Finite Group Databasebeta

← #10784667

Subgroups — #10784667

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#215C211yesmaximal, characteristic
2#83558C211yesmaximal, characteristic, complemented
3#81037C211yesmaximal, characteristic, complemented
4#80448C211yesmaximal, characteristic
5C3×#1651C211yesmaximal, characteristic, complemented
6#71342C211yesmaximal, characteristic, complemented
7#71335C211yesmaximal, characteristic, complemented
8#71120C211yesmaximal, characteristic, complemented
9#71038C211yesmaximal, characteristic, complemented
10#68464C211yesmaximal, characteristic, complemented
11#68443C211yesmaximal, characteristic, complemented
12#68265C211yesmaximal, characteristic, complemented
13#68216C211yesmaximal, characteristic, complemented
14#68182C211yesmaximal, characteristic
15#68150C211yesmaximal, characteristic, complemented
16#19764—31nomaximal, Sylow
17C3×D4×Q8C2211yescharacteristic
18C22×#215—21no—
19C2×D3×SD16—21no—
20#5727C2211yescharacteristic
21#5567C2211yescharacteristic, complemented
22D4×D12C2211yescharacteristic
23C3×#498C2211yescharacteristic, complemented
24C3×#475C2211yescharacteristic, complemented
25C3×#461C2211yescharacteristic
26#5227—21no—
27#5169—21no—
28#5094C2211yescharacteristic, complemented
29#5093C2211yescharacteristic, complemented
30#5091—21no—
31#5085C2211yescharacteristic
32#5085C2211yescharacteristic
33#5083C2211yescharacteristic
34#5081—21no—
35#5078C2211yescharacteristic, complemented
36#5048—21no—
37#5046—21no—
38#5027C2211yescharacteristic
39C4×#215C2211yescharacteristic
40#5024C2211yescharacteristic, complemented
41#5018—21no—
42C4×#214C2211yescharacteristic
43D4×C3⋊C8C2211yescharacteristic
44#5009—21no—
45#5007C2211yescharacteristic
46#4854—21no—
47#4852—21no—
48#4848C2211yescharacteristic
49#4841C2211yescharacteristic
50#4834C2211yescharacteristic
51#4830C2211yescharacteristic, complemented
52#4818—21no—
53#4807—21no—
54#4806—21no—
55#4792—21no—
56#4694C2211yescharacteristic
57C3×#332C2211yescharacteristic, complemented
58C3×Q8⋊C8C2211yescharacteristic, complemented
59C3×D4⋊C8C2211yescharacteristic
60#4547C2211yescharacteristic, complemented
61#4486C2211yescharacteristic, complemented
62#4485C2211yescharacteristic
63#4480C2211yescharacteristic, complemented
64#4479C2211yescharacteristic, complemented
65#4459C2211yescharacteristic
66#4456C2211yescharacteristic, complemented
67#4454C2211yescharacteristic, complemented
68#3386—31no—
69#3359—31no—
70D4×SD16—31no—
71D4×SD16—31no—
72#1714—31no—
73#1686—31no—
74#1679—31no—
75#1673—31no—
76#1666—31no—
77#1651—31no—
78#1651D311yescharacteristic, complemented
79#1639—31no—
80#1626—31no—
81#1582—31no—
82SD16⋊C8—31no—
83C22×C3×Q8—21no—
84C2×D3×Q8—21no—
85C2×D3×D4—21no—
86C22×D12—21no—
87C2×C3×SD16—21no—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—41no—
93C3×C4×Q8C2311yescharacteristic
94C3×C4×D4—21no—
95C3×C4×D4C2311yescharacteristic
96#946—21no—
97#944—21no—
98C2×#215—21no—
99C2×#215—21no—
100C2×#215—21no—
101C2×#215—21no—
102C2×#215—21no—
103C2×#215—21no—
104C2×#215—44no—
105C2×#215D411yescharacteristic
106C2×#214D411yescharacteristic
107#930—21no—
108C22×C3⋊C8—21no—
109D3×SD16—44no—
110C2×#204—21no—
111C2×#204—21no—
112C2×D3×C8—21no—
113#896—41no—
114#895—21no—
115#893—21no—
116#889—41no—
117#888—21no—
118#887—21no—
119#882—41no—
120#874C2311yescharacteristic
121C4×D12C2311yescharacteristic
122#870—21no—
123#869C2311yescharacteristic
124C4×Dic6—21no—
125C4×Dic6C2311yescharacteristic
126C3×#106—21no—
127C3×#106—41no—
128C3×C4⋊C8C2311yescharacteristic
129C3×C4⋊C8C2311yescharacteristic
130C3×#103—21no—
131C3×#103—21no—
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835D411yescharacteristic
135#832—21no—
136#832D411yescharacteristic
137#830—21no—
138#821—21no—
139#821—41no—
140(C2×D3)⋊C8—21no—
141#816—21no—
142Dic3⋊C8—21no—
143C8×Dic3—21no—
144#810—21no—
145#810—21no—
146#810—21no—
147#810D411yescharacteristic
148#809—21no—
149#809D411yescharacteristic
150#808—21no—
151#808D411yescharacteristic
152#808D411yescharacteristic
153#807—21no—
154#807—21no—
155#807—21no—
156C4⋊(C3⋊C8)—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160C4×C3⋊C8C2311yescharacteristic
161C4×C3⋊C8C2311yescharacteristic
162C22×SD16—61no—
163C22×SD16—61no—
164#557—31no—
165D4×Q8—31no—
166D4×Q8C2×D311yescharacteristic
167D42—31no—
168#501—61no—
169#500—31no—
170#498C2×D311yescharacteristic, complemented
171#492—31no—
172#492—31no—
173#487—31no—
174#487—61no—
175#486—31no—
176#478—61no—
177#477—61no—
178#475—31no—
179#475—61no—
180#475C2×D311yescharacteristic, complemented
181#474—31no—
182#465—61no—
183#465—61no—
184#461—31no—
185#461—31no—
186#461—61no—
187#461C2×D311yescharacteristic
188#460—31no—
189#460—31no—
190#450—61no—
191#450—61no—
192#448—61no—
193#448—61no—
194#446—31no—
195C8×Q8—31no—
196#440—61no—
197C4×SD16—31no—
198C4×SD16—31no—
199C4×SD16—61no—
200C8×D4—31no—
201C8×D4—31no—
202#335—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332C2×D311yescharacteristic, complemented
207#331—31no—
208Q8⋊C8—31no—
209Q8⋊C8C2×D311yescharacteristic, complemented
210D4⋊C8—31no—
211D4⋊C8C2×D311yescharacteristic
212#322—31no—
213C23×D3—41no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—41no—
217C2×C3×Q8—42no—
218C2×C3×Q8—81no—
219C2×C3×Q8C2×D411yescharacteristic
220C2×C3×D4C2×D411yescharacteristic
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C2×#67—41no—
224D3×Q8—42no—
225D3×Q8—81no—
226D3×D4—42no—
227D3×D4—81no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—42no—
232C2×D12—41no—
233C2×D12—81no—
234C2×D12C2×D411yescharacteristic
235C2×C4×D3—21no—
236C2×C4×D3—41no—
237C2×Dic6—21no—
238C2×Dic6—41no—
239C2×Dic6C2×D411yescharacteristic
240C3×SD16—42no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4C2×D411yescharacteristic
251C3×C4⋊C4C2×D411yescharacteristic
252C3×C22⋊C4—21no—
253C3×C22⋊C4—41no—
254C3×C42—21no—
255C3×C42C2411yescharacteristic
256#215—22no—
257#215—42no—
258#215—42no—
259#215—42no—
260#215—42no—
261#215—81no—
262#215—81no—
263#214SD1614yescomplemented
264(C2×D3)⋊C4—21no—
265(C2×D3)⋊C4—41no—
266(C2×D3)⋊C4—41no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3C2×D411yescharacteristic
270C4⋊Dic3C2×D411yescharacteristic
271Dic3⋊C4—21no—
272Dic3⋊C4—41no—
273Dic3⋊C4—41no—
274Dic3⋊C4—41no—
275Dic3⋊C4—41no—
276C4×Dic3—21no—
277C4×Dic3—41no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—42no—
281C2×C3⋊C8C2×D411yescharacteristic
282C2×C3⋊C8C2×D411yescharacteristic
283C2×C3⋊C8C2×D411yescharacteristic
284C2×C3⋊C8C2×D411yescharacteristic
285#204—42no—
286#204—42no—
287D3×C8—42no—
288C22×Q8—21no—
289C22×Q8—61no—
290C22×D4—61no—
291C22×D4—61no—
292C2×SD16—21no—
293C2×SD16—31no—
294C2×SD16—31no—
295C2×SD16—61no—
296C2×SD16—61no—
297C2×SD16—61no—
298C2×SD16—61no—
299C2×SD16—61no—
300C2×SD16—61no—
301C2×SD16—61no—
302C2×SD16—61no—
303C2×SD16—124no—
304C2×SD16—124no—
305C22×C8—61no—
306C22×C8—61no—
307C4⋊Q8—21no—
308C4⋊Q8—31no—
309C4⋊Q8—61no—
310C4⋊Q8—61no—
311C4⋊Q8C22×D311yescharacteristic
312C4⋊Q8C22×D311yescharacteristic
313#127—31no—
314#125—61no—
315#125—121no—
316C22⋊Q8—21no—
317C22⋊Q8—41no—
318C22⋊Q8—61no—
319C22⋊Q8—121no—
320#121—61no—
321#121—61no—
322#120—121no—
323C4×Q8—31no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×Q8C22×D311yescharacteristic
327C4×D4—21no—
328C4×D4—31no—
329C4×D4—61no—
330C4×D4C22×D311yescharacteristic
331#107—61no—
332#107—61no—
333#107—61no—
334#106—21no—
335#106—31no—
336#106—31no—
337#106—41no—
338#106—61no—
339C4⋊C8—31no—
340C4⋊C8—31no—
341C4⋊C8—61no—
342C4⋊C8—61no—
343C4⋊C8C22×D311yescharacteristic
344C4⋊C8C22×D311yescharacteristic
345#103—21no—
346#103—21no—
347#103—31no—
348#103—31no—
349#103—61no—
350#103—61no—
351#103—61no—
352#103—61no—
353#103—61no—
354#102—31no—
355#102—31no—
356#102—61no—
357#102—61no—
358#102—61no—
359#102—121no—
360C22⋊C8—61no—
361C22⋊C8—61no—
362#97—61no—
363C4×C8—31no—
364C4×C8—31no—
365C4×C8—61no—
366C4×C8C22×D311yescharacteristic
367C23×C3—21no—
368C22×D3—21no—
369C22×D3—41no—
370C22×D3—82no—
371C22×D3—82no—
372C22×D3—161no—
373C3×Q8—21no—
374C3×Q8—21no—
375C3×Q8—41no—
376C3×Q8—41no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—82no—
380C3×D4—21no—
381C3×D4C2×SD1612yes—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—82no—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393#67—82no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398D12—21no—
399D12—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403D12—81no—
404D12—82no—
405D12AGL(1,ℤ/8)12yes—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—82no—
409Dic6—21no—
410Dic6—41no—
411Dic6—41no—
412Dic6—82no—
413Dic6C2×SD1612yescomplemented
414C3×C8—41no—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—21no—
418C3⋊C8—21no—
419C3⋊C8—21no—
420C3⋊C8—41no—
421C3⋊C8—41no—
422C3⋊C8C2×SD1612yescomplemented
423C24—121no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—31no—
427C2×Q8—41no—
428C2×Q8—42no—
429C2×Q8—61no—
430C2×Q8—81no—
431C2×Q8—121no—
432C2×Q8—122no—
433C2×Q8—241no—
434C2×Q8D3×D411yescharacteristic
435C2×D4—31no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—121no—
440C2×D4—122no—
441C2×D4—121no—
442C2×D4—122no—
443C2×D4—241no—
444C2×D4—241no—
445C2×D4D3×D411yescharacteristic
446C22×C4—21no—
447C22×C4—41no—
448C22×C4—61no—
449C22×C4—121no—
450SD16—34no—
451SD16—42no—
452SD16—62no—
453SD16—122no—
454SD16—122no—
455SD16—122no—
456SD16—122no—
457SD16—122no—
458SD16—122no—
459SD16—241no—
460SD16—241no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—31no—
465C2×C8—31no—
466C2×C8—31no—
467C2×C8—31no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—122no—
471C2×C8—122no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—31no—
478C4⋊C4—31no—
479C4⋊C4—41no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—121no—
484C4⋊C4—121no—
485C4⋊C4—121no—
486C4⋊C4—121no—
487C4⋊C4D3×D411yescharacteristic
488C4⋊C4D3×D411yescharacteristic
489C22⋊C4—21no—
490C22⋊C4—41no—
491C22⋊C4—61no—
492C22⋊C4—121no—
493C22⋊C4—121no—
494C42—21no—
495C42—61no—
496C42—121no—
497C42C23×D311yescharacteristic
498C22×C3—22no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3D4211yescharacteristic
502C2×D3—22no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—164no—
509C3×C4—21no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4C22×SD1611yescharacteristic
516C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
517C3×C4#57711yescharacteristic
518C3×C4#57711yescharacteristic
519Dic3—22no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523C23—21no—
524C23—61no—
525C23—121no—
526C23—242no—
527C23—242no—
528C23—481no—
529Q8—21no—
530Q8—21no—
531Q8—32no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—61no—
537Q8—82no—
538Q8—121no—
539Q8—121no—
540Q8—242no—
541D4—21no—
542D4—32no—
543D4—61no—
544D4—121no—
545D4—121no—
546D4—121no—
547D4—121no—
548D4—241no—
549D4—242no—
550D4—242no—
551D4D3×SD1612yes—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—82no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—242no—
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4#100411yescharacteristic
570C8—32no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—61no—
575C8—61no—
576C8—61no—
577C8—121no—
578C8—121no—
579C2×C3—42no—
580C2×C3#326511yescharacteristic
581C2×C3D4×SD1611yescharacteristic
582C2×C3#328911yescharacteristic
583D3—42no—
584D3—161no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—62no—
589C22—121no—
590C22—121no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22—484no—
595C22#559011yescharacteristic
596C4—21no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—62no—
601C4—81no—
602C4—81no—
603C4—121no—
604C4—121no—
605C4—241no—
606C4C2×D3×SD1611yescharacteristic
607C4#576211yescharacteristic
608C4D3×AGL(1,ℤ/8)11yescharacteristic
609C4#577811yescharacteristic
610C3#1976411yescharacteristic, complemented, Sylow
611C2—42no—
612C2—122no—
613C2—481no—
614C2#8006311yescharacteristic
615C2#8015611yescharacteristic
616C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1651C2
socleC22×C3D42
prime radicalD4×D12C22
hypercenter#1651D3
hypocenterC3#19764
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1651

Complements

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

NH ≅ G/Ntype
2. #83558C2semidirect
3. #81037C2semidirect
5. C3×#1651C2semidirect
6. #71342C2semidirect
7. #71335C2semidirect
8. #71120C2semidirect
9. #71038C2semidirect
10. #68464C2semidirect
11. #68443C2semidirect
12. #68265C2semidirect
13. #68216C2semidirect
15. #68150C2semidirect
21. #5567C22semidirect
23. C3×#498C22semidirect
24. C3×#475C22semidirect
28. #5094C22semidirect
29. #5093C22semidirect
35. #5078C22semidirect
40. #5024C22semidirect
51. #4830C22semidirect
57. C3×#332C22semidirect
58. C3×Q8⋊C8C22semidirect
60. #4547C22semidirect
61. #4486C22semidirect
63. #4480C22semidirect
64. #4479C22semidirect
66. #4456C22semidirect
67. #4454C22semidirect
78. #1651D3semidirect
170. #498C2×D3semidirect
180. #475C2×D3semidirect
206. #332C2×D3semidirect
209. Q8⋊C8C2×D3semidirect
263. #214SD16semidirect
413. Dic6C2×SD16semidirect
422. C3⋊C8C2×SD16semidirect
610. C3#19764semidirect

Maximal subgroups

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

subgroupindex
D4×#2152
#835582
#810372
#804482
C3×#16512
#713422
#713352
#711202
#710382
#684642
#684432
#682652
#682162
#681822
#681502
#197643