristic
628C8—21no—
629C8—21no—
630C8—22no—
631C8—21no—
632C8—22no—
633C8C2×H(5,𝔽2)11yescharacteristic
634C8#58511yescharacteristic
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—84no—
645C22—84no—
646C22—81no—
647C22—84no—
648C22—81no—
649C22—164no—
650C22—324no—
651C22C2×#50812yes—
652C22C23×D811yescharacteristic
653C4—22no—
654C4—21no—
655C4—22no—
656C4—21no—
657C4—22no—
658C4—21no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4C2×#50812yes—
666C4C2×#50812yes—
667C4#348811yescharacteristic
668C4#358511yescharacteristic
669C2—22no—
670C2—42no—
671C2—82no—
672C2—161no—
673C2—164no—
674C2#6194711yescharacteristic
675C2C22×#50811yescharacteristic
676C2#6263611yescharacteristic
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. #61427C2semidirect
3. #33961C2semidirect
4. #33928C2semidirect
5. #33872C2semidirect
6. #33863C2semidirect
7. #33850C2semidirect
8. #33846C2semidirect
9. #33815C2semidirect
10. #33813C2semidirect
11. #33778C2semidirect
12. #33774C2semidirect
13. #33666C2semidirect
14. #33661C2semidirect
15. #33579C2semidirect
16. C2×#2206C2semidirect
17. #33502C2semidirect
20. C2×#508C22semidirect
23. #3396C22semidirect
24. #3393C22semidirect
25. #3380C22semidirect
27. #3341C22semidirect
32. #3287C22semidirect
33. #3283C22semidirect
41. #3169C22semidirect
42. #3168C22semidirect
43. #3147C22semidirect
48. C4×#135C22semidirect
50. #2249C22semidirect
51. #2248C22semidirect
52. #2242C22semidirect
53. #2240C22semidirect
55. #2234C22semidirect
56. #2228C22semidirect
57. #2227C22semidirect
58. #2227C22semidirect
59. #2226C22semidirect
60. #2226C22semidirect
61. #2217C22semidirect
63. #2215C22semidirect
64. #2211C22semidirect
65. #2210C22semidirect
66. #2210C22semidirect
67. #2209C22semidirect
68. #2209C22semidirect
69. #2208C22semidirect
70. #2208C22semidirect
71. #2207C22semidirect
72. #2207C22semidirect
77. #2189C22semidirect
80. #2186C22semidirect
81. #2186C22semidirect
82. Q8×C16C22semidirect
83. C4×SD32C22semidirect
84. C4×SD32C22semidirect
85. C4×SD32C22semidirect
86. C4×D16C22semidirect
88. D4×C16C22semidirect
90. #2162C22semidirect
92. C2×C4⋊C16C22semidirect
93. #2138C22semidirect
618. C2×C4#508semidirect

Maximal subgroups

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

subgroupindex
#624172
#614272
#339612
#339282
#338722
#338632
#338502
#338462
#338152
#338132
#337782
#337742
#336662
#336612
#335792
C2×#22062
#335022