Finite Group Databasebeta

← #10766177

Subgroups — #10766177

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#83140C211yesmaximal, characteristic
2#83129C211yesmaximal, characteristic, complemented
3#79328C211yesmaximal, characteristic, complemented
4C3×#1494C211yesmaximal, characteristic, complemented
5#70796C211yesmaximal, characteristic, complemented
6#69734C211yesmaximal, characteristic, complemented
7#69725C211yesmaximal, characteristic, complemented
8#68227C211yesmaximal, characteristic, complemented
9#68221C211yesmaximal, characteristic, complemented
10#68220C211yesmaximal, characteristic, complemented
11#68214C211yesmaximal, characteristic, complemented
12#68162C211yesmaximal, characteristic, complemented
13#68158C211yesmaximal, characteristic, complemented
14#68156C211yesmaximal, characteristic, complemented
15#68151C211yesmaximal, characteristic, complemented
16#18002—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#432C2211yescharacteristic
20C2×C3×C4×C8C2211yescharacteristic
21#5066C411yescharacteristic
22#5063C411yescharacteristic
23#5057C411yescharacteristic
24#5054C411yescharacteristic
25#5047C411yescharacteristic
26#5045C411yescharacteristic
27#5037C411yescharacteristic
28#5035C411yescharacteristic
29C4×#216C2211yescharacteristic
30C4×#215C2211yescharacteristic
31#5024C2211yescharacteristic
32Q8×C3⋊C8C2211yescharacteristic
33C4×#214C2211yescharacteristic
34C4×#213C2211yescharacteristic
35#5012—21no—
36#5012C2211yescharacteristic
37#5011—21no—
38D4×C3⋊C8C2211yescharacteristic
39#5000—21no—
40#4997—21no—
41#4925C2211yescharacteristic
42#4925C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4834C2211yescharacteristic
45#4834C2211yescharacteristic
46#4831C2211yescharacteristic
47#4831C2211yescharacteristic
48#4728—21no—
49#4728—21no—
50#4705—21no—
51D3×#97—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#4456C2211yescharacteristic
63#4456C2211yescharacteristic
64#4455C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67#4454C2211yescharacteristic
68#3102—31no—
69#2970—31no—
70#2969—31no—
71#1593—31no—
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#1494D311yescharacteristic, complemented
81#1476—31no—
82#1451—31no—
83C2×#235C2×C411yescharacteristic
84C22×C3×C8—21no—
85C3×#126—21no—
86C3×#125C2×C411yescharacteristic
87C3×#124C2×C411yescharacteristic
88C3×#123—21no—
89C3×C22⋊Q8C2×C411yescharacteristic
90C3×#121C2×C411yescharacteristic
91C3×C4×Q8C2311yescharacteristic
92C3×C4×D4—21no—
93C3×C4×D4C2311yescharacteristic
94C3×#117—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#216C2×C411yescharacteristic
97C2×#215C2×C411yescharacteristic
98C2×#214C2×C411yescharacteristic
99C2×#213C2×C411yescharacteristic
100C4×#67—21no—
101C4×#67—21no—
102#926C2×C411yescharacteristic
103C2×#208—21no—
104C2×#208D411yescharacteristic
105C2×#208D411yescharacteristic
106C22×C3⋊C8—21no—
107C2×D3×C8—21no—
108C2×D3×C8—21no—
109C4×D12C2×C411yescharacteristic
110C4×D12C2311yescharacteristic
111#872—21no—
112C42×D3—21no—
113C4×Dic6C2×C411yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×C4⋊C8C2311yescharacteristic
116C3×C4⋊C8C2311yescharacteristic
117C3×C22⋊C8—21no—
118C3×#97—21no—
119C3×C4×C8—21no—
120C3×C4×C8C2311yescharacteristic
121C3×C4×C8C2311yescharacteristic
122#835C2×C411yescharacteristic
123#835C2×C411yescharacteristic
124#832C2×C411yescharacteristic
125#832C2×C411yescharacteristic
126#830—21no—
127#830—21no—
128#830—21no—
129#830C2×C411yescharacteristic
130#830C2×C411yescharacteristic
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8—21no—
133#815—21no—
134#815—21no—
135Dic3⋊C8—21no—
136Dic3⋊C8—21no—
137#810C2×C411yescharacteristic
138#810C2×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)C2×C411yescharacteristic
148C4⋊(C3⋊C8)C2×C411yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151#803—21no—
152#803D411yescharacteristic
153#803D411yescharacteristic
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529C2×D311yescharacteristic
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#446—31no—
172#446—31no—
173C8×Q8—31no—
174C4×Q16—31no—
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—61no—
188#432C2×D311yescharacteristic
189#431—31no—
190#431—31no—
191#405—61no—
192C4×M16—31no—
193C2×#97—61no—
194C2×C4×C8C2×D311yescharacteristic
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8C2×D311yescharacteristic
202Q8⋊C8C2×D311yescharacteristic
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322—31no—
210#322—31no—
211C2×C3×Q8C22×C411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4C22×C411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4111yescharacteristic
218C2×#67—21no—
219#235—22no—
220#235—41no—
221C2×D12C22×C411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×Dic6C22×C411yescharacteristic
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—42no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4C22×C411yescharacteristic
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C22⋊C4—21no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C42—21no—
243C3×C42C22×C411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42#4111yescharacteristic
246C3×C42C2411yescharacteristic
247#217—21no—
248#216—41no—
249#215—22no—
250#214—41no—
251#213—22no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254C4⋊Dic3C22×C411yescharacteristic
255C4⋊Dic3C22×C411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261#208—41no—
262#208—41no—
263#208—42no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—42no—
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
275D3×C8—42no—
276D3×C8—42no—
277C2×#41—31no—
278C2×Q16—31no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×D8—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—61no—
285C22×C8—21no—
286C22×C8—61no—
287C22×C8—61no—
288C22×C8—61no—
289#126—21no—
290#125C4×D311yescharacteristic
291#124C4×D311yescharacteristic
292#123—21no—
293C22⋊Q8C4×D311yescharacteristic
294#121C4×D311yescharacteristic
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8C22×D311yescharacteristic
298C4×D4—21no—
299C4×D4—31no—
300C4×D4—31no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4C22×D311yescharacteristic
304#117—21no—
305#117—31no—
306#117—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—31no—
310#107—31no—
311#106—31no—
312#106—31no—
313C4⋊C8—31no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—31no—
324#103—31no—
325#103—31no—
326#103—31no—
327#102—31no—
328#102—31no—
329#102—31no—
330#102—31no—
331C22⋊C8—21no—
332C22⋊C8—31no—
333C22⋊C8—31no—
334C22⋊C8—61no—
335C22⋊C8—61no—
336C22⋊C8—61no—
337C22⋊C8—61no—
338C22⋊C8—61no—
339#97—21no—
340#97—31no—
341#97—31no—
342#97—61no—
343#97—61no—
344#97—61no—
345C4×C8—21no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—61no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C23×C3C4×D411yescharacteristic
356C22×D3—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—81no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4C4×D411yescharacteristic
373C2×C3×C4C4×D411yescharacteristic
374C2×C3×C4C4×D411yescharacteristic
375C2×C3×C4C23×C411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378#67—42no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382D12—21no—
383D12#13112yes—
384C4×D3—42no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—81no—
388Dic6—21no—
389Dic6—21no—
390C3×C8—22no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—21no—
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×Q8—31no—
407C2×Q8C2×C4×D311yescharacteristic
408C2×D4—21no—
409C2×D4—31no—
410C2×D4—61no—
411C2×D4C2×C4×D311yescharacteristic
412C22×C4—21no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4C2×C4×D311yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4#23711yescharacteristic
419Q16—121no—
420SD16—62no—
421SD16—121no—
422D8—62no—
423M16—121no—
424M16—121no—
425M16—122no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—31no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—42no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—122no—
442C2×C8—122no—
443C2×C8—122no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—31no—
449C4⋊C4—31no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4C2×C4×D311yescharacteristic
453C4⋊C4C2×C4×D311yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C42—21no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42C2×C4×D311yescharacteristic
467C42D3×D411yescharacteristic
468C42#23711yescharacteristic
469C42C23×D311yescharacteristic
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3C2×C4×D411yescharacteristic
477C2×D3—22no—
478C2×D3—41no—
479C2×D3—41no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4C2×#13111yescharacteristic
488C3×C4#56811yescharacteristic
489C3×C4#57511yescharacteristic
490C3×C4#57511yescharacteristic
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494C23—21no—
495C23—61no—
496C23#67⋊C411yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—61no—
500Q8—61no—
501D4—21no—
502D4—21no—
503D4—32no—
504D4—61no—
505D4—81no—
506D4—122no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—241no—
524C2×C4#67⋊C411yescharacteristic
525C2×C4#67⋊C411yescharacteristic
526C2×C4#67⋊C411yescharacteristic
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×#23711yescharacteristic
530C8—22no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—61no—
535C8—61no—
536C8—61no—
537C8—61no—
538C8—61no—
539C8—61no—
540C8—61no—
541C8—61no—
542C8—121no—
543C8—121no—
544C2×C3—41no—
545C2×C3—81no—
546C2×C3#293311yescharacteristic
547C2×C3#294311yescharacteristic
548C2×C3#295611yescharacteristic
549D3—42no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—62no—
557C22—121no—
558C22—121no—
559C22C2×#67⋊C411yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4D3×#13111yescharacteristic
571C4#575011yescharacteristic
572C4#576811yescharacteristic
573C4#576811yescharacteristic
574C3#1800211yescharacteristic, complemented, Sylow
575C2—41no—
576C2—81no—
577C2—122no—
578C2#7865911yescharacteristic
579C2#7881911yescharacteristic
580C2#7898011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1494C2
socleC22×C3C2×C4×D4
prime radical#5037C4
cosocleC3×C42C24
hypercenter#1494D3
hypocenterC3#18002
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #83129C2semidirect
3. #79328C2semidirect
4. C3×#1494C2semidirect
5. #70796C2semidirect
6. #69734C2semidirect
7. #69725C2semidirect
8. #68227C2semidirect
9. #68221C2semidirect
10. #68220C2semidirect
11. #68214C2semidirect
12. #68162C2semidirect
13. #68158C2semidirect
14. #68156C2semidirect
15. #68151C2semidirect
80. #1494D3semidirect
574. C3#18002semidirect

Maximal subgroups

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

subgroupindex
#831402
#831292
#793282
C3×#14942
#707962
#697342
#697252
#682272
#682212
#682202
#682142
#681622
#681582
#681562
#681512
#180023