Finite Group Databasebeta

← #10765607

Subgroups — #10765607

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#83299C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic, complemented
3#79014C211yesmaximal, characteristic, complemented
4C3×#1480C211yesmaximal, characteristic, complemented
5#70763C211yesmaximal, characteristic, complemented
6#70744C211yesmaximal, characteristic, complemented
7#70423C211yesmaximal, characteristic, complemented
8#68163C211yesmaximal, characteristic, complemented
9#68162C211yesmaximal, characteristic, complemented
10#68161C211yesmaximal, characteristic, complemented
11#68160C211yesmaximal, characteristic, complemented
12#68157C211yesmaximal, characteristic, complemented
13#68156C211yesmaximal, characteristic, complemented
14#68154C211yesmaximal, characteristic, complemented
15#68152C211yesmaximal, characteristic, complemented
16#17948—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#431C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21#5234—21no—
22#5234—21no—
23#5111—21no—
24#5111—21no—
25#5091C411yescharacteristic
26#5084C411yescharacteristic
27#5075C411yescharacteristic
28#5072C411yescharacteristic
29#5037C411yescharacteristic
30#5037C411yescharacteristic
31#5035C411yescharacteristic
32#5035C411yescharacteristic
33C4×#214C2211yescharacteristic
34C4×#214C2211yescharacteristic
35C4×#213C2211yescharacteristic
36C4×#213C2211yescharacteristic
37#5012C2211yescharacteristic
38#5012C2211yescharacteristic
39#5011—21no—
40#5011—21no—
41D4×C3⋊C8C2211yescharacteristic
42D4×C3⋊C8C2211yescharacteristic
43#4926C2211yescharacteristic
44#4925C2211yescharacteristic
45#4925C2211yescharacteristic
46#4838C2211yescharacteristic
47#4838C2211yescharacteristic
48#4837—21no—
49#4835—21no—
50#4831C2211yescharacteristic
51#4831C2211yescharacteristic
52C3×D4⋊C8C2211yescharacteristic
53C3×D4⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×D4⋊C8C2211yescharacteristic
56#4480C2211yescharacteristic
57#4480C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4459C2211yescharacteristic
61#4459C2211yescharacteristic
62#4456C2211yescharacteristic
63#4456C2211yescharacteristic
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#1488—31no—
81#1480D311yescharacteristic, complemented
82#1454—31no—
83C2×C3×#41—21no—
84C2×#235C2×C411yescharacteristic
85C22×C3×C8—21no—
86C3×C4⋊Q8C2×C411yescharacteristic
87C3×#127C2×C411yescharacteristic
88C3×#124—21no—
89C3×#121—21no—
90C3×#121C2×C411yescharacteristic
91C3×#121C2×C411yescharacteristic
92C3×C4×D4—21no—
93C3×C4×D4C2311yescharacteristic
94C3×C4×D4C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#214C2×C411yescharacteristic
97C2×#214C2×C411yescharacteristic
98C2×#213C2×C411yescharacteristic
99C2×#213C2×C411yescharacteristic
100C4×#67—21no—
101C4×#67—21no—
102#926C2×C411yescharacteristic
103C2×#208—21no—
104C2×#208—21no—
105C22×C3⋊C8—21no—
106C22×C3⋊C8—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)—21no—
109C4×D12C2×C411yescharacteristic
110C4×D12C2311yescharacteristic
111#872—21no—
112C42×D3—21no—
113C4×Dic6C2×C411yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×C22⋊C8—21no—
120C3×C4×C8C2311yescharacteristic
121C3×C4×C8C2311yescharacteristic
122#832C2×C411yescharacteristic
123#832C2×C411yescharacteristic
124#832C2×C411yescharacteristic
125#832C2×C411yescharacteristic
126#830—21no—
127#830—21no—
128#830—21no—
129#830—21no—
130#830—21no—
131#830C2×C411yescharacteristic
132#830C2×C411yescharacteristic
133#830D411yescharacteristic
134#830D411yescharacteristic
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137#815—21no—
138#815—21no—
139Dic3⋊C8—21no—
140Dic3⋊C8—21no—
141#810C2×C411yescharacteristic
142#810C2×C411yescharacteristic
143#809C2×C411yescharacteristic
144#809C2×C411yescharacteristic
145#808C2×C411yescharacteristic
146#808C2×C411yescharacteristic
147#807C2×C411yescharacteristic
148#807C2×C411yescharacteristic
149C4⋊(C3⋊C8)C2×C411yescharacteristic
150C4⋊(C3⋊C8)C2×C411yescharacteristic
151C4⋊(C3⋊C8)D411yescharacteristic
152C4⋊(C3⋊C8)D411yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532C2×D311yescharacteristic
161C4×#41—31no—
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—31no—
187#432—61no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#423—61no—
192C2×C4⋊C8C2×D311yescharacteristic
193#408—61no—
194#408—61no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8C2×D311yescharacteristic
204D4⋊C8C2×D311yescharacteristic
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322—31no—
210C82—31no—
211C3×#41—81no—
212C3×#41—81no—
213C2×C3×Q8—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C22×C411yescharacteristic
219C2×C3×D4C22×C411yescharacteristic
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C22×C411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4#4111yescharacteristic
225C2×#67—21no—
226#235—41no—
227#235—41no—
228C2×D12C22×C411yescharacteristic
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×Dic6C22×C411yescharacteristic
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—81no—
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×C22⋊C4—21no—
245C3×C42C22×C411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C3×C42#4111yescharacteristic
248C3×C42C2411yescharacteristic
249#217—21no—
250#214—41no—
251#214—41no—
252#213—41no—
253#213—41no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—21no—
256C4⋊Dic3C22×C411yescharacteristic
257C4⋊Dic3C22×C411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263#208—81no—
264#208—81no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—81no—
268C2×C3⋊C8—81no—
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277ΓL(1,𝔽25)—81no—
278ΓL(1,𝔽25)—81no—
279C2×#41—21no—
280C2×#41—31no—
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—21no—
290C22×C8—61no—
291C22×C8—61no—
292C4⋊Q8C4×D311yescharacteristic
293#127C4×D311yescharacteristic
294#124—21no—
295#121—21no—
296#121C4×D311yescharacteristic
297#121C4×D311yescharacteristic
298C4×Q8—31no—
299C4×Q8—31no—
300C4×D4—21no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—31no—
308#117—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—31no—
312#107—31no—
313#106—31no—
314#106—31no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
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—31no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—31no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347#97—61no—
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×C3—21no—
358C23×C3C4×D411yescharacteristic
359C22×D3—21no—
360C3×Q8—81no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—81no—
366C3×D4—81no—
367C3×D4—81no—
368C3×D4—81no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4C4×D411yescharacteristic
381C2×C3×C4C4×D411yescharacteristic
382C2×C3×C4C4×D411yescharacteristic
383C2×C3×C4C23×C411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386#67—42no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390D12—21no—
391D12—21no—
392C4×D3—42no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—81no—
396Dic6—21no—
397Dic6—21no—
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—81no—
413#41—81no—
414#41—121no—
415#41—121no—
416C2×Q8—21no—
417C2×Q8—31no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—31no—
423C2×D4—61no—
424C2×D4C2×C4×D311yescharacteristic
425C2×D4C2×C4×D311yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—61no—
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—81no—
457C2×C8—241no—
458C2×C8—241no—
459C4⋊C4—21no—
460C4⋊C4—31no—
461C4⋊C4—31no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4C2×C4×D311yescharacteristic
465C4⋊C4C2×C4×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C42—61no—
474C42—61no—
475C42—61no—
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×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3C2×C4×D411yescharacteristic
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#56811yescharacteristic
500C3×C4#56811yescharacteristic
501C3×C4#57611yescharacteristic
502C3×C4#57711yescharacteristic
503Dic3—41no—
504Dic3—41no—
505Dic3—41no—
506C23—21no—
507C23—21no—
508C23—61no—
509C23#67⋊C411yescharacteristic
510Q8—61no—
511Q8—61no—
512Q8—81no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—21no—
517D4—61no—
518D4—61no—
519D4—81no—
520D4—81no—
521D4—81no—
522D4—81no—
523D4—122no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—241no—
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—81no—
565C2×C3#293011yescharacteristic
566C2×C3#294611yescharacteristic
567C2×C3#295711yescharacteristic
568D3—81no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22C2×#67⋊C411yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—41no—
587C4—41no—
588C4—121no—
589C4—121no—
590C4—121no—
591C4#575011yescharacteristic
592C4#575011yescharacteristic
593C4#575711yescharacteristic
594C4#576211yescharacteristic
595C3#1794811yescharacteristic, complemented, Sylow
596C2—41no—
597C2—81no—
598C2—81no—
599C2—241no—
600C2#7865611yescharacteristic
601C2#7872611yescharacteristic
602C2#7899011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1480C2
socleC22×C3C2×C4×D4
prime radical#83299C2
cosocleC3×C42C24
hypercenter#1480D3
hypocenterC3#17948
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #83280C2semidirect
3. #79014C2semidirect
4. C3×#1480C2semidirect
5. #70763C2semidirect
6. #70744C2semidirect
7. #70423C2semidirect
8. #68163C2semidirect
9. #68162C2semidirect
10. #68161C2semidirect
11. #68160C2semidirect
12. #68157C2semidirect
13. #68156C2semidirect
14. #68154C2semidirect
15. #68152C2semidirect
81. #1480D3semidirect
595. C3#17948semidirect

Maximal subgroups

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

subgroupindex
#832992
#832802
#790142
C3×#14802
#707632
#707442
#704232
#681632
#681622
#681612
#681602
#681572
#681562
#681542
#681522
#179483