Finite Group Databasebeta

← #11664551

Subgroups — #11664551

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87293C211yesmaximal, characteristic
2#85890C211yesmaximal, characteristic, complemented
3C3×#2898C211yesmaximal, characteristic, complemented
4#84111C211yesmaximal, characteristic, complemented
5#83797C211yesmaximal, characteristic, complemented
6Dic3×#131C211yesmaximal, characteristic, complemented
7#82563C211yesmaximal, characteristic, complemented
8#78954C213yesmaximal, complemented
9#78952C213yesmaximal, complemented
10#78879C213yesmaximal, complemented
11#78876C213yesmaximal, complemented
12#78827C213yesmaximal, complemented
13#78825C213yesmaximal, complemented
14#78785C213yesmaximal, complemented
15C2×#4800C213yesmaximal, complemented
16#60624—31nomaximal, Sylow
17C2×C3×2−1+4C2211yescharacteristic
18C2×#1009C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#520C2211yescharacteristic
21#5832C2213yes—
22#5830C2213yes—
23#5829—23no—
24#5829—23no—
25#5827C2211yescharacteristic
26Dic3×#41C2211yescharacteristic
27C2×#948C2211yescharacteristic
28#5817—21no—
29#5817—21no—
30#5816C2213yes—
31C2×#947C2213yes—
32#5570—21no—
33#5569—21no—
34#5543—23no—
35#5539—23no—
36#5519C2213yescomplemented
37#5517C2213yescomplemented
38#5500C2213yescomplemented
39C2×#890C2213yescomplemented
40C3×#419C2213yescomplemented
41C3×#417C2213yescomplemented
42C3×#416C2213yescomplemented
43C2×C3×#103C2213yescomplemented
44#5233C2213yescomplemented
45#5232C2213yescomplemented
46#5225C2213yescomplemented
47C2×#835C2213yescomplemented
48#5133C2213yes—
49#5131C2213yes—
50#5122C2213yescomplemented
51Dic3×M16C2213yescomplemented
52#5113C2213yes—
53C2×#821C2213yes—
54#5101C2213yescomplemented
55C2×C8×Dic3C2213yescomplemented
56#5002C2213yes—
57#5001C2213yes—
58#4966C2213yes—
59C2×#809C2213yes—
60#4800C2216yescomplemented
61#4800C2216yescomplemented
62#4800C2213yescomplemented
63#4800C2213yescomplemented
64#4800C2216yescomplemented
65#4797C2216yescomplemented
66#4794C2216yescomplemented
67#4794C2216yescomplemented
68#4792C2213yescomplemented
69#4792C2213yescomplemented
70#4768C2213yescomplemented
71#4768C2211yescharacteristic, complemented
72#4764C2216yescomplemented
73#4760C2213yescomplemented
74#4760C2213yescomplemented
75#3504—31no—
76#3500—31no—
77#3142—33no—
78#3140—33no—
79#3137—33no—
80#3137—33no—
81#3136—33no—
82#3136—33no—
83#3131—33no—
84C2×#488—33no—
85#2898—31no—
86#2898D311yescharacteristic, complemented
87#2897—31no—
88#2897—31no—
89C4×#131—31no—
90C3×2−1+4—24no—
91C3×2−1+4—24no—
92C2×C3×#41—23no—
93C2×C3×#41C2313yes—
94C2×C3×#41C2311yescharacteristic
95C22×C3×Q8—21no—
96C22×C3×Q8C2313yes—
97#1009—24no—
98#1009—24no—
99C2×#239—21no—
100C2×D3×D4—23no—
101C2×#235C2313yes—
102C22×D12C2313yes—
103C3×#131—24no—
104C2×C3×M16C2313yes—
105C22×C3×C8C2313yes—
106C3×C4×Q8—21no—
107C3×C4×D4—23no—
108C3×#117C2313yes—
109C2×C3×C4⋊C4C2313yes—
110#948—24no—
111#947—23no—
112#947—23no—
113#947C2313yes—
114#947C2316yes—
115#946—23no—
116#946—23no—
117Q8×Dic3—21no—
118Q8×Dic3D411yescharacteristic
119Q8×Dic3D411yescharacteristic
120#944C2313yes—
121#940C2313yes—
122#939—23no—
123#939—23no—
124#937—26no—
125#936C2311yescharacteristic
126D4×Dic3—23no—
127D4×Dic3D413yes—
128D4×Dic3D413yes—
129#930—23no—
130#930—23no—
131#929—26no—
132C4×#67—23no—
133C4×#67—23no—
134C2×(C2×D3)⋊C4—23no—
135C2×(C2×D3)⋊C4—23no—
136#926C2313yes—
137#926C2313yes—
138C2×C4⋊Dic3—23no—
139C2×C4⋊Dic3C2313yes—
140#924—26no—
141C2×Dic3⋊C4—23no—
142C2×C4×Dic3C2313yes—
143C2×#208C2313yes—
144C22×C3⋊C8C2313yes—
145#890—23no—
146#890—23no—
147#890—23no—
148#890C2313yes—
149#890C2316yes—
150#890C2311yescharacteristic
151#888—23no—
152#888C2313yes—
153#888C2313yes—
154#878—23no—
155#878—23no—
156#878—26no—
157#870—23no—
158#869—23no—
159C4×Dic6—23no—
160C4×Dic6—23no—
161C3×#103C2313yes—
162C3×#103C2316yes—
163C3×#103C2313yes—
164C3×#102C2311yescharacteristic
165C3×#102C2313yes—
166#835C2313yes—
167#835C2316yes—
168#835C2313yes—
169#832C2313yes—
170#832C2311yescharacteristic
171#821C2313yes—
172#821C2316yes—
173#821C2313yes—
174#816C2313yes—
175#816C2311yescharacteristic
176#815C2316yes—
177C8×Dic3C2313yes—
178C8×Dic3C2313yes—
179C8×Dic3C2311yescharacteristic
180C8×Dic3C2313yes—
181#810C2313yes—
182#810C2311yescharacteristic
183#809C2313yes—
184#809C2316yes—
185#809C2313yes—
186C2×2−1+4C2×D311yescharacteristic
187C2×H(5,𝔽2)—31no—
188C2×#131—31no—
189C2×#131C2×D311yescharacteristic
190#557—61no—
191#557—61no—
192#554—63no—
193#554—63no—
194#548—61no—
195#548—61no—
196#546—63no—
197#546—63no—
198#544—33no—
199#544—33no—
200#536—33no—
201#535—33no—
202#533—33no—
203#532—33no—
204C2×#125—33no—
205C2×#124—33no—
206#520—31no—
207#520C2×D311yescharacteristic
208C4×#41—31no—
209#491—36no—
210#490—36no—
211#489—36no—
212#489—36no—
213#488—33no—
214#488—36no—
215#488—36no—
216#488—36no—
217#488—33no—
218#488—31no—
219#488—33no—
220#487—33no—
221#487—33no—
222#486—33no—
223#486—33no—
224#419—33no—
225#419C2×D313yescomplemented
226#418—33no—
227#418—33no—
228#417—33no—
229#417—33no—
230#417—33no—
231#417C2×D313yescomplemented
232#416—33no—
233#416—33no—
234#416—33no—
235#416C2×D313yescomplemented
236C2×#103—33no—
237C2×#103C2×D313yescomplemented
238C2×#102—33no—
239C2×#102—33no—
240#405—33no—
241#405—33no—
242C4×M16—33no—
243C2×C4×C8—33no—
244C23×D3—23no—
245C3×#41—26no—
246C3×#41—26no—
247C3×#41—412no—
248C3×#41#4114yes—
249C3×#41#4114yes—
250C2×C3×Q8—23no—
251C2×C3×Q8—26no—
252C2×C3×Q8—26no—
253C2×C3×Q8—44no—
254C2×C3×Q8C2×D411yescharacteristic
255C2×C3×Q8C2413yes—
256C2×C3×D4—23no—
257C2×C3×D4C2×D413yes—
258C2×C3×D4C2411yescharacteristic
259C22×C3×C4—23no—
260C22×C3×C4—23no—
261C22×C3×C4—23no—
262C22×C3×C4C2413yes—
263C2×#67—23no—
264C2×#67—23no—
265C2×#67—23no—
266C22×Dic3—23no—
267C22×Dic3C2×D413yes—
268C22×Dic3C2×D413yes—
269#239—44no—
270D3×D4—412no—
271#235—26no—
272#235—26no—
273C2×D12—26no—
274C2×D12—26no—
275C2×D12—23no—
276C2×D12C2413yes—
277C2×C4×D3—23no—
278C2×Dic6—23no—
279C2×Dic6—23no—
280C2×Dic6C2411yescharacteristic
281C3×M16—26no—
282C2×C3×C8—26no—
283C2×C3×C8C2411yescharacteristic
284C2×C3×C8C2413yes—
285C3×C4⋊C4—23no—
286C3×C4⋊C4C2411yescharacteristic
287C3×C4⋊C4C2413yes—
288C3×C22⋊C4—23no—
289C3×C42—23no—
290#217—23no—
291#217—21no—
292#217—21no—
293#217C2×D416yes—
294(C2×D3)⋊C4—23no—
295(C2×D3)⋊C4—23no—
296(C2×D3)⋊C4—26no—
297(C2×D3)⋊C4—26no—
298C4⋊Dic3—23no—
299C4⋊Dic3—21no—
300C4⋊Dic3—23no—
301C4⋊Dic3—26no—
302C4⋊Dic3C2×D413yes—
303C4⋊Dic3C2×D413yes—
304C4⋊Dic3C2411yescharacteristic
305C4⋊Dic3C2413yes—
306Dic3⋊C4—21no—
307Dic3⋊C4—23no—
308Dic3⋊C4—23no—
309Dic3⋊C4—23no—
310Dic3⋊C4—26no—
311Dic3⋊C4—23no—
312Dic3⋊C4—23no—
313C4×Dic3—23no—
314C4×Dic3C2×D413yes—
315C4×Dic3C2×D413yes—
316C4×Dic3C2411yescharacteristic
317C4×Dic3C2413yes—
318#208—26no—
319C2×C3⋊C8—26no—
320C2×C3⋊C8C2411yescharacteristic
321C2×C3⋊C8C2413yes—
3222−1+4—24no—
3232−1+4—24no—
324H(5,𝔽2)—64no—
325H(5,𝔽2)—64no—
326C2×#41—23no—
327C2×#41—33no—
328C2×#41—61no—
329C2×#41C22×D313yes—
330C2×#41C22×D311yescharacteristic
331C22×Q8—21no—
332C22×Q8C22×D313yes—
333C22×D4—33no—
334C22×D4—63no—
335#131—24no—
336#131—64no—
337C2×M16—33no—
338C2×M16C22×D313yes—
339C22×C8—33no—
340C22×C8C22×D313yes—
341C4⋊Q8—33no—
342C4⋊Q8—33no—
343C4⋊Q8—33no—
344C4⋊Q8—63no—
345C4⋊Q8—63no—
346#127—33no—
347#125—33no—
348#125—36no—
349#125—31no—
350#125—63no—
351#125—63no—
352#125—63no—
353#125—63no—
354#124—33no—
355#124—36no—
356#124—31no—
357#124—63no—
358#124—63no—
359#123—66no—
360C22⋊Q8—66no—
361C22⋊Q8—63no—
362C22⋊Q8—63no—
363C22⋊Q8—66no—
364C22⋊Q8—63no—
365C22⋊Q8—63no—
366#121—63no—
367#121—63no—
368#121—63no—
369#121—63no—
370#120—66no—
371C4×Q8—21no—
372C4×Q8—31no—
373C4×Q8—31no—
374C4×Q8—63no—
375C4×Q8—63no—
376C4×Q8—61no—
377C4×D4—23no—
378C4×D4—33no—
379C4×D4—33no—
380C4×D4—63no—
381C4×D4—63no—
382C4×D4—63no—
383#117—33no—
384#117—33no—
385#117C22×D313yes—
386C2×C4⋊C4—33no—
387C2×C4⋊C4—63no—
388C2×C4⋊C4—63no—
389C2×C4⋊C4C22×D313yes—
390C2×C22⋊C4—63no—
391C2×C22⋊C4—63no—
392C2×C42—33no—
393#103—33no—
394#103—36no—
395#103—33no—
396#103—31no—
397#103—33no—
398#103—31no—
399#103—33no—
400#103C22×D313yes—
401#103C22×D316yes—
402#103C22×D313yes—
403#102—31no—
404#102—33no—
405#102—33no—
406#102—36no—
407#102—33no—
408#102—33no—
409#102—36no—
410#102—33no—
411#102C22×D311yescharacteristic
412#102C22×D313yes—
413#97—36no—
414C4×C8—31no—
415C4×C8—33no—
416C4×C8—33no—
417C4×C8—33no—
418C23×C3—21no—
419C23×C3C22×D413yes—
420C22×D3—23no—
421C22×D3—46no—
422C22×D3—83no—
423C22×D3—83no—
424C3×Q8—23no—
425C3×Q8—23no—
426C3×Q8—46no—
427C3×Q8#12414yes—
428C3×D4—21no—
429C3×D4—21no—
430C3×D4—46no—
431C3×D4#124112yes—
432C2×C3×C4—23no—
433C2×C3×C4—21no—
434C2×C3×C4—23no—
435C2×C3×C4—46no—
436C2×C3×C4—46no—
437C2×C3×C4—83no—
438C2×C3×C4C22×D413yes—
439C2×C3×C4C2×#4116yes—
440C2×C3×C4C2×#4116yes—
441C2×C3×C4C2511yescharacteristic
442#67—46no—
443#67—83no—
444#67—83no—
445C2×Dic3—21no—
446C2×Dic3—23no—
447C2×Dic3—21no—
448C2×Dic3—43no—
449C2×Dic3—43no—
450C2×Dic3—83no—
451C2×Dic3C22×D411yescharacteristic
452C2×Dic3C22×D411yescharacteristic
453C2×Dic3C22×D413yes—
454C2×Dic3C22×D413yes—
455D12—23no—
456D12—23no—
457D12—46no—
458C4×D3—46no—
459Dic6—21no—
460Dic6—21no—
461Dic6—83no—
462Dic6—83no—
463C3×C8—21no—
464C3×C8—23no—
465C3⋊C8—21no—
466C3⋊C8—23no—
467C24—63no—
468#41—26no—
469#41—26no—
470#41—412no—
471#41—66no—
472#41—66no—
473#41—124no—
474#41#23514yes—
475#41#23714yes—
476C2×Q8—23no—
477C2×Q8—26no—
478C2×Q8—26no—
479C2×Q8—31no—
480C2×Q8—44no—
481C2×Q8—63no—
482C2×Q8—63no—
483C2×Q8D3×D411yescharacteristic
484C2×Q8C23×D313yes—
485C2×D4—23no—
486C2×D4—33no—
487C2×D4—63no—
488C2×D4—63no—
489C2×D4—63no—
490C2×D4—66no—
491C2×D4—63no—
492C2×D4—66no—
493C2×D4—1212no—
494C2×D4D3×D413yes—
495C2×D4C23×D311yescharacteristic
496C22×C4—23no—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—33no—
500C22×C4—33no—
501C22×C4—63no—
502C22×C4—63no—
503C22×C4C23×D313yes—
504M16—26no—
505M16—66no—
506C2×C8—26no—
507C2×C8—31no—
508C2×C8—33no—
509C2×C8—66no—
510C2×C8C23×D311yescharacteristic
511C2×C8C23×D313yes—
512C4⋊C4—23no—
513C4⋊C4—33no—
514C4⋊C4—33no—
515C4⋊C4—33no—
516C4⋊C4—31no—
517C4⋊C4—63no—
518C4⋊C4—63no—
519C4⋊C4—66no—
520C4⋊C4—66no—
521C4⋊C4—63no—
522C4⋊C4—63no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—63no—
526C4⋊C4—63no—
527C4⋊C4—63no—
528C4⋊C4C23×D313yes—
529C4⋊C4C23×D311yescharacteristic
530C22⋊C4—23no—
531C22⋊C4—36no—
532C22⋊C4—63no—
533C22⋊C4—63no—
534C22⋊C4—66no—
535C22⋊C4—66no—
536C22⋊C4—61no—
537C22⋊C4—61no—
538C22⋊C4—63no—
539C42—23no—
540C42—33no—
541C42—33no—
542C42—31no—
543C42—33no—
544C42—63no—
545C22×C3—23no—
546C22×C3—23no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3C2×#12416yes—
551C22×C3C23×D411yescharacteristic
552C2×D3—43no—
553C2×D3—43no—
554C2×D3—43no—
555C2×D3—166no—
556C3×C4—41no—
557C3×C4—43no—
558C3×C4—43no—
559C3×C4C2×#12416yes—
560C3×C4C22×#4111yescharacteristic
561C3×C4C22×#4111yescharacteristic
562Dic3—23no—
563Dic3—23no—
564Dic3—21no—
565Dic3—21no—
566Dic3—41no—
567Dic3—43no—
568Dic3—41no—
569C23—21no—
570C23—63no—
571C23—126no—
572C23—243no—
573C23—243no—
574C23C2×D3×D413yes—
575Q8—23no—
576Q8—23no—
577Q8—46no—
578Q8—61no—
579Q8—61no—
580Q8—243no—
581Q8—243no—
582Q8#88514yes—
583D4—21no—
584D4—21no—
585D4—46no—
586D4—63no—
587D4—63no—
588D4—126no—
589D4—126no—
590D4—243no—
591D4—243no—
592D4#885112yes—
593C2×C4—21no—
594C2×C4—23no—
595C2×C4—23no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—33no—
599C2×C4—33no—
600C2×C4—46no—
601C2×C4—46no—
602C2×C4—61no—
603C2×C4—63no—
604C2×C4—61no—
605C2×C4—83no—
606C2×C4—123no—
607C2×C4—123no—
608C2×C4—126no—
609C2×C4—243no—
610C2×C4C2×#23516yes—
611C2×C4C2×D3×D413yes—
612C2×C4C2×#23716yes—
613C2×C4C24×D311yescharacteristic
614C8—21no—
615C8—23no—
616C8—61no—
617C8—63no—
618C2×C3—26no—
619C2×C3—81no—
620C2×C3C22×#12411yescharacteristic
621C2×C3#358411yescharacteristic
622C2×C3#358411yescharacteristic
623D3—83no—
624C22—23no—
625C22—23no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—123no—
630C22—123no—
631C22—123no—
632C22—486no—
633C22C2×#88516yes—
634C22C22×D3×D411yescharacteristic
635C4—43no—
636C4—41no—
637C4—43no—
638C4—61no—
639C4—63no—
640C4—61no—
641C4—63no—
642C4—121no—
643C4—123no—
644C4—121no—
645C4C2×#88516yes—
646C4C22×#23511yescharacteristic
647C4C22×#23711yescharacteristic
648C3#6062411yescharacteristic, complemented, Sylow
649C2—26no—
650C2—81no—
651C2—243no—
652C2C22×#88511yescharacteristic
653C2#8720611yescharacteristic
654C2#8721011yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2898C2
socleC22×C3C23×D4
prime radical#87293C2
hypercenter#2898D3
hypocenterC3#60624
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2898

Complements

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

NH ≅ G/Ntype
2. #85890C2semidirect
3. C3×#2898C2semidirect
4. #84111C2semidirect
5. #83797C2semidirect
6. Dic3×#131C2semidirect
7. #82563C2semidirect
8. #78954C2semidirect
9. #78952C2semidirect
10. #78879C2semidirect
11. #78876C2semidirect
12. #78827C2semidirect
13. #78825C2semidirect
14. #78785C2semidirect
15. C2×#4800C2semidirect
36. #5519C22semidirect
37. #5517C22semidirect
38. #5500C22semidirect
39. C2×#890C22semidirect
40. C3×#419C22semidirect
41. C3×#417C22semidirect
42. C3×#416C22semidirect
43. C2×C3×#103C22semidirect
44. #5233C22semidirect
45. #5232C22semidirect
46. #5225C22semidirect
47. C2×#835C22semidirect
50. #5122C22semidirect
51. Dic3×M16C22semidirect
54. #5101C22semidirect
55. C2×C8×Dic3C22semidirect
60. #4800C22semidirect
61. #4800C22semidirect
62. #4800C22semidirect
63. #4800C22semidirect
64. #4800C22semidirect
65. #4797C22semidirect
66. #4794C22semidirect
67. #4794C22semidirect
68. #4792C22semidirect
69. #4792C22semidirect
70. #4768C22semidirect
71. #4768C22semidirect
72. #4764C22semidirect
73. #4760C22semidirect
74. #4760C22semidirect
86. #2898D3semidirect
225. #419C2×D3semidirect
231. #417C2×D3semidirect
235. #416C2×D3semidirect
237. C2×#103C2×D3semidirect
648. C3#60624semidirect

Maximal subgroups

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

subgroupindex
#872932
#858902
C3×#28982
#841112
#837972
Dic3×#1312
#825632
#789542
#789522
#788792
#788762
#788272
#788252
#787852
C2×#48002
#606243