Finite Group Databasebeta

← #7624555

Subgroups — #7624555

Order 512.

1357 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62418C211yesmaximal, characteristic, complemented
2#61431C211yesmaximal, characteristic, complemented
3#33961C212yesmaximal, complemented
4#33927C212yesmaximal, complemented
5#33898C212yesmaximal, complemented
6#33893C214yesmaximal, complemented
7#33841C214yesmaximal, complemented
8#33838C212yesmaximal, complemented
9#33819C211yesmaximal, characteristic, complemented
10#33818C212yesmaximal, complemented
11#33778C212yesmaximal, complemented
12#33772C211yesmaximal, characteristic, complemented
13#33666C211yesmaximal, characteristic, complemented
14#33661C212yesmaximal, complemented
15#33583C211yesmaximal, characteristic, complemented
16#33580C212yesmaximal, complemented
17#33505C211yesmaximal, characteristic, complemented
18#3511—21no—
19#3481—21no—
20C2×#510C2212yes—
21C2×#509C2214yes—
22#3396C2211yescharacteristic
23#3393—21no—
24#3393C2212yescomplemented
25#3380C2211yescharacteristic
26#3361—21no—
27#3339C2212yes—
28#3338C2211yescharacteristic
29#3297—22no—
30#3295—21no—
31#3285—22no—
32#3285C2214yescomplemented
33#3283—21no—
34#3283C2212yes—
35#3280C2212yes—
36D4×D8C2211yescharacteristic
37#3210—22no—
38#3206—21no—
39#3170C2212yescomplemented
40#3169C2211yescharacteristic
41#3169C2211yescharacteristic, complemented
42#3168C2212yes—
43#3147C2212yescomplemented
44C2×#494C2211yescharacteristic, complemented
45#3106—21no—
46#3105—22no—
47C8×#41C2211yescharacteristic
48C4×#135C2211yescharacteristic
49C2×C4×D8C2212yes—
50#2249C2214yescomplemented
51#2248C2212yes—
52#2242C2214yescomplemented
53#2241C2212yescomplemented
54#2235C2214yescomplemented
55#2234C2212yescomplemented
56#2229C2214yescomplemented
57#2227C2211yescharacteristic, complemented
58#2226C2211yescharacteristic
59#2226C2212yescomplemented
60#2226C2212yescomplemented
61#2220C2214yes—
62#2219C2214yescomplemented
63#2218C2214yescomplemented
64#2211C2214yescomplemented
65#2210C2214yes—
66#2210C2214yescomplemented
67#2209C2212yescomplemented
68#2209C2212yescomplemented
69#2208C2212yes—
70#2208C2212yes—
71#2208C2212yescomplemented
72#2208C2212yescomplemented
73#2207C2211yescharacteristic, complemented
74#2206C2212yes—
75#2206C2211yescharacteristic
76#2206C2212yescomplemented
77#2189C2214yescomplemented
78#2187C2214yes—
79#2186C2214yes—
80#2186C2212yes—
81#2185C2214yescomplemented
82Q8×C16C2212yescomplemented
83C4×SD32C2214yes—
84C4×SD32C2214yescomplemented
85C4×SD32C2212yescomplemented
86C4×D16C2212yes—
87#2169C2212yes—
88#2169C2214yescomplemented
89#2162C2212yescomplemented
90#2162C2211yescharacteristic, complemented
91#2150C2212yescomplemented
92#2150C2211yescharacteristic, complemented
93#2141C2212yescomplemented
94#2139C2214yescomplemented
95C2×#135—22no—
96C2×#135C2311yescharacteristic
97C2×#135C2314yes—
98C22×Q16—21no—
99C22×Q16C2312yes—
100C22×D8C2312yes—
101#559—21no—
102#556—22no—
103#555—41no—
104#554—21no—
105#554—21no—
106#554—41no—
107#553—22no—
108#553—42no—
109#553—41no—
110#551—42no—
111#548—24no—
112#547—22no—
113#546—22no—
114#546—41no—
115D42—21no—
116#542—42no—
117#537—41no—
118#533—22no—
119#532—21no—
120#532—21no—
121#532—22no—
122C2×C4⋊Q8—21no—
123#529—22no—
124#529—41no—
125#525—42no—
126#520—41no—
127C4×#41—21no—
128C4×#41D411yescharacteristic
129C2×C4×D4—22no—
130C2×C4×D4—41no—
131#510—44no—
132#509—48no—
133C2×Q32C2314yes—
134C2×SD32C2314yes—
135C2×SD32C2314yes—
136C2×SD32C2314yes—
137C2×SD32C2312yes—
138C2×D16C2312yes—
139C2×D16C2314yes—
140C2×M32C2314yes—
141C2×M32C2312yes—
142#500C2311yescharacteristic
143#500C2311yescharacteristic
144#499C2312yes—
145#495—22no—
146#495C2314yes—
147#495C2312yes—
148#494—21no—
149#494—21no—
150#494C2312yes—
151#494C2312yes—
152#493C2311yescharacteristic
153#493C2311yescharacteristic
154#492—22no—
155#488—22no—
156#487—22no—
157#486—21no—
158#486—21no—
159#484—22no—
160#482—22no—
161#481—22no—
162#478—22no—
163#478—22no—
164#477—21no—
165#476—21no—
166#467—22no—
167#467C2312yes—
168#467C2314yes—
169#466—21no—
170#466—21no—
171#466C2312yes—
172#466C2312yes—
173#466C2312yes—
174#465—22no—
175#464—24no—
176#464—24no—
177#463—22no—
178#463—22no—
179#461—22no—
180#461—22no—
181#460—21no—
182#459—21no—
183#452—24no—
184#452—24no—
185#450—22no—
186#449—22no—
187#448—24no—
188#447—22no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—22no—
192C4×Q16C2311yescharacteristic
193C4×Q16C2314yes—
194C4×Q16C2311yescharacteristic
195C4×SD16—21no—
196C4×SD16—22no—
197C4×SD16—21no—
198C4×D8—21no—
199C4×D8—22no—
200C4×D8—22no—
201C4×D8—21no—
202C4×D8C2312yes—
203C4×D8C2312yes—
204C4×D8C2311yescharacteristic
205C4×D8C2312yes—
206C4×D8C2312yes—
207C4×D8C2311yescharacteristic
208C8×D4D411yescharacteristic
209C8×D4D412yes—
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#367C2314yes—
221#367C2312yes—
222#366C2312yes—
223C4⋊C16C2311yescharacteristic
224C4⋊C16C2312yes—
225C4⋊C16C2312yes—
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2314yes—
229#358C2314yes—
230#358C2314yes—
231#358C2314yes—
232#358C2312yes—
233#357C2312yes—
234#357C2314yes—
235#357C2312yes—
236#357C2312yes—
237#357C2314yes—
238C22⋊C16C2314yes—
239C22⋊C16C2312yes—
240C4×C16C2312yes—
241C4×C16C2314yes—
242C2×#41—21no—
243C2×#41—24no—
244C2×#41—41no—
245C2×#41—41no—
246C2×#41D811yescharacteristic
247C22×Q8—22no—
248C22×D4—22no—
249C22×D4—41no—
250C23×C4—42no—
251#135—22no—
252#135—41no—
253#135—44no—
254#135—44no—
255#135—84no—
256C2×Q16—22no—
257C2×Q16—22no—
258C2×Q16—22no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×Q16—82no—
262C2×Q16C2414yes—
263C2×Q16C2411yescharacteristic
264C2×SD16—21no—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—22no—
268C2×D8—24no—
269C2×D8—22no—
270C2×D8—21no—
271C2×D8—21no—
272C2×D8—42no—
273C2×D8C2412yes—
274C2×D8C2412yes—
275C2×D8C2411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—41no—
285C4⋊Q8—41no—
286#127—21no—
287#127—21no—
288#126—42no—
289#126—44no—
290#126—42no—
291#125—22no—
292#125—41no—
293#125—42no—
294#124—22no—
295#124—24no—
296#124—41no—
297#124—44no—
298#124—41no—
299#124—41no—
300#123—42no—
301#123—42no—
302#123—42no—
303#123—42no—
304#123—44no—
305C22⋊Q8—22no—
306C22⋊Q8—24no—
307C22⋊Q8—41no—
308C22⋊Q8—42no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C22⋊Q8—44no—
312C22⋊Q8—42no—
313#121—22no—
314#121—22no—
315#121—22no—
316#121—42no—
317#121—42no—
318#121—44no—
319#121—42no—
320#121—41no—
321#121—41no—
322#121—42no—
323#121—41no—
324#121—42no—
325#120—42no—
326#120—42no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—24no—
330C4×Q8—42no—
331C4×Q8D811yescharacteristic
332C4×Q8C2×D411yescharacteristic
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—41no—
340C4×D4—42no—
341C4×D4—41no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—42no—
345C4×D4—44no—
346C4×D4—42no—
347C4×D4—42no—
348C4×D4—41no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—41no—
352C4×D4D811yescharacteristic
353C4×D4D812yes—
354C4×D4C2×D411yescharacteristic
355C4×D4C2×D412yes—
356#117—21no—
357#117—42no—
358#117—42no—
359#117—41no—
360#117—42no—
361#117D811yescharacteristic
362#117D812yes—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—42no—
370C2×C42—41no—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373Q32—44no—
374SD32—28no—
375SD32—44no—
376SD32—42no—
377SD32—44no—
378D16—24no—
379D16—44no—
380M32—44no—
381M32—42no—
382C2×C16C2412yes—
383C2×C16C2414yes—
384C2×C16C2412yes—
385#107—22no—
386#107—21no—
387#107—21no—
388#107C2412yes—
389#107C2411yescharacteristic
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—24no—
402#103—24no—
403#102—21no—
404#102—22no—
405#102—22no—
406#102—22no—
407#102—21no—
408#102—22no—
409#102—22no—
410#102—22no—
411#102—22no—
412C22⋊C8C2×D412yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C4×C8C2×D411yescharacteristic
416C4×C8C2×D412yes—
417C4×C8C2×D411yescharacteristic
418C4×C8C2×D412yes—
419C4×C8C2411yescharacteristic
420C4×C8C2412yes—
421C4×C8C2411yescharacteristic
422C24—42no—
423#41—42no—
424#41—42no—
425#41—81no—
426#41—84no—
427#41—84no—
428#41—84no—
429#41—162no—
430C2×Q8—21no—
431C2×Q8—24no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8C2×D811yescharacteristic
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—44no—
442C2×D4—41no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—41no—
448C2×D4—42no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—82no—
453C2×D4—84no—
454C2×D4C2×D811yescharacteristic
455C2×D4C2×D812yes—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—164no—
467C22×C4C2×D811yescharacteristic
468C22×C4C2×D811yescharacteristic
469C22×C4C2×D812yes—
470C22×C4C2×D812yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473Q16—24no—
474Q16—21no—
475Q16—21no—
476Q16—24no—
477Q16—44no—
478Q16—82no—
479Q16—82no—
480SD16—42no—
481SD16—82no—
482SD16—84no—
483SD16—84no—
484D8—22no—
485D8—22no—
486D8—21no—
487D8—22no—
488D8—22no—
489D8—44no—
490D8—42no—
491D8—81no—
492D8H(5,𝔽2)12yes—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—22no—
496C2×C8—21no—
497C2×C8—41no—
498C2×C8—42no—
499C2×C8C22×D411yescharacteristic
500C2×C8C22×D411yescharacteristic
501C2×C8C22×D412yes—
502C2×C8C22×D411yescharacteristic
503C2×C8C22×D412yes—
504C2×C8C2511yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4C2×D811yescharacteristic
524C4⋊C4C2×D811yescharacteristic
525C4⋊C4C2×D812yes—
526C4⋊C4C2×D812yes—
527C22⋊C4—42no—
528C22⋊C4—41no—
529C22⋊C4—44no—
530C22⋊C4—41no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—41no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4C2×D812yes—
544C22⋊C4C2×D812yes—
545C22⋊C4C2×D812yes—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—44no—
550C42C2×D811yescharacteristic
551C42C2×D811yescharacteristic
552C42C2×D812yes—
553C42C2×D812yes—
554C42C22×D411yescharacteristic
555C42C22×D411yescharacteristic
556C42C22×D412yes—
557C16—22no—
558C16—24no—
559C16—22no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—84no—
564C23—84no—
565C23—162no—
566C23C22×D812yes—
567C23C22×D811yescharacteristic
568Q8—22no—
569Q8—41no—
570Q8—44no—
571Q8—41no—
572Q8—44no—
573Q8—82no—
574Q8—162no—
575Q8—161no—
576D4—22no—
577D4—22no—
578D4—42no—
579D4—41no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—84no—
585D4—82no—
586D4—82no—
587D4—84no—
588D4—84no—
589D4—82no—
590D4—82no—
591D4—161no—
592D4—164no—
593D4—162no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—82no—
612C2×C4—84no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4—164no—
616C2×C4—162no—
617C2×C4—162no—
618C2×C4C22×D812yes—
619C2×C4C22×D811yescharacteristic
620C2×C4C22×D811yescharacteristic
621C2×C4C22×D812yes—
622C2×C4C22×D812yes—
623C2×C4C22×D811yescharacteristic
624C2×C4C22×D811yescharacteristic
625C2×C4C22×D811yescharacteristic
626C2×C4C23×D411yescharacteristic
627C8—21no—
628C8—21no—
629C8—22no—
630C8—22no—
631C8—21no—
632C8C2×H(5,𝔽2)11yescharacteristic
633C8#58511yescharacteristic
634C22—21no—
635C22—22no—
636C22—22no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—168no—
650C22C23×D811yescharacteristic
651C4—21no—
652C4—22no—
653C4—22no—
654C4—21no—
655C4—21no—
656C4—22no—
657C4—81no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4—84no—
663C4#341412yes—
664C4#341612yes—
665C4#348811yescharacteristic
666C4#358511yescharacteristic
667C2—41no—
668C2—42no—
669C2—82no—
670C2—162no—
671C2—162no—
672C2#6194711yescharacteristic
673C2C2×#341411yescharacteristic
674C2C2×#341611yescharacteristic
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. #62418C2semidirect
2. #61431C2semidirect
3. #33961C2semidirect
4. #33927C2semidirect
5. #33898C2semidirect
6. #33893C2semidirect
7. #33841C2semidirect
8. #33838C2semidirect
9. #33819C2semidirect
10. #33818C2semidirect
11. #33778C2semidirect
12. #33772C2semidirect
13. #33666C2semidirect
14. #33661C2semidirect
15. #33583C2semidirect
16. #33580C2semidirect
17. #33505C2semidirect
24. #3393C22semidirect
32. #3285C22semidirect
39. #3170C22semidirect
41. #3169C22semidirect
43. #3147C22semidirect
44. C2×#494C22semidirect
50. #2249C22semidirect
52. #2242C22semidirect
53. #2241C22semidirect
54. #2235C22semidirect
55. #2234C22semidirect
56. #2229C22semidirect
57. #2227C22semidirect
59. #2226C22semidirect
60. #2226C22semidirect
62. #2219C22semidirect
63. #2218C22semidirect
64. #2211C22semidirect
66. #2210C22semidirect
67. #2209C22semidirect
68. #2209C22semidirect
71. #2208C22semidirect
72. #2208C22semidirect
73. #2207C22semidirect
76. #2206C22semidirect
77. #2189C22semidirect
81. #2185C22semidirect
82. Q8×C16C22semidirect
84. C4×SD32C22semidirect
85. C4×SD32C22semidirect
88. #2169C22semidirect
89. #2162C22semidirect
90. #2162C22semidirect
91. #2150C22semidirect
92. #2150C22semidirect
93. #2141C22semidirect
94. #2139C22semidirect

Maximal subgroups

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

subgroupindex
#624182
#614312
#339612
#339272
#338982
#338932
#338412
#338382
#338192
#338182
#337782
#337722
#336662
#336612
#335832
#335802
#335052