Finite Group Databasebeta

← #10766178

Subgroups — #10766178

Order 768.

610 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79943C211yesmaximal, characteristic, complemented
2#79942C211yesmaximal, characteristic, complemented
3#79746C211yesmaximal, characteristic
4C3×#1593C211yesmaximal, characteristic, complemented
5#70954C211yesmaximal, characteristic, complemented
6#70954C211yesmaximal, characteristic, complemented
7#70904C211yesmaximal, characteristic, complemented
8#68533C211yesmaximal, characteristic, complemented
9#68521C211yesmaximal, characteristic, complemented
10#68241C211yesmaximal, characteristic, complemented
11#68229C211yesmaximal, characteristic, complemented
12#68226C211yesmaximal, characteristic, complemented
13#68214C211yesmaximal, characteristic, complemented
14#68077C211yesmaximal, characteristic, complemented
15#68070C211yesmaximal, characteristic, complemented
16#18002—31nomaximal, Sylow
17#5580C2211yescharacteristic
18C4×#239C2211yescharacteristic
19C3×#446C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×C4×Q16C2211yescharacteristic
22#5187C411yescharacteristic
23#5185C411yescharacteristic
24C4×#215C2211yescharacteristic
25C4×#215C2211yescharacteristic
26#5024C2211yescharacteristic
27Q8×C3⋊C8C2211yescharacteristic
28#4891C411yescharacteristic
29#4885C411yescharacteristic
30#4840—21no—
31#4838—21no—
32#4838C2211yescharacteristic
33#4836—21no—
34#4835—21no—
35#4835C2211yescharacteristic
36#4834C2211yescharacteristic
37#4833C2211yescharacteristic
38#4819C411yescharacteristic
39#4814C411yescharacteristic
40#4810C411yescharacteristic
41#4796C411yescharacteristic
42#4708—21no—
43#4706—21no—
44#4706—21no—
45D3×#97—21no—
46C4×D24C2211yescharacteristic
47C4×ΓL(1,𝔽25)C2211yescharacteristic
48C8×D12C2211yescharacteristic
49C8×D12C2211yescharacteristic
50#4685C2211yescharacteristic
51C4×D3×C8C2211yescharacteristic
52C3×#334C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×Q8⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56#4535C2211yescharacteristic
57#4535C2211yescharacteristic
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic
60#4479C2211yescharacteristic
61#4479C2211yescharacteristic
62#4459C2211yescharacteristic
63#4459C2211yescharacteristic
64#4457C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67#4454C2211yescharacteristic
68#3102—31no—
69#2970—31no—
70#2969—31no—
71#1593D311yescharacteristic, complemented
72#1592—31no—
73#1587—31no—
74#1586—31no—
75#1585—31no—
76#1585—31no—
77#1580—31no—
78SD16⋊C8—31no—
79#1572—31no—
80#1494—31no—
81#1476—31no—
82#1451—31no—
83C2×#239C2×C411yescharacteristic
84C2×C3×Q16C2×C411yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86C3×C4×Q8C2311yescharacteristic
87#947C2×C411yescharacteristic
88#946C2×C411yescharacteristic
89Q8×Dic3C2×C411yescharacteristic
90C2×#215C2×C411yescharacteristic
91C2×#215C2×C411yescharacteristic
92C2×D24C2×C411yescharacteristic
93C2×ΓL(1,𝔽25)—21no—
94C2×ΓL(1,𝔽25)D411yescharacteristic
95C2×ΓL(1,𝔽25)D411yescharacteristic
96C2×D3×C8—21no—
97C2×D3×C8—21no—
98C2×D3×C8—21no—
99C2×D3×C8—21no—
100#898—21no—
101#895C2×C411yescharacteristic
102#894—21no—
103#893—21no—
104#893C2×C411yescharacteristic
105#892—21no—
106#892C2×C411yescharacteristic
107#890C2×C411yescharacteristic
108C4×D12—21no—
109C4×D12—21no—
110C4×D12C2311yescharacteristic
111C4×D12C2311yescharacteristic
112#872—21no—
113C42×D3—21no—
114C42×D3C2311yescharacteristic
115C3×#107C2×C411yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×#103C2×C411yescharacteristic
121C3×#103C2×C411yescharacteristic
122C3×#97—21no—
123C3×#97—21no—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127#835C2×C411yescharacteristic
128#835C2×C411yescharacteristic
129#821C2×C411yescharacteristic
130#821C2×C411yescharacteristic
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8C2×C411yescharacteristic
138(C2×D3)⋊C8C2×C411yescharacteristic
139#818C2×C411yescharacteristic
140#815—21no—
141#815D411yescharacteristic
142#815D411yescharacteristic
143Dic3⋊C8C2×C411yescharacteristic
144Dic3⋊C8C2×C411yescharacteristic
145C8×Dic3—21no—
146C8×Dic3—21no—
147#809C2×C411yescharacteristic
148#809C2×C411yescharacteristic
149#808C2×C411yescharacteristic
150#808C2×C411yescharacteristic
151C4⋊(C3⋊C8)—21no—
152C4⋊(C3⋊C8)—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529—31no—
161C4×#41—31no—
162#485—31no—
163#482—31no—
164#479—31no—
165#478—31no—
166#464—31no—
167#463—31no—
168#452—31no—
169#449—31no—
170#446—31no—
171#446C2×D311yescharacteristic
172#446C2×D311yescharacteristic
173C8×Q8—31no—
174C4×Q16C2×D311yescharacteristic
175C4×SD16—31no—
176C4×SD16—31no—
177C4×D8—31no—
178#436—31no—
179#436—61no—
180#435—61no—
181#435—61no—
182#435—61no—
183C8×D4—31no—
184C8×D4—31no—
185C8×D4—31no—
186#433—61no—
187#432—31no—
188#432—61no—
189#431—31no—
190#431—31no—
191#405—61no—
192C4×M16—31no—
193C2×#97—61no—
194C2×C4×C8—31no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334C2×D311yescharacteristic
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8C2×D311yescharacteristic
202Q8⋊C8C2×D311yescharacteristic
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8—31no—
207#322—31no—
208#322—31no—
209#322—31no—
210#322C2×D311yescharacteristic
211C2×C3×Q8C22×C411yescharacteristic
212C2×C3×Q8C22×C411yescharacteristic
213#239—22no—
214#239—41no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12C22×C411yescharacteristic
218C2×D12C22×C411yescharacteristic
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3C22×C411yescharacteristic
224C2×C4×D3C2×D411yescharacteristic
225C2×C4×D3#4111yescharacteristic
226C3×Q16—41no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8C22×C411yescharacteristic
231C2×C3×C8C22×C411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C4⋊C4C22×C411yescharacteristic
238C3×C42—21no—
239C3×C42—21no—
240C3×C42C2411yescharacteristic
241#215—22no—
242#215—41no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3C22×C411yescharacteristic
254C4⋊Dic3C22×C411yescharacteristic
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3C22×C411yescharacteristic
260C4×Dic3C2×D411yescharacteristic
261C4×Dic3#4111yescharacteristic
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8C22×C411yescharacteristic
266C2×C3⋊C8C22×C411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269D24—22no—
270ΓL(1,𝔽25)—41no—
271ΓL(1,𝔽25)—41no—
272ΓL(1,𝔽25)—42no—
273D3×C8—42no—
274D3×C8—42no—
275D3×C8—42no—
276D3×C8—42no—
277C2×#41—31no—
278C2×Q16C4×D311yescharacteristic
279C2×SD16—31no—
280C2×SD16—31no—
281C2×D8—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—61no—
285C22×C8—61no—
286C22×C8—61no—
287C22×C8—61no—
288C22×C8—61no—
289#126—61no—
290#125—31no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294#121—31no—
295C4×Q8—31no—
296C4×Q8C22×D311yescharacteristic
297C4×Q8C22×D311yescharacteristic
298C4×D4—31no—
299C4×D4—31no—
300C4×D4—31no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4—61no—
304#117—31no—
305#117—61no—
306#117—61no—
307C2×C42—31no—
308C2×C42—61no—
309#107—31no—
310#107C4×D311yescharacteristic
311#106—31no—
312#106—31no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—31no—
324#103—31no—
325#103C4×D311yescharacteristic
326#103C4×D311yescharacteristic
327#102—31no—
328#102—31no—
329#102—31no—
330#102—31no—
331C22⋊C8—31no—
332C22⋊C8—31no—
333C22⋊C8—61no—
334C22⋊C8—61no—
335C22⋊C8—61no—
336C22⋊C8—61no—
337C22⋊C8—61no—
338C22⋊C8—61no—
339#97—21no—
340#97—21no—
341#97—31no—
342#97—31no—
343#97—61no—
344#97—61no—
345C4×C8—31no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C22×D3—21no—
355C22×D3—21no—
356C22×D3C4×D411yescharacteristic
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4C23×C411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3C4×D411yescharacteristic
371C2×Dic3C4×D411yescharacteristic
372C2×Dic3C4×D411yescharacteristic
373D12—21no—
374D12—21no—
375D12—21no—
376D12—42no—
377D12—81no—
378D12#13112yes—
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—41no—
384C4×D3—42no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—81no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—22no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—21no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404#41—62no—
405#41—121no—
406C2×Q8C2×C4×D311yescharacteristic
407C2×Q8C2×C4×D311yescharacteristic
408C2×D4—31no—
409C2×D4—31no—
410C2×D4—61no—
411C2×D4—61no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419Q16—41no—
420SD16—62no—
421SD16—121no—
422D8—62no—
423M16—121no—
424M16—121no—
425M16—122no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—122no—
437C2×C8—122no—
438C2×C8—122no—
439C2×C8—122no—
440C2×C8C2×C4×D311yescharacteristic
441C2×C8C2×C4×D311yescharacteristic
442C2×C8D3×D411yescharacteristic
443C2×C8D3×D411yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—31no—
447C4⋊C4—31no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4C2×C4×D311yescharacteristic
453C4⋊C4C2×C4×D311yescharacteristic
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C42—21no—
463C42—21no—
464C42—31no—
465C42—31no—
466C42—31no—
467C42—61no—
468C42—61no—
469C42C23×D311yescharacteristic
470C22×C3C2×C4×D411yescharacteristic
471C2×D3—21no—
472C2×D3—22no—
473C2×D3—21no—
474C2×D3—21no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4C2×#13111yescharacteristic
486C3×C4#56811yescharacteristic
487C3×C4#57511yescharacteristic
488C3×C4#57511yescharacteristic
489Dic3—21no—
490Dic3—21no—
491Dic3—21no—
492Dic3—41no—
493Dic3—41no—
494C23—31no—
495C23—61no—
496C23—61no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501D4—32no—
502D4—61no—
503D4—61no—
504D4—61no—
505D4—122no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—121no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—241no—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4D3×#4111yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—62no—
538C8—61no—
539C8—61no—
540C8—61no—
541C8—61no—
542C8—121no—
543C8—121no—
544C2×C3#293311yescharacteristic
545C2×C3#294311yescharacteristic
546C2×C3#295611yescharacteristic
547D3—41no—
548D3—42no—
549D3—81no—
550C22—61no—
551C22—61no—
552C22—62no—
553C22—61no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22C4×D3×D411yescharacteristic
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—61no—
566C4—61no—
567C4—61no—
568C4—121no—
569C4—121no—
570C4#574511yescharacteristic
571C4D3×#13111yescharacteristic
572C4#577711yescharacteristic
573C4#577711yescharacteristic
574C3#1800211yescharacteristic, complemented, Sylow
575C2—121no—
576C2—122no—
577C2—241no—
578C2#7953911yescharacteristic
579C2#7982811yescharacteristic
580C2#7989811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1593C2
socleC22×C3C2×C4×D4
prime radical#4810C4
cosocleC3×C42C24
hypercenter#1593D3
hypocenterC3#18002
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1593

Complements

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

NH ≅ G/Ntype
1. #79943C2semidirect
2. #79942C2semidirect
4. C3×#1593C2semidirect
5. #70954C2semidirect
6. #70954C2semidirect
7. #70904C2semidirect
8. #68533C2semidirect
9. #68521C2semidirect
10. #68241C2semidirect
11. #68229C2semidirect
12. #68226C2semidirect
13. #68214C2semidirect
14. #68077C2semidirect
15. #68070C2semidirect
71. #1593D3semidirect
574. C3#18002semidirect

Maximal subgroups

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

subgroupindex
#799432
#799422
#797462
C3×#15932
#709542
#709042
#685332
#685212
#682412
#682292
#682262
#682142
#680772
#680702
#180023