Finite Group Databasebeta

← #10951976

Subgroups — #10951976

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84010C211yesmaximal, characteristic, complemented
2#84007C211yesmaximal, characteristic, complemented
3#82036C211yesmaximal, characteristic, complemented
4#80941C211yesmaximal, characteristic, complemented
5C3×#2192C211yesmaximal, characteristic, complemented
6#71514C211yesmaximal, characteristic, complemented
7#71471C211yesmaximal, characteristic, complemented
8#71464C211yesmaximal, characteristic, complemented
9#71092C211yesmaximal, characteristic, complemented
10#69263C211yesmaximal, characteristic, complemented
11#69260C211yesmaximal, characteristic, complemented
12#69242C211yesmaximal, characteristic, complemented
13#69238C211yesmaximal, characteristic, complemented
14#69126C211yesmaximal, characteristic, complemented
15#69122C211yesmaximal, characteristic, complemented
16#33858—31nomaximal, Sylow
17C3×#576C2211yescharacteristic, complemented
18#5835—21no—
19#5774—21no—
20#5769C2211yescharacteristic, complemented
21#5757C2211yescharacteristic, complemented
22#5752C2211yescharacteristic, complemented
23C3×#509C2211yescharacteristic, complemented
24C2×C3×D16C2211yescharacteristic, complemented
25C3×#472C2211yescharacteristic, complemented
26#5241—21no—
27#5199—21no—
28#5198—21no—
29#5198—21no—
30#5196C2211yescharacteristic, complemented
31#5194—21no—
32#5194C2211yescharacteristic
33#5193—21no—
34#5193C2211yescharacteristic, complemented
35#5160C2211yescharacteristic, complemented
36#5151C2211yescharacteristic, complemented
37#5151C2211yescharacteristic, complemented
38#5147—21no—
39#5147C2211yescharacteristic, complemented
40C2×#826—21no—
41C2×#826C2211yescharacteristic, complemented
42C2×#826C2211yescharacteristic, complemented
43#5144C2211yescharacteristic
44#5142C2211yescharacteristic, complemented
45#5140C2211yescharacteristic, complemented
46#4916—21no—
47#4914—21no—
48#4911—21no—
49D3×D16—21no—
50#4908C2211yescharacteristic, complemented
51#4907C2211yescharacteristic, complemented
52C2×D48C2211yescharacteristic, complemented
53#4897C2211yescharacteristic, complemented
54#4895C2211yescharacteristic, complemented
55#4749—21no—
56#4747—21no—
57C3×#361C2211yescharacteristic, complemented
58C3×#359C2211yescharacteristic, complemented
59C3×#350C2211yescharacteristic, complemented
60#4563C2211yescharacteristic, complemented
61#4516C2211yescharacteristic, complemented
62#4511C2211yescharacteristic, complemented
63#4510C2211yescharacteristic, complemented
64#4508C2211yescharacteristic, complemented
65Dic3×C16C2211yescharacteristic, complemented
66#4495C2211yescharacteristic, complemented
67#4493C2211yescharacteristic, complemented
68#3415—31no—
69#3415—31no—
70#3292—31no—
71#3292—31no—
72#2246—31no—
73#2221—31no—
74#2221—31no—
75#2213—31no—
76#2213—31no—
77#2192—31no—
78#2192—31no—
79#2192—31no—
80#2192D311yescharacteristic, complemented
81#2178—31no—
82#2170—31no—
83C3×H(5,𝔽2)—21no—
84#1009—21no—
85#1009—21no—
86#1004—21no—
87C3×AGL(1,ℤ/8)—21no—
88C3×AGL(1,ℤ/8)—21no—
89C3×AGL(1,ℤ/8)—41no—
90C3×#135—21no—
91C3×#135C2311yescharacteristic
92C2×C3×D8—21no—
93C2×C3×D8C2311yescharacteristic
94C2×C3×D8C2311yescharacteristic
95C3×#131C2311yescharacteristic
96#950—41no—
97#950D411yescharacteristic
98#949—21no—
99#949—21no—
100#949—41no—
101#948—21no—
102#940—21no—
103#940—21no—
104#932—21no—
105#932—21no—
106#932—41no—
107C2×#213—21no—
108C2×#213—21no—
109C2×#213—41no—
110#919—21no—
111#916—41no—
112#914—41no—
113#914—41no—
114#912—21no—
115#911—41no—
116#911—41no—
117D3×D8—21no—
118D3×D8—21no—
119D3×D8—41no—
120#908—21no—
121#908—21no—
122#908—41no—
123#907—21no—
124#904—21no—
125#904C2311yescharacteristic
126C2×D24—21no—
127C2×D24C2311yescharacteristic
128C2×D24C2311yescharacteristic
129#901C2311yescharacteristic
130C3×SD32—21no—
131C3×D16—21no—
132C3×D16—41no—
133C3×M32C2311yescharacteristic
134C2×C3×C16C2311yescharacteristic
135C3×#108C2311yescharacteristic
136C3×#100—21no—
137#837—21no—
138#837D411yescharacteristic
139#833—21no—
140#829D411yescharacteristic, complemented
141#828—21no—
142#828D411yescharacteristic, complemented
143#827—21no—
144#827D411yescharacteristic, complemented
145#826—21no—
146#826—21no—
147#826—21no—
148#826—21no—
149#826—21no—
150#826—21no—
151#826D411yescharacteristic, complemented
152#825—21no—
153#825D411yescharacteristic
154#823—21no—
155#823—21no—
156#822D411yescharacteristic
157#819C2311yescharacteristic
158C8×Dic3C2311yescharacteristic
159#812—21no—
160#812C2311yescharacteristic
161C2×C3⋊C16—21no—
162C2×C3⋊C16C2311yescharacteristic
163#800—21no—
164D48—21no—
165D48—41no—
166#798—21no—
167D3×C16—21no—
168#576—31no—
169#576—31no—
170#576—31no—
171#576—61no—
172#576—61no—
173#576C2×D311yescharacteristic, complemented
174#509—31no—
175#509—31no—
176#509—31no—
177#509—61no—
178#509—61no—
179#509—61no—
180#509—61no—
181#509C2×D311yescharacteristic, complemented
182#508—31no—
183#508—61no—
184#508—61no—
185C2×D16—31no—
186C2×D16—31no—
187C2×D16—31no—
188C2×D16—61no—
189C2×D16—61no—
190C2×D16C2×D311yescharacteristic, complemented
191#504—31no—
192#504—31no—
193#493—31no—
194#493—31no—
195#472—31no—
196#472—31no—
197#472—31no—
198#472—61no—
199#472—61no—
200#472C2×D311yescharacteristic, complemented
201#453—61no—
202#453—61no—
203#453—61no—
204#453—61no—
205#443—31no—
206#443—31no—
207#443—31no—
208#368—31no—
209#364—31no—
210#361—31no—
211#361—31no—
212#361—31no—
213#361C2×D311yescharacteristic, complemented
214#359—31no—
215#359C2×D311yescharacteristic, complemented
216#350—31no—
217#350C2×D311yescharacteristic, complemented
218C4×C16—31no—
219C3×#41—21no—
220C3×#41—41no—
221C3×#41—41no—
222C3×#41C2×D411yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4—81no—
228C2×#67—81no—
229C2×#67—81no—
230C2×#67—81no—
231#239—41no—
232#239—41no—
233#237—41no—
234#237—41no—
235D3×D4—41no—
236D3×D4—41no—
237D3×D4—41no—
238D3×D4—41no—
239D3×D4—81no—
240D3×D4—81no—
241#235—21no—
242#235—41no—
243#235—41no—
244#235C2×D411yescharacteristic
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—41no—
248C2×D12—41no—
249C3×Q16C2×D411yescharacteristic
250C3×SD16—21no—
251C3×SD16—41no—
252C3×SD16—41no—
253C3×D8—21no—
254C3×D8—21no—
255C3×D8—21no—
256C3×D8—21no—
257C3×D8—41no—
258C3×D8—41no—
259C3×D8C2×D411yescharacteristic
260C3×M16—21no—
261C3×M16—21no—
262C3×M16C2×D411yescharacteristic
263C2×C3×C8—21no—
264C2×C3×C8C2411yescharacteristic
265#216D811yescharacteristic
266#215—41no—
267#215—41no—
268#215D811yescharacteristic
269#214—41no—
270#214—41no—
271#214D811yescharacteristic
272#213—41no—
273#213—41no—
274#213—41no—
275#213—41no—
276#213—41no—
277#213—41no—
278#213—41no—
279#213—41no—
280#213D811yescharacteristic
281C4×Dic3C2×D411yescharacteristic
282#208—21no—
283#208—41no—
284#208C2×D411yescharacteristic
285C2×C3⋊C8—41no—
286C2×C3⋊C8C2×D411yescharacteristic
287Dic12C2×D411yescharacteristic
288D24—21no—
289D24—21no—
290D24—21no—
291D24—21no—
292D24—41no—
293D24—41no—
294D24C2×D411yescharacteristic
295#204—21no—
296#204—41no—
297#204—41no—
298ΓL(1,𝔽25)—21no—
299ΓL(1,𝔽25)—41no—
300D3×C8—21no—
301D3×C8—41no—
302C3×C16—21no—
303C3×C16—21no—
304C3⋊C16—21no—
305C3⋊C16C2×D411yescharacteristic
306C3⋊C16C2×D411yescharacteristic
307H(5,𝔽2)—21no—
308H(5,𝔽2)—61no—
309H(5,𝔽2)—61no—
310H(5,𝔽2)—61no—
311AGL(1,ℤ/8)—21no—
312AGL(1,ℤ/8)—21no—
313AGL(1,ℤ/8)—41no—
314AGL(1,ℤ/8)—61no—
315AGL(1,ℤ/8)—61no—
316AGL(1,ℤ/8)—61no—
317AGL(1,ℤ/8)—61no—
318AGL(1,ℤ/8)—61no—
319AGL(1,ℤ/8)—61no—
320AGL(1,ℤ/8)—121no—
321AGL(1,ℤ/8)—121no—
322AGL(1,ℤ/8)—121no—
323AGL(1,ℤ/8)—121no—
324AGL(1,ℤ/8)—121no—
325AGL(1,ℤ/8)—121no—
326AGL(1,ℤ/8)—121no—
327#135—21no—
328#135—31no—
329#135—31no—
330#135—61no—
331#135—61no—
332#135—61no—
333#135—121no—
334#135—121no—
335#135C22×D311yescharacteristic
336C2×D8—21no—
337C2×D8—31no—
338C2×D8—31no—
339C2×D8—61no—
340C2×D8—61no—
341C2×D8—61no—
342C2×D8—61no—
343C2×D8—61no—
344C2×D8—121no—
345C2×D8—121no—
346C2×D8C22×D311yescharacteristic
347C2×D8C22×D311yescharacteristic
348#131—31no—
349#131—61no—
350#131—61no—
351#131C22×D311yescharacteristic
352#127—61no—
353#127—61no—
354Q32—31no—
355SD32—21no—
356SD32—31no—
357SD32—31no—
358SD32—61no—
359SD32—61no—
360SD32—61no—
361D16—21no—
362D16—31no—
363D16—41no—
364D16—61no—
365D16—61no—
366D16—61no—
367D16—61no—
368D16—61no—
369D16—61no—
370D16—61no—
371D16—121no—
372M32—31no—
373M32—61no—
374M32—61no—
375M32C22×D311yescharacteristic
376C2×C16—31no—
377C2×C16—61no—
378C2×C16—61no—
379C2×C16C22×D311yescharacteristic
380#108—31no—
381#108—31no—
382#108C22×D311yescharacteristic
383#104—31no—
384#104—31no—
385#104—61no—
386#104—61no—
387#100—21no—
388#100—61no—
389#100—61no—
390#100—61no—
391C4×C8—31no—
392C23×C3—41no—
393C23×C3—41no—
394C23×C3—81no—
395C22×D3—41no—
396C22×D3—41no—
397C22×D3—81no—
398C22×D3—81no—
399C22×D3—81no—
400C3×Q8—21no—
401C3×Q8C2×D811yescharacteristic
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—41no—
405C3×D4—41no—
406C3×D4—41no—
407C3×D4—41no—
408C3×D4—41no—
409C3×D4—81no—
410C3×D4—81no—
411C3×D4C2×D811yescharacteristic
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—81no—
415C2×C3×C4C22×D411yescharacteristic
416#67—21no—
417#67—41no—
418#67—81no—
419#67—81no—
420#67—81no—
421#67—81no—
422#67—162no—
423#67—162no—
424C2×Dic3—21no—
425C2×Dic3—81no—
426D12—21no—
427D12—41no—
428D12—41no—
429D12—41no—
430D12—41no—
431D12—81no—
432D12—81no—
433D12C2×D811yescharacteristic
434C4×D3—21no—
435C4×D3—41no—
436C4×D3—81no—
437C4×D3—81no—
438Dic6—21no—
439Dic6C2×D811yescharacteristic
440C3×C8—21no—
441C3×C8—41no—
442C3×C8C22×D411yescharacteristic
443C3×C8H(5,𝔽2)11yescharacteristic
444C3⋊C8—21no—
445C3⋊C8—41no—
446C3⋊C8C2×D811yescharacteristic
447C3⋊C8C2×D811yescharacteristic
448#41—21no—
449#41—31no—
450#41—41no—
451#41—41no—
452#41—61no—
453#41—121no—
454#41—121no—
455#41—121no—
456#41—121no—
457#41—121no—
458#41—121no—
459#41D3×D411yescharacteristic
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—81no—
467C2×D4—121no—
468C2×D4—121no—
469C2×D4—121no—
470C2×D4—121no—
471C2×D4—121no—
472C2×D4—121no—
473C2×D4—241no—
474C2×D4—241no—
475C2×D4—241no—
476C2×D4—241no—
477C2×D4—241no—
478Q16—31no—
479Q16—31no—
480Q16D3×D411yescharacteristic
481SD16—21no—
482SD16—31no—
483SD16—31no—
484SD16—41no—
485SD16—41no—
486SD16—61no—
487SD16—121no—
488SD16—121no—
489SD16—121no—
490SD16—121no—
491SD16—121no—
492SD16—121no—
493D8—21no—
494D8—21no—
495D8—21no—
496D8—21no—
497D8—31no—
498D8—31no—
499D8—41no—
500D8—41no—
501D8—61no—
502D8—61no—
503D8—61no—
504D8—61no—
505D8—121no—
506D8—121no—
507D8—121no—
508D8—121no—
509D8—121no—
510D8—121no—
511D8—121no—
512D8—121no—
513D8—121no—
514D8—121no—
515D8D3×D411yescharacteristic
516M16—21no—
517M16—21no—
518M16—31no—
519M16—61no—
520M16—61no—
521M16—121no—
522M16—121no—
523M16D3×D411yescharacteristic
524C2×C8—21no—
525C2×C8—31no—
526C2×C8—61no—
527C2×C8—121no—
528C2×C8—121no—
529C2×C8C23×D311yescharacteristic
530C42—31no—
531C16—21no—
532C16—21no—
533C16—31no—
534C16—31no—
535C16—61no—
536C22×C3—21no—
537C22×C3—41no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—161no—
541C22×C3—161no—
542C22×C3—321no—
543C22×C3#57611yescharacteristic
544C2×D3—21no—
545C2×D3—41no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—161no—
549C2×D3—161no—
550C2×D3—321no—
551C2×D3—321no—
552C2×D3—321no—
553C3×C4—21no—
554C3×C4—41no—
555C3×C4D4211yescharacteristic
556C3×C4C22×D811yescharacteristic
557Dic3—22no—
558Dic3—21no—
559Dic3—41no—
560C23—41no—
561C23—41no—
562C23—81no—
563C23—121no—
564C23—121no—
565C23—241no—
566C23—241no—
567C23—241no—
568Q8—21no—
569Q8—31no—
570Q8—61no—
571Q8D3×D811yescharacteristic
572D4—21no—
573D4—21no—
574D4—31no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—61no—
581D4—61no—
582D4—81no—
583D4—81no—
584D4—121no—
585D4—121no—
586D4—121no—
587D4—121no—
588D4—121no—
589D4—241no—
590D4—241no—
591D4—241no—
592D4—241no—
593D4—241no—
594D4—241no—
595D4—482no—
596D4—482no—
597D4D3×D811yescharacteristic
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—81no—
603C2×C4—121no—
604C2×C4—241no—
605C2×C4—241no—
606C2×C4—241no—
607C2×C4C2×D3×D411yescharacteristic
608C8—21no—
609C8—31no—
610C8—31no—
611C8—41no—
612C8—61no—
613C8—121no—
614C8C2×D3×D411yescharacteristic
615C8#100411yescharacteristic
616C2×C3—21no—
617C2×C3—41no—
618C2×C3—81no—
619C2×C3—161no—
620C2×C3—161no—
621C2×C3D4×D811yescharacteristic
622D3—41no—
623D3—81no—
624D3—161no—
625D3—161no—
626C22—21no—
627C22—41no—
628C22—61no—
629C22—81no—
630C22—81no—
631C22—121no—
632C22—161no—
633C22—161no—
634C22—241no—
635C22—241no—
636C22—321no—
637C22—481no—
638C22—481no—
639C22—961no—
640C22—961no—
641C22—961no—
642C22#577411yescharacteristic
643C4—21no—
644C4—41no—
645C4—62no—
646C4—61no—
647C4—121no—
648C4#559011yescharacteristic
649C4C2×D3×D811yescharacteristic
650C3#3385811yescharacteristic, complemented, Sylow
651C2—21no—
652C2—41no—
653C2—81no—
654C2—121no—
655C2—161no—
656C2—161no—
657C2—241no—
658C2—481no—
659C2—481no—
660C2#8002411yescharacteristic
subgroupG/H
centerC2#80024
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2192C2
socleC2×C3D4×D8
hypercenter#2192D3
hypocenterC3#33858
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #2192

Complements

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

NH ≅ G/Ntype
1. #84010C2semidirect
2. #84007C2semidirect
3. #82036C2semidirect
4. #80941C2semidirect
5. C3×#2192C2semidirect
6. #71514C2semidirect
7. #71471C2semidirect
8. #71464C2semidirect
9. #71092C2semidirect
10. #69263C2semidirect
11. #69260C2semidirect
12. #69242C2semidirect
13. #69238C2semidirect
14. #69126C2semidirect
15. #69122C2semidirect
17. C3×#576C22semidirect
20. #5769C22semidirect
21. #5757C22semidirect
22. #5752C22semidirect
23. C3×#509C22semidirect
24. C2×C3×D16C22semidirect
25. C3×#472C22semidirect
30. #5196C22semidirect
34. #5193C22semidirect
35. #5160C22semidirect
36. #5151C22semidirect
37. #5151C22semidirect
39. #5147C22semidirect
41. C2×#826C22semidirect
42. C2×#826C22semidirect
44. #5142C22semidirect
45. #5140C22semidirect
50. #4908C22semidirect
51. #4907C22semidirect
52. C2×D48C22semidirect
53. #4897C22semidirect
54. #4895C22semidirect
57. C3×#361C22semidirect
58. C3×#359C22semidirect
59. C3×#350C22semidirect
60. #4563C22semidirect
61. #4516C22semidirect
62. #4511C22semidirect
63. #4510C22semidirect
64. #4508C22semidirect
65. Dic3×C16C22semidirect
66. #4495C22semidirect
67. #4493C22semidirect
80. #2192D3semidirect
140. #829D4semidirect
142. #828D4semidirect
144. #827D4semidirect
151. #826D4semidirect
173. #576C2×D3semidirect
181. #509C2×D3semidirect
190. C2×D16C2×D3semidirect
200. #472C2×D3semidirect
213. #361C2×D3semidirect
215. #359C2×D3semidirect
217. #350C2×D3semidirect
650. C3#33858semidirect

Maximal subgroups

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

subgroupindex
#840102
#840072
#820362
#809412
C3×#21922
#715142
#714712
#714642
#710922
#692632
#692602
#692422
#692382
#691262
#691222
#338583