Finite Group Databasebeta

← #10765263

Subgroups — #10765263

Order 768.

596 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#79920C211yesmaximal, characteristic, complemented
2#79908C211yesmaximal, characteristic
3#79723C211yesmaximal, characteristic
4C3×#1579C211yesmaximal, characteristic, complemented
5C8×#216C211yesmaximal, characteristic, complemented
6C8×#215C211yesmaximal, characteristic, complemented
7#70903C211yesmaximal, characteristic, complemented
8#68535C211yesmaximal, characteristic, complemented
9#68520C211yesmaximal, characteristic, complemented
10#68236C211yesmaximal, characteristic, complemented
11#68233C211yesmaximal, characteristic
12#68223C211yesmaximal, characteristic, complemented
13#68222C211yesmaximal, characteristic, complemented
14#68076C211yesmaximal, characteristic, complemented
15#68074C211yesmaximal, characteristic, complemented
16#17919—31nomaximal, Sylow
17#5580C2211yescharacteristic
18C4×D3×Q8C2211yescharacteristic
19C3×C8×Q8C2211yescharacteristic
20C3×C8×Q8C2211yescharacteristic
21C3×C4×Q16C2211yescharacteristic
22#5186C411yescharacteristic
23#5185C411yescharacteristic
24C4×#216C2211yescharacteristic
25C4×#215C2211yescharacteristic
26#5024C2211yescharacteristic
27Q8×C3⋊C8C2211yescharacteristic
28#4882C411yescharacteristic
29#4877C411yescharacteristic
30#4839—21no—
31#4838C2211yescharacteristic
32#4837C2211yescharacteristic
33#4836—21no—
34#4833—21no—
35#4833C2211yescharacteristic
36#4830—21no—
37#4830C2211yescharacteristic
38#4806C411yescharacteristic
39#4805C411yescharacteristic
40#4799C411yescharacteristic
41#4798C411yescharacteristic
42C4×#204C2211yescharacteristic
43#4690—21no—
44#4688—21no—
45C8×D12C2211yescharacteristic
46#4685—21no—
47#4685C2211yescharacteristic
48#4685C2211yescharacteristic
49#4685C2211yescharacteristic
50C8×Dic6—21no—
51C8×Dic6C2211yescharacteristic
52C3×#334C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×Q8⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56#4535C2211yescharacteristic
57#4535C2211yescharacteristic
58#4484C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4479C2211yescharacteristic
62#4459C2211yescharacteristic
63#4458C2211yescharacteristic
64#4456C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3108—31no—
69#2993—31no—
70#2982—31no—
71#1588—31no—
72#1587—31no—
73#1584—31no—
74#1583—31no—
75#1579D311yescharacteristic, complemented
76SD16⋊C8—31no—
77C8×Q16—31no—
78C8×SD16—31no—
79#1574—31no—
80#1487—31no—
81#1482—31no—
82#1454—31no—
83C2×D3×Q8C2×C411yescharacteristic
84C2×C3×Q16C2×C411yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86C3×C4×Q8C2311yescharacteristic
87#947C2×C411yescharacteristic
88#946C2×C411yescharacteristic
89Q8×Dic3C2×C411yescharacteristic
90C2×#216C2×C411yescharacteristic
91C2×#215C2×C411yescharacteristic
92C2×#204C2×C411yescharacteristic
93C2×ΓL(1,𝔽25)—21no—
94C2×ΓL(1,𝔽25)—21no—
95C2×D3×C8—21no—
96#898—21no—
97#895C2×C411yescharacteristic
98#894—21no—
99D3×C4⋊C4—21no—
100D3×C4⋊C4C2×C411yescharacteristic
101#890C2×C411yescharacteristic
102#887—21no—
103#887C2×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×C4×C8—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
136Dic3⋊C8—21no—
137Dic3⋊C8—21no—
138Dic3⋊C8—21no—
139Dic3⋊C8—21no—
140Dic3⋊C8C2×C411yescharacteristic
141Dic3⋊C8C2×C411yescharacteristic
142Dic3⋊C8D411yescharacteristic
143Dic3⋊C8D411yescharacteristic
144C8×Dic3—21no—
145C8×Dic3—21no—
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
155#803—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529—31no—
161C2×C4×Q8—31no—
162#483—31no—
163#480—31no—
164#479—31no—
165#479—31no—
166#464—31no—
167#464—31no—
168#451—31no—
169#448—31no—
170#446—31no—
171C8×Q8—31no—
172C8×Q8—31no—
173C8×Q8—31no—
174C8×Q8—61no—
175C8×Q8—61no—
176C8×Q8C2×D311yescharacteristic
177C8×Q8C2×D311yescharacteristic
178C4×Q16—31no—
179C4×Q16C2×D311yescharacteristic
180C4×SD16—31no—
181C4×SD16—31no—
182#436—31no—
183#436—61no—
184#435—61no—
185C8×D4—31no—
186#433—61no—
187#431—31no—
188#431—31no—
189#431—31no—
190#431—31no—
191#431—61no—
192#431—61no—
193#424—61no—
194#423—31no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334C2×D311yescharacteristic
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
213D3×Q8—22no—
214D3×Q8—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—
270ΓL(1,𝔽25)—81no—
271ΓL(1,𝔽25)—81no—
272D3×C8—81no—
273C22×Q8—31no—
274C2×Q16—31no—
275C2×Q16C4×D311yescharacteristic
276C2×SD16—31no—
277C2×SD16—31no—
278C2×M16—61no—
279C2×M16—61no—
280C22×C8—61no—
281#126—61no—
282#125—31no—
283#124—31no—
284#123—61no—
285C22⋊Q8—31no—
286#121—31no—
287C4×Q8—31no—
288C4×Q8—31no—
289C4×Q8—31no—
290C4×Q8—61no—
291C4×Q8—61no—
292C4×Q8C22×D311yescharacteristic
293C4×Q8C22×D311yescharacteristic
294C4×D4—31no—
295C4×D4—61no—
296#117—61no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—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—61no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—31no—
341C4×C8—31no—
342C4×C8—31no—
343C4×C8—61no—
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—21no—
370D12—21no—
371D12—81no—
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—24no—
382Dic6—21no—
383Dic6#13112yes—
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—
398C2×Q8—31no—
399C2×Q8—61no—
400C2×Q8—62no—
401C2×Q8—62no—
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—
416M16—241no—
417M16—241no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
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—21no—
463C2×D3—21no—
464C2×D3—41no—
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—22no—
478Dic3—21no—
479Dic3—21no—
480Dic3—41no—
481Dic3—41no—
482C23—31no—
483C23—61no—
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—32no—
488Q8—64no—
489Q8—61no—
490Q8#90112yes—
491D4—61no—
492D4—61no—
493D4—241no—
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—81no—
535C22—61no—
536C22—61no—
537C22—61no—
538C22—121no—
539C22—121no—
540C22—121no—
541C22C4×D3×D411yescharacteristic
542C4—21no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—61no—
548C4—61no—
549C4—62no—
550C4—61no—
551C4—121no—
552C4—121no—
553C4C2×#90111yescharacteristic
554C4D3×#13111yescharacteristic
555C4#577811yescharacteristic
556C4#577911yescharacteristic
557C3#1791911yescharacteristic, complemented, Sylow
558C2—121no—
559C2—241no—
560C2#7942111yescharacteristic
561C2#7983011yescharacteristic
562C2#7989911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1579C2
socleC22×C3C2×C4×D4
prime radical#895C2×C4
cosocleC3×C42C24
hypercenter#1579D3
hypocenterC3#17919
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
1. #79920C2semidirect
4. C3×#1579C2semidirect
5. C8×#216C2semidirect
6. C8×#215C2semidirect
7. #70903C2semidirect
8. #68535C2semidirect
9. #68520C2semidirect
10. #68236C2semidirect
12. #68223C2semidirect
13. #68222C2semidirect
14. #68076C2semidirect
15. #68074C2semidirect
75. #1579D3semidirect
557. C3#17919semidirect

Maximal subgroups

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

subgroupindex
#799202
#799082
#797232
C3×#15792
C8×#2162
C8×#2152
#709032
#685352
#685202
#682362
#682332
#682232
#682222
#680762
#680742
#179193