Finite Group Databasebeta

← #10765299

Subgroups — #10765299

Order 768.

598 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#79965C211yesmaximal, characteristic, complemented
2#79906C211yesmaximal, characteristic
3#79747C211yesmaximal, characteristic
4C3×#1584C211yesmaximal, characteristic, complemented
5#70954C211yesmaximal, characteristic, complemented
6#70935C211yesmaximal, characteristic, complemented
7#70902C211yesmaximal, characteristic, complemented
8#68574C211yesmaximal, characteristic, complemented
9#68558C211yesmaximal, characteristic, complemented
10#68237C211yesmaximal, characteristic
11#68232C211yesmaximal, characteristic, complemented
12#68213C211yesmaximal, characteristic, complemented
13#68212C211yesmaximal, characteristic, complemented
14#68079C211yesmaximal, characteristic, complemented
15#68074C211yesmaximal, characteristic, complemented
16#17921—31nomaximal, Sylow
17#5578C2211yescharacteristic
18C4×#239C2211yescharacteristic
19C3×#446C2211yescharacteristic
20C3×C8×Q8C2211yescharacteristic
21C3×C4×Q16C2211yescharacteristic
22#5187C411yescharacteristic
23#5182C411yescharacteristic
24C4×#216C2211yescharacteristic
25C4×#215C2211yescharacteristic
26Q8×C3⋊C8C2211yescharacteristic
27Q8×C3⋊C8C2211yescharacteristic
28#4890C411yescharacteristic
29#4888C411yescharacteristic
30#4840—21no—
31#4835C2211yescharacteristic
32#4834—21no—
33#4834C2211yescharacteristic
34D3×C4⋊C8—21no—
35D3×C4⋊C8C2211yescharacteristic
36#4831—21no—
37#4830C2211yescharacteristic
38#4818C411yescharacteristic
39#4815C411yescharacteristic
40#4808C411yescharacteristic
41#4791C411yescharacteristic
42#4709—21no—
43#4701—21no—
44C4×#204C2211yescharacteristic
45C8×D12—21no—
46C8×D12C2211yescharacteristic
47#4685—21no—
48#4685C2211yescharacteristic
49#4685C2211yescharacteristic
50#4685C2211yescharacteristic
51#4679C2211yescharacteristic
52C3×#335C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×Q8⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56#4535C2211yescharacteristic
57#4535C2211yescharacteristic
58#4484C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4479C2211yescharacteristic
62#4459C2211yescharacteristic
63#4457C2211yescharacteristic
64#4456C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3107—31no—
69#2988—31no—
70#2985—31no—
71#1587—31no—
72#1585—31no—
73#1584D311yescharacteristic, complemented
74#1582—31no—
75#1579—31no—
76SD16⋊C8—31no—
77C8×Q16—31no—
78C8×SD16—31no—
79#1573—31no—
80#1488—31no—
81#1481—31no—
82#1454—31no—
83C2×#239C2×C411yescharacteristic
84C2×C3×Q16C2×C411yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86C3×C4×Q8C2311yescharacteristic
87#946C2×C411yescharacteristic
88Q8×Dic3C2×C411yescharacteristic
89#944C2×C411yescharacteristic
90C2×#216C2×C411yescharacteristic
91C2×#215C2×C411yescharacteristic
92C2×#204C2×C411yescharacteristic
93C2×D3×C8—21no—
94C2×D3×C8—21no—
95C2×D3×C8—21no—
96#897C2×C411yescharacteristic
97#896—21no—
98#893—21no—
99#893C2×C411yescharacteristic
100#892—21no—
101#892C2×C411yescharacteristic
102#889—21no—
103#888C2×C411yescharacteristic
104C4×D12—21no—
105C4×D12C2311yescharacteristic
106#872—21no—
107C42×D3—21no—
108C42×D3C2311yescharacteristic
109C4×Dic6—21no—
110C4×Dic6C2311yescharacteristic
111C3×#107C2×C411yescharacteristic
112C3×C4⋊C8—21no—
113C3×C4⋊C8—21no—
114C3×C4⋊C8C2311yescharacteristic
115C3×C4⋊C8C2311yescharacteristic
116C3×#103C2×C411yescharacteristic
117C3×#103C2×C411yescharacteristic
118C3×#97—21no—
119C3×C4×C8—21no—
120C3×C4×C8C2311yescharacteristic
121C3×C4×C8C2311yescharacteristic
122C3×C4×C8C2311yescharacteristic
123#835C2×C411yescharacteristic
124#835C2×C411yescharacteristic
125#821C2×C411yescharacteristic
126(C2×D3)⋊C8—21no—
127(C2×D3)⋊C8—21no—
128(C2×D3)⋊C8—21no—
129(C2×D3)⋊C8—21no—
130(C2×D3)⋊C8C2×C411yescharacteristic
131(C2×D3)⋊C8C2×C411yescharacteristic
132(C2×D3)⋊C8D411yescharacteristic
133(C2×D3)⋊C8D411yescharacteristic
134#817C2×C411yescharacteristic
135#816C2×C411yescharacteristic
136#815—21no—
137#815—21no—
138Dic3⋊C8—21no—
139Dic3⋊C8—21no—
140Dic3⋊C8—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8C2×C411yescharacteristic
143Dic3⋊C8C2×C411yescharacteristic
144Dic3⋊C8D411yescharacteristic
145Dic3⋊C8D411yescharacteristic
146C8×Dic3—21no—
147#810C2×C411yescharacteristic
148#809C2×C411yescharacteristic
149#808C2×C411yescharacteristic
150#807C2×C411yescharacteristic
151C4⋊(C3⋊C8)—21no—
152C4⋊(C3⋊C8)—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4×C3⋊C8—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161C4×#41—31no—
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—61no—
172#446—61no—
173#446C2×D311yescharacteristic
174C8×Q8—31no—
175C8×Q8—31no—
176C8×Q8—31no—
177C8×Q8C2×D311yescharacteristic
178C4×Q16—31no—
179C4×Q16C2×D311yescharacteristic
180C4×SD16—31no—
181C4×SD16—31no—
182C8×D4—31no—
183C8×D4—31no—
184C8×D4—61no—
185C8×D4—61no—
186#433—61no—
187#433—61no—
188#432—31no—
189#431—31no—
190#431—31no—
191#431—31no—
192#431—61no—
193C2×C4⋊C8—31no—
194C2×C4⋊C8—61no—
195#335—31no—
196#335C2×D311yescharacteristic
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⋊C8—31no—
206D4⋊C8—31no—
207#322—31no—
208#322—31no—
209#322C2×D311yescharacteristic
210C82—31no—
211C2×C3×Q8C22×C411yescharacteristic
212C2×C3×Q8C22×C411yescharacteristic
213#239—22no—
214#239—22no—
215C2×D12—21no—
216C2×D12C22×C411yescharacteristic
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3C22×C411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3#4111yescharacteristic
223C2×Dic6—21no—
224C2×Dic6C22×C411yescharacteristic
225C3×Q16—22no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8C22×C411yescharacteristic
230C2×C3×C8C22×C411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4C22×C411yescharacteristic
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C42—21no—
238C3×C42—21no—
239C3×C42C2411yescharacteristic
240#216—22no—
241#215—22no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3C22×C411yescharacteristic
250C4⋊Dic3C22×C411yescharacteristic
251Dic3⋊C4—21no—
252Dic3⋊C4—21no—
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256C4×Dic3—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
269#204—22no—
270D3×C8—81no—
271D3×C8—81no—
272D3×C8—81no—
273C2×#41—31no—
274C2×Q16—31no—
275C2×Q16C4×D311yescharacteristic
276C2×SD16—31no—
277C2×SD16—31no—
278C22×C8—61no—
279C22×C8—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×Q8C22×D311yescharacteristic
291C4×Q8C22×D311yescharacteristic
292C4×D4—31no—
293C4×D4—31no—
294C4×D4—61no—
295C4×D4—61no—
296#117—31no—
297#117—61no—
298#117—61no—
299C2×C42—31no—
300C2×C42—61no—
301#107—31no—
302#107C4×D311yescharacteristic
303#106—31no—
304#106—31no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—31no—
308C4⋊C8—31no—
309C4⋊C8—31no—
310C4⋊C8—31no—
311C4⋊C8—31no—
312C4⋊C8—31no—
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#103C4×D311yescharacteristic
326#103C4×D311yescharacteristic
327#102—31no—
328#102—31no—
329C22⋊C8—31no—
330C22⋊C8—31no—
331C22⋊C8—31no—
332C22⋊C8—31no—
333C22⋊C8—61no—
334C22⋊C8—61no—
335C22⋊C8—61no—
336C22⋊C8—61no—
337#97—21no—
338#97—61no—
339#97—61no—
340C4×C8—21no—
341C4×C8—31no—
342C4×C8—31no—
343C4×C8—31no—
344C4×C8—61no—
345C4×C8—61no—
346C4×C8—61no—
347C4×C8C22×D311yescharacteristic
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C22×D3—21no—
351C22×D3C4×D411yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8#13112yes—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4C23×C411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3C4×D411yescharacteristic
367C2×Dic3C4×D411yescharacteristic
368C2×Dic3C4×D411yescharacteristic
369D12—24no—
370D12—21no—
371D12#13112yes—
372C4×D3—24no—
373C4×D3—21no—
374C4×D3—21no—
375C4×D3—21no—
376C4×D3—21no—
377C4×D3—41no—
378C4×D3—41no—
379C4×D3—41no—
380C4×D3—81no—
381Dic6—21no—
382Dic6—21no—
383Dic6—81no—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—62no—
399#41—62no—
400C2×Q8—31no—
401C2×Q8—61no—
402C2×Q8C2×C4×D311yescharacteristic
403C2×Q8C2×C4×D311yescharacteristic
404C2×D4—31no—
405C2×D4—61no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412Q16—22no—
413Q16—62no—
414SD16—62no—
415SD16—62no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—31no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—241no—
427C2×C8—241no—
428C2×C8—241no—
429C2×C8C2×C4×D311yescharacteristic
430C2×C8C2×C4×D311yescharacteristic
431C2×C8D3×D411yescharacteristic
432C2×C8D3×D411yescharacteristic
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—61no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C42—21no—
452C42—21no—
453C42—31no—
454C42—31no—
455C42—31no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42C23×D311yescharacteristic
460C22×C3C2×C4×D411yescharacteristic
461C2×D3—21no—
462C2×D3—22no—
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—41no—
466C2×D3—41no—
467C3×C4—22no—
468C3×C4—21no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4C2×#13111yescharacteristic
473C3×C4C2×#13111yescharacteristic
474C3×C4#57711yescharacteristic
475C3×C4#57811yescharacteristic
476Dic3—21no—
477Dic3—21no—
478Dic3—21no—
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482C23—31no—
483C23—61no—
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—61no—
488Q8—61no—
489Q8—241no—
490Q8#90712yes—
491D4—32no—
492D4—64no—
493D4—61no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—64no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—241no—
513C2×C4C22×C4×D311yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—61no—
524C8—61no—
525C8—61no—
526C8—61no—
527C8—121no—
528C8—121no—
529C8—121no—
530C2×C3#292911yescharacteristic
531C2×C3#294711yescharacteristic
532C2×C3#295711yescharacteristic
533D3—41no—
534D3—42no—
535C22—61no—
536C22—61no—
537C22—62no—
538C22—61no—
539C22—121no—
540C22—121no—
541C22C4×D3×D411yescharacteristic
542C4—21no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—61no—
548C4—61no—
549C4—61no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4C2×#90711yescharacteristic
554C4D3×#13111yescharacteristic
555C4#576611yescharacteristic
556C4#577011yescharacteristic
557C3#1792111yescharacteristic, complemented, Sylow
558C2—121no—
559C2—122no—
560C2#7947811yescharacteristic
561C2#7971911yescharacteristic
562C2#7989911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1584C2
socleC22×C3C2×C4×D4
prime radicalC2×#239C2×C4
cosocleC3×C42C24
hypercenter#1584D3
hypocenterC3#17921
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
1. #79965C2semidirect
4. C3×#1584C2semidirect
5. #70954C2semidirect
6. #70935C2semidirect
7. #70902C2semidirect
8. #68574C2semidirect
9. #68558C2semidirect
11. #68232C2semidirect
12. #68213C2semidirect
13. #68212C2semidirect
14. #68079C2semidirect
15. #68074C2semidirect
73. #1584D3semidirect
557. C3#17921semidirect

Maximal subgroups

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

subgroupindex
#799652
#799062
#797472
C3×#15842
#709542
#709352
#709022
#685742
#685582
#682372
#682322
#682132
#682122
#680792
#680742
#179213