Finite Group Databasebeta

← #10765610

Subgroups — #10765610

Order 768.

568 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82534C211yesmaximal, characteristic
2#82521C211yesmaximal, characteristic
3#79316C211yesmaximal, characteristic, complemented
4C3×#1488C211yesmaximal, characteristic
5#70797C211yesmaximal, characteristic, complemented
6#70768C211yesmaximal, characteristic, complemented
7#70744C211yesmaximal, characteristic, complemented
8#68227C211yesmaximal, characteristic, complemented
9#68225C211yesmaximal, characteristic, complemented
10#68222C211yesmaximal, characteristic, complemented
11#68219C211yesmaximal, characteristic, complemented
12#68162C211yesmaximal, characteristic, complemented
13#68161C211yesmaximal, characteristic, complemented
14#68159C211yesmaximal, characteristic, complemented
15#68155C211yesmaximal, characteristic, complemented
16#17949—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5470C2211yescharacteristic
19C3×#432C2211yescharacteristic
20C3×#431C2211yescharacteristic
21#5239C411yescharacteristic
22#5239C411yescharacteristic
23#5031C411yescharacteristic
24C4×#216C2211yescharacteristic
25C4×#216C2211yescharacteristic
26#5024C2211yescharacteristic
27#5024C2211yescharacteristic
28#5020C411yescharacteristic
29#5019C411yescharacteristic
30C4×#214C2211yescharacteristic
31C4×#214C2211yescharacteristic
32#5012C2211yescharacteristic
33#5012C2211yescharacteristic
34#5011—21no—
35#5011—21no—
36#5007C411yescharacteristic
37#4996C411yescharacteristic
38#4996C411yescharacteristic
39#4926—21no—
40#4925C2211yescharacteristic
41#4925C2211yescharacteristic
42#4924—21no—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4831C2211yescharacteristic
45#4831C2211yescharacteristic
46#4830C2211yescharacteristic
47#4830C2211yescharacteristic
48#4720—21no—
49#4720—21no—
50#4701—21no—
51#4701—21no—
52C3×Q8⋊C8C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×D4⋊C8C2211yescharacteristic
56#4480C2211yescharacteristic
57#4480C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4456C2211yescharacteristic
61#4456C2211yescharacteristic
62#4456C2211yescharacteristic
63#4456C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3107—31no—
69#2989—31no—
70#2985—31no—
71#1593—31no—
72#1592—31no—
73#1591—31no—
74#1590—31no—
75#1588—31no—
76#1587—31no—
77#1584—31no—
78#1583—31no—
79#1573—31no—
80#1488D311yescharacteristic
81#1481—31no—
82#1454—31no—
83C2×C3×#41C2×C411yescharacteristic
84C3×C4×Q8C2×C411yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86C3×C4×D4—21no—
87C3×C4×D4—21no—
88C3×C4×D4C2×C411yescharacteristic
89C3×C4×D4C2311yescharacteristic
90C3×#117—21no—
91C3×#117C2×C411yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#216C2×C411yescharacteristic
95C2×#216C2×C411yescharacteristic
96C2×#214C2×C411yescharacteristic
97C2×#214C2×C411yescharacteristic
98#926—21no—
99#924—21no—
100#924C2×C411yescharacteristic
101#924C2×C411yescharacteristic
102C2×#208—21no—
103C2×#208—21no—
104C22×C3⋊C8—21no—
105#870—21no—
106#869C2×C411yescharacteristic
107#869C2×C411yescharacteristic
108C4×Dic6—21no—
109C4×Dic6C2311yescharacteristic
110C4×Dic6C2311yescharacteristic
111C3×C4⋊C8—21no—
112C3×C4⋊C8C2311yescharacteristic
113C3×C4⋊C8C2311yescharacteristic
114C3×C22⋊C8—21no—
115C3×C22⋊C8—21no—
116C3×#97—21no—
117C3×C4×C8C2311yescharacteristic
118C3×C4×C8C2311yescharacteristic
119#835C2×C411yescharacteristic
120#835C2×C411yescharacteristic
121#832C2×C411yescharacteristic
122#832C2×C411yescharacteristic
123#830—21no—
124#830—21no—
125#830C2×C411yescharacteristic
126#830C2×C411yescharacteristic
127#830D411yescharacteristic
128#830D411yescharacteristic
129#815—21no—
130#815—21no—
131Dic3⋊C8—21no—
132Dic3⋊C8—21no—
133Dic3⋊C8—21no—
134Dic3⋊C8—21no—
135C8×Dic3—21no—
136C8×Dic3—21no—
137#810C2×C411yescharacteristic
138#810C2×C411yescharacteristic
139#810C2×C411yescharacteristic
140#810C2×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#533—31no—
161C4×#41C2×D311yescharacteristic
162#485—31no—
163#485—31no—
164#475—31no—
165#474—31no—
166#462—31no—
167#461—31no—
168#452—31no—
169#452—31no—
170#446—31no—
171#446—31no—
172#446—31no—
173#446—31no—
174#446—61no—
175#446—61no—
176C8×Q8—31no—
177C8×Q8—31no—
178C4×Q16—31no—
179C4×Q16—31no—
180C4×SD16—31no—
181C4×SD16—31no—
182#436—31no—
183#436—31no—
184#435—61no—
185#435—61no—
186#433—61no—
187#433—61no—
188#432—61no—
189#432C2×D311yescharacteristic
190#431—31no—
191#431—31no—
192#431C2×D311yescharacteristic
193#423—61no—
194C2×C4⋊C8—31no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203Q8⋊C8C2×D311yescharacteristic
204Q8⋊C8C2×D311yescharacteristic
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×Dic6—21no—
223C2×Dic6C22×C411yescharacteristic
224C2×Dic6C22×C411yescharacteristic
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C22×C411yescharacteristic
232C3×C4⋊C4C22×C411yescharacteristic
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42C22×C411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42#4111yescharacteristic
242C3×C42C2411yescharacteristic
243#217—21no—
244#217—21no—
245#216—41no—
246#216—41no—
247#214—41no—
248#214—41no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3C22×C411yescharacteristic
253C4⋊Dic3C22×C411yescharacteristic
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260#208—81no—
261#208—81no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—81no—
265C2×C3⋊C8C22×C411yescharacteristic
266C2×C3⋊C8C22×C411yescharacteristic
267C2×C3⋊C8C22×C411yescharacteristic
268C2×C3⋊C8C22×C411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×#41C4×D311yescharacteristic
274C2×Q16—31no—
275C2×Q16—31no—
276C2×SD16—31no—
277C2×SD16—31no—
278C2×M16—61no—
279C2×M16—61no—
280C22×C8—61no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283#125—61no—
284C22⋊Q8—31no—
285C22⋊Q8—31no—
286C22⋊Q8—61no—
287C4×Q8—31no—
288C4×Q8—31no—
289C4×Q8—61no—
290C4×Q8C4×D311yescharacteristic
291C4×Q8C22×D311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C4×D311yescharacteristic
295C4×D4C22×D311yescharacteristic
296#117—21no—
297#117—61no—
298#117C4×D311yescharacteristic
299C2×C42—21no—
300C2×C42C22×D311yescharacteristic
301#107—31no—
302#107—31no—
303#106—31no—
304#106—31no—
305C4⋊C8—21no—
306C4⋊C8—31no—
307C4⋊C8—31no—
308C4⋊C8—31no—
309C4⋊C8—31no—
310C4⋊C8—31no—
311C4⋊C8—31no—
312C4⋊C8—61no—
313C4⋊C8—61no—
314C4⋊C8—61no—
315C4⋊C8—61no—
316C4⋊C8—61no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321#103—31no—
322#103—31no—
323#103—31no—
324#103—31no—
325#103—31no—
326#103—31no—
327#102—31no—
328#102—31no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—31no—
332C22⋊C8—31no—
333C22⋊C8—31no—
334C22⋊C8—31no—
335C22⋊C8—61no—
336C22⋊C8—61no—
337#97—21no—
338#97—61no—
339#97—61no—
340#97—61no—
341#97—61no—
342C4×C8—31no—
343C4×C8—31no—
344C4×C8—31no—
345C4×C8—31no—
346C4×C8—61no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C23×C3—21no—
351C23×C3C4×D411yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—42no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—81no—
369C2×C3×C4C4×D411yescharacteristic
370C2×C3×C4C4×D411yescharacteristic
371C2×C3×C4C4×D411yescharacteristic
372C2×C3×C4C23×C411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—21no—
383Dic6—81no—
384C3×C8—41no—
385C3×C8—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—41no—
399#41—41no—
400C2×Q8—31no—
401C2×Q8—31no—
402C2×Q8—61no—
403C2×Q8C2×C4×D311yescharacteristic
404C2×D4—21no—
405C2×D4C2×C4×D311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4C2×C4×D311yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4#23711yescharacteristic
412Q16—121no—
413Q16—121no—
414SD16—121no—
415SD16—121no—
416M16—241no—
417M16—241no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—31no—
429C2×C8—31no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—241no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4C2×C4×D311yescharacteristic
445C4⋊C4C2×C4×D311yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—61no—
455C42—61no—
456C42C2×C4×D311yescharacteristic
457C42D3×D411yescharacteristic
458C42#23711yescharacteristic
459C42C23×D311yescharacteristic
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3C2×C4×D411yescharacteristic
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4#56811yescharacteristic
475C3×C4#56811yescharacteristic
476C3×C4#57711yescharacteristic
477C3×C4#57811yescharacteristic
478Dic3—41no—
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482C23—21no—
483C23#67⋊C411yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—61no—
487Q8—61no—
488Q8—61no—
489Q8—61no—
490Q8—241no—
491D4—21no—
492D4—21no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—81no—
510C2×C4#67⋊C411yescharacteristic
511C2×C4#67⋊C411yescharacteristic
512C2×C4#67⋊C411yescharacteristic
513C2×C4C22×C4×D311yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×#23711yescharacteristic
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—61no—
521C8—61no—
522C8—61no—
523C8—61no—
524C8—61no—
525C8—61no—
526C8—61no—
527C8—61no—
528C8—121no—
529C8—121no—
530C2×C3—41no—
531C2×C3—81no—
532C2×C3#293011yescharacteristic
533C2×C3#294711yescharacteristic
534C2×C3#295711yescharacteristic
535C22—21no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22C2×#67⋊C411yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—121no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4#574511yescharacteristic
554C4#574511yescharacteristic
555C4#577011yescharacteristic
556C4#577111yescharacteristic
557C3#1794911yescharacteristic, complemented, Sylow
558C2—41no—
559C2—81no—
560C2#7862711yescharacteristic
561C2#7882111yescharacteristic
562C2#7899011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1488C2
socleC22×C3C2×C4×D4
prime radicalC2×C3×#41C2×C4
cosocleC3×C42C24
hypercenter#1488D3
hypocenterC3#17949
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
3. #79316C2semidirect
5. #70797C2semidirect
6. #70768C2semidirect
7. #70744C2semidirect
8. #68227C2semidirect
9. #68225C2semidirect
10. #68222C2semidirect
11. #68219C2semidirect
12. #68162C2semidirect
13. #68161C2semidirect
14. #68159C2semidirect
15. #68155C2semidirect
557. C3#17949semidirect

Maximal subgroups

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

subgroupindex
#825342
#825212
#793162
C3×#14882
#707972
#707682
#707442
#682272
#682252
#682222
#682192
#681622
#681612
#681592
#681552
#179493