Finite Group Databasebeta

← #10765595

Subgroups — #10765595

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82529C211yesmaximal, characteristic
2#82521C211yesmaximal, characteristic, complemented
3#79338C211yesmaximal, characteristic, complemented
4C3×#1488C211yesmaximal, characteristic, complemented
5#70797C211yesmaximal, characteristic, complemented
6#70758C211yesmaximal, characteristic, complemented
7#70744C211yesmaximal, characteristic, complemented
8#68226C211yesmaximal, characteristic, complemented
9#68224C211yesmaximal, characteristic, complemented
10#68220C211yesmaximal, characteristic, complemented
11#68216C211yesmaximal, characteristic, complemented
12#68163C211yesmaximal, characteristic, complemented
13#68160C211yesmaximal, characteristic, complemented
14#68158C211yesmaximal, characteristic, complemented
15#68153C211yesmaximal, characteristic, complemented
16#17948—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5477C2211yescharacteristic
19C3×#432C2211yescharacteristic
20C3×#431C2211yescharacteristic
21#5238C411yescharacteristic
22#5238C411yescharacteristic
23#5027C411yescharacteristic
24C4×#215C2211yescharacteristic
25C4×#215C2211yescharacteristic
26#5024C2211yescharacteristic
27#5024C2211yescharacteristic
28#5021C411yescharacteristic
29#5015C411yescharacteristic
30C4×#213C2211yescharacteristic
31C4×#213C2211yescharacteristic
32#5012C2211yescharacteristic
33#5012C2211yescharacteristic
34#5011—21no—
35#5011—21no—
36#5008C411yescharacteristic
37#5003C411yescharacteristic
38#5003C411yescharacteristic
39#4926—21no—
40#4925C2211yescharacteristic
41#4925C2211yescharacteristic
42#4924—21no—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4838C2211yescharacteristic
45#4838C2211yescharacteristic
46#4834C2211yescharacteristic
47#4834C2211yescharacteristic
48#4727—21no—
49#4727—21no—
50#4706—21no—
51#4706—21no—
52C3×Q8⋊C8C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×D4⋊C8C2211yescharacteristic
56#4480C2211yescharacteristic
57#4480C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4459C2211yescharacteristic
61#4459C2211yescharacteristic
62#4459C2211yescharacteristic
63#4459C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3106—31no—
69#2989—31no—
70#2981—31no—
71#1594—31no—
72#1592—31no—
73#1590—31no—
74#1589—31no—
75#1586—31no—
76#1585—31no—
77#1582—31no—
78#1581—31no—
79#1573—31no—
80#1488D311yescharacteristic, complemented
81#1480—31no—
82#1454—31no—
83C2×C3×#41C2×C411yescharacteristic
84C2×#235—21no—
85C3×C4×Q8C2×C411yescharacteristic
86C3×C4×Q8C2311yescharacteristic
87C3×C4×D4—21no—
88C3×C4×D4—21no—
89C3×C4×D4C2×C411yescharacteristic
90C3×C4×D4C2311yescharacteristic
91C3×#117—21no—
92C3×#117C2×C411yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#215C2×C411yescharacteristic
96C2×#215C2×C411yescharacteristic
97C2×#213C2×C411yescharacteristic
98C2×#213C2×C411yescharacteristic
99#930—21no—
100#930C2×C411yescharacteristic
101#930C2×C411yescharacteristic
102C2×#208—21no—
103C2×#208—21no—
104C22×C3⋊C8—21no—
105C2×ΓL(1,𝔽25)—21no—
106C2×ΓL(1,𝔽25)—21no—
107C2×D3×C8—21no—
108C2×D3×C8—21no—
109#875—21no—
110#874C2×C411yescharacteristic
111C4×D12—21no—
112C4×D12C2311yescharacteristic
113C4×D12C2311yescharacteristic
114#869C2×C411yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C22⋊C8—21no—
119C3×C22⋊C8—21no—
120C3×#97—21no—
121C3×C4×C8C2311yescharacteristic
122C3×C4×C8C2311yescharacteristic
123#835C2×C411yescharacteristic
124#835C2×C411yescharacteristic
125#832C2×C411yescharacteristic
126#832C2×C411yescharacteristic
127#830—21no—
128#830—21no—
129#830C2×C411yescharacteristic
130#830C2×C411yescharacteristic
131#830D411yescharacteristic
132#830D411yescharacteristic
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137#809C2×C411yescharacteristic
138#809C2×C411yescharacteristic
139#809C2×C411yescharacteristic
140#809C2×C411yescharacteristic
141#808C2×C411yescharacteristic
142#808C2×C411yescharacteristic
143#807C2×C411yescharacteristic
144#807C2×C411yescharacteristic
145C4⋊(C3⋊C8)—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2×C411yescharacteristic
149C4⋊(C3⋊C8)C2×C411yescharacteristic
150C4⋊(C3⋊C8)D411yescharacteristic
151C4⋊(C3⋊C8)D411yescharacteristic
152C4⋊(C3⋊C8)C2311yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154#803—21no—
155#803—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C4×#41C2×D311yescharacteristic
162#482—31no—
163#482—31no—
164#477—31no—
165#476—31no—
166#460—31no—
167#459—31no—
168#449—31no—
169#449—31no—
170#446—31no—
171#446—31no—
172C4×SD16—31no—
173C4×SD16—31no—
174C4×D8—31no—
175C4×D8—31no—
176#436—31no—
177#436—31no—
178#436—31no—
179#436—31no—
180#435—61no—
181#435—61no—
182#435—61no—
183#435—61no—
184C8×D4—31no—
185C8×D4—31no—
186#432—61no—
187#432C2×D311yescharacteristic
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#423—61no—
192C2×C4⋊C8—31no—
193#408—61no—
194#408—61no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8C2×D311yescharacteristic
200Q8⋊C8C2×D311yescharacteristic
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322—31no—
210C82—31no—
211C3×#41—41no—
212C3×#41—41no—
213C2×C3×Q8C22×C411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C22×C411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C22×C411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×#67—21no—
223C2×#67—21no—
224#235—81no—
225#235—81no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12C22×C411yescharacteristic
229C2×D12C22×C411yescharacteristic
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×Dic6—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C22×C411yescharacteristic
240C3×C4⋊C4C22×C411yescharacteristic
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42C22×C411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42#4111yescharacteristic
250C3×C42C2411yescharacteristic
251#215—41no—
252#215—41no—
253#213—41no—
254#213—41no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C22×C411yescharacteristic
261C4⋊Dic3C22×C411yescharacteristic
262#208—81no—
263#208—81no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—81no—
267C2×C3⋊C8C22×C411yescharacteristic
268C2×C3⋊C8C22×C411yescharacteristic
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275ΓL(1,𝔽25)—81no—
276ΓL(1,𝔽25)—81no—
277D3×C8—81no—
278D3×C8—81no—
279C2×#41—61no—
280C2×#41C4×D311yescharacteristic
281C2×SD16—31no—
282C2×SD16—31no—
283C2×D8—31no—
284C2×D8—31no—
285C2×M16—61no—
286C2×M16—61no—
287C2×M16—61no—
288C2×M16—61no—
289C22×C8—61no—
290C22×C8—61no—
291C22×C8—61no—
292C4⋊Q8—31no—
293#127—31no—
294#124—61no—
295#121—31no—
296#121—31no—
297#121—61no—
298C4×Q8C4×D311yescharacteristic
299C4×Q8C22×D311yescharacteristic
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—31no—
303C4×D4—31no—
304C4×D4—61no—
305C4×D4C4×D311yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—21no—
308#117C4×D311yescharacteristic
309C2×C42—21no—
310C2×C42C22×D311yescharacteristic
311#107—31no—
312#107—31no—
313#106—31no—
314#106—31no—
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103—31no—
329#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102—31no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—31no—
340C22⋊C8—31no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347#97—21no—
348#97—61no—
349#97—61no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—31no—
353C4×C8—31no—
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C23×C3C4×D411yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—42no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—81no—
377C2×C3×C4C4×D411yescharacteristic
378C2×C3×C4C4×D411yescharacteristic
379C2×C3×C4C4×D411yescharacteristic
380C2×C3×C4C23×C411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383#67—81no—
384#67—81no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387D12—21no—
388D12—21no—
389D12—21no—
390D12—21no—
391D12—81no—
392D12—81no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—41no—
396C4×D3—41no—
397Dic6—81no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—21no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—41no—
413#41—41no—
414#41—241no—
415#41—241no—
416C2×Q8—61no—
417C2×Q8C2×C4×D311yescharacteristic
418C2×D4—21no—
419C2×D4—31no—
420C2×D4—31no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4C2×C4×D311yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4C2×C4×D311yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4#23711yescharacteristic
434SD16—121no—
435SD16—121no—
436D8—121no—
437D8—121no—
438M16—241no—
439M16—241no—
440M16—241no—
441M16—241no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—31no—
451C2×C8—31no—
452C2×C8—31no—
453C2×C8—31no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—241no—
457C2×C8—241no—
458C2×C8—241no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—31no—
462C4⋊C4—31no—
463C4⋊C4—61no—
464C4⋊C4C2×C4×D311yescharacteristic
465C4⋊C4C2×C4×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42C2×C4×D311yescharacteristic
477C42D3×D411yescharacteristic
478C42#23711yescharacteristic
479C42C23×D311yescharacteristic
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C4×D411yescharacteristic
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4#56811yescharacteristic
501C3×C4#56811yescharacteristic
502C3×C4#57611yescharacteristic
503C3×C4#57711yescharacteristic
504Dic3—41no—
505Dic3—41no—
506C23—21no—
507C23—61no—
508C23—61no—
509C23#67⋊C411yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—241no—
513D4—21no—
514D4—21no—
515D4—42no—
516D4—61no—
517D4—61no—
518D4—61no—
519D4—61no—
520D4—241no—
521D4—241no—
522D4—241no—
523D4—241no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—81no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4#67⋊C411yescharacteristic
543C2×C4#67⋊C411yescharacteristic
544C2×C4#67⋊C411yescharacteristic
545C2×C4C22×C4×D311yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×#23711yescharacteristic
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—61no—
553C8—61no—
554C8—61no—
555C8—61no—
556C8—61no—
557C8—61no—
558C8—61no—
559C8—61no—
560C8—121no—
561C8—121no—
562C2×C3—41no—
563C2×C3—81no—
564C2×C3#293011yescharacteristic
565C2×C3#294611yescharacteristic
566C2×C3#295711yescharacteristic
567D3—81no—
568D3—81no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22C2×#67⋊C411yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—121no—
590C4—121no—
591C4#574511yescharacteristic
592C4#574511yescharacteristic
593C4#576911yescharacteristic
594C4#577011yescharacteristic
595C3#1794811yescharacteristic, complemented, Sylow
596C2—41no—
597C2—81no—
598C2—241no—
599C2—241no—
600C2#7862711yescharacteristic
601C2#7882011yescharacteristic
602C2#7899011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1488C2
socleC22×C3C2×C4×D4
prime radical#82529C2
cosocleC3×C42C24
hypercenter#1488D3
hypocenterC3#17948
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
2. #82521C2semidirect
3. #79338C2semidirect
4. C3×#1488C2semidirect
5. #70797C2semidirect
6. #70758C2semidirect
7. #70744C2semidirect
8. #68226C2semidirect
9. #68224C2semidirect
10. #68220C2semidirect
11. #68216C2semidirect
12. #68163C2semidirect
13. #68160C2semidirect
14. #68158C2semidirect
15. #68153C2semidirect
80. #1488D3semidirect
595. C3#17948semidirect

Maximal subgroups

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

subgroupindex
#825292
#825212
#793382
C3×#14882
#707972
#707582
#707442
#682262
#682242
#682202
#682162
#681632
#681602
#681582
#681532
#179483