Finite Group Databasebeta

← #10765681

Subgroups — #10765681

Order 768.

590 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#83131C211yesmaximal, characteristic, complemented
3#79014C211yesmaximal, characteristic, complemented
4C3×#1483C211yesmaximal, characteristic, complemented
5#70422C211yesmaximal, characteristic, complemented
6#69854C211yesmaximal, characteristic, complemented
7#69848C211yesmaximal, characteristic, complemented
8#68226C211yesmaximal, characteristic, complemented
9#68225C211yesmaximal, characteristic, complemented
10#68223C211yesmaximal, characteristic, complemented
11#68218C211yesmaximal, characteristic, complemented
12#68162C211yesmaximal, characteristic, complemented
13#68160C211yesmaximal, characteristic, complemented
14#68156C211yesmaximal, characteristic, complemented
15#68152C211yesmaximal, characteristic, complemented
16#17954—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×C4×M16C2211yescharacteristic
21#5111—21no—
22#5111—21no—
23#5066C411yescharacteristic
24#5063C411yescharacteristic
25#5057C411yescharacteristic
26#5054C411yescharacteristic
27#5047C411yescharacteristic
28#5045C411yescharacteristic
29#5037C411yescharacteristic
30#5035C411yescharacteristic
31C4×#216C2211yescharacteristic
32C4×#215C2211yescharacteristic
33#5024C2211yescharacteristic
34Q8×C3⋊C8C2211yescharacteristic
35C4×#214C2211yescharacteristic
36C4×#213C2211yescharacteristic
37#5012—21no—
38#5012C2211yescharacteristic
39#5011—21no—
40D4×C3⋊C8C2211yescharacteristic
41#5000—21no—
42#4999—21no—
43#4925C2211yescharacteristic
44#4925C2211yescharacteristic
45#4924C2211yescharacteristic
46#4838C2211yescharacteristic
47#4838C2211yescharacteristic
48#4837—21no—
49#4835—21no—
50#4831C2211yescharacteristic
51#4831C2211yescharacteristic
52C3×Q8⋊C8C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×D4⋊C8C2211yescharacteristic
56#4479C2211yescharacteristic
57#4479C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4459C2211yescharacteristic
61#4459C2211yescharacteristic
62#4456C2211yescharacteristic
63#4456C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3102—31no—
69#2993—31no—
70#2989—31no—
71#1592—31no—
72#1592—31no—
73#1591—31no—
74#1589—31no—
75#1588—31no—
76#1585—31no—
77#1584—31no—
78#1581—31no—
79#1571—31no—
80#1489—31no—
81#1483D311yescharacteristic, complemented
82#1453—31no—
83C2×#235C2×C411yescharacteristic
84C2×C3×M16—21no—
85C22×C3×C8—21no—
86C3×#126—21no—
87C3×#125C2×C411yescharacteristic
88C3×#124C2×C411yescharacteristic
89C3×#123—21no—
90C3×C22⋊Q8C2×C411yescharacteristic
91C3×#121C2×C411yescharacteristic
92C3×C4×Q8C2311yescharacteristic
93C3×C4×D4—21no—
94C3×C4×D4C2311yescharacteristic
95C3×#117—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#216C2×C411yescharacteristic
98C2×#215C2×C411yescharacteristic
99C2×#214C2×C411yescharacteristic
100C2×#213C2×C411yescharacteristic
101C4×#67—21no—
102C4×#67—21no—
103#926C2×C411yescharacteristic
104C2×#208—21no—
105C2×#208—21no—
106C22×C3⋊C8—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)—21no—
109C4×D12C2×C411yescharacteristic
110C4×D12C2311yescharacteristic
111#872—21no—
112C42×D3—21no—
113C4×Dic6C2×C411yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×#97—21no—
119C3×C4×C8C2311yescharacteristic
120C3×C4×C8C2311yescharacteristic
121#835C2×C411yescharacteristic
122#835C2×C411yescharacteristic
123#832C2×C411yescharacteristic
124#832C2×C411yescharacteristic
125#830—21no—
126#830—21no—
127#830C2×C411yescharacteristic
128#830C2×C411yescharacteristic
129#830D411yescharacteristic
130#830D411yescharacteristic
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)—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—
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—31no—
180#436—31no—
181#436—61no—
182#435—61no—
183#435—61no—
184#435—61no—
185C8×D4—31no—
186#433—61no—
187#432—61no—
188#431—31no—
189#431—31no—
190#424—61no—
191#423—31no—
192#423—61no—
193C2×C4⋊C8C2×D311yescharacteristic
194C4×M16C2×D311yescharacteristic
195#335—31no—
196#335—31no—
197#335—31no—
198#335—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—
210C82—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—41no—
220#235—41no—
221C2×D12C22×C411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×Dic6C22×C411yescharacteristic
226C3×M16—42no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—81no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C4⋊C4C22×C411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C42—21no—
244C3×C42C22×C411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42#4111yescharacteristic
247C3×C42C2411yescharacteristic
248#217—21no—
249#216—41no—
250#215—41no—
251#214—41no—
252#213—41no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255C4⋊Dic3C22×C411yescharacteristic
256C4⋊Dic3C22×C411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262#208—81no—
263#208—81no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—81no—
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
275ΓL(1,𝔽25)—81no—
276ΓL(1,𝔽25)—81no—
277C2×#41—31no—
278C2×Q16—31no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×D8—31no—
282C2×M16—21no—
283C2×M16—61no—
284C2×M16—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—21no—
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—21no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103—31no—
329#103—31no—
330#103—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102—31no—
335C22⋊C8—31no—
336C22⋊C8—31no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—21no—
344#97—61no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
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—21no—
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—121no—
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—121no—
421SD16—121no—
422D8—121no—
423M16—42no—
424M16—241no—
425M16—241no—
426M16—241no—
427M16—241no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—31no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—31no—
439C2×C8—31no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—81no—
443C2×C8—241no—
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—41no—
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×C4#56811yescharacteristic
488C3×C4#56811yescharacteristic
489C3×C4#57611yescharacteristic
490C3×C4#57811yescharacteristic
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—61no—
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#293011yescharacteristic
547C2×C3#295111yescharacteristic
548C2×C3#295611yescharacteristic
549D3—81no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—121no—
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—
570C4#575011yescharacteristic
571C4#575011yescharacteristic
572C4#577411yescharacteristic
573C4#577911yescharacteristic
574C3#1795411yescharacteristic, complemented, Sylow
575C2—41no—
576C2—81no—
577C2—241no—
578C2#7865611yescharacteristic
579C2#7886811yescharacteristic
580C2#7898011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1483C2
socleC22×C3C2×C4×D4
prime radical#5037C4
cosocleC3×C42C24
hypercenter#1483D3
hypocenterC3#17954
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1483

Complements

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

NH ≅ G/Ntype
2. #83131C2semidirect
3. #79014C2semidirect
4. C3×#1483C2semidirect
5. #70422C2semidirect
6. #69854C2semidirect
7. #69848C2semidirect
8. #68226C2semidirect
9. #68225C2semidirect
10. #68223C2semidirect
11. #68218C2semidirect
12. #68162C2semidirect
13. #68160C2semidirect
14. #68156C2semidirect
15. #68152C2semidirect
81. #1483D3semidirect
574. C3#17954semidirect

Maximal subgroups

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

subgroupindex
#831402
#831312
#790142
C3×#14832
#704222
#698542
#698482
#682262
#682252
#682232
#682182
#681622
#681602
#681562
#681522
#179543