Finite Group Databasebeta

← #10765686

Subgroups — #10765686

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
1C3×#2993C211yesmaximal, characteristic, complemented
2#83140C211yesmaximal, characteristic
3#79292C211yesmaximal, characteristic, complemented
4#71751C211yesmaximal, characteristic, complemented
5#70952C211yesmaximal, characteristic, complemented
6#70950C211yesmaximal, characteristic, complemented
7#70935C211yesmaximal, characteristic, complemented
8#70933C211yesmaximal, characteristic, complemented
9#70866C211yesmaximal, characteristic, complemented
10#70861C211yesmaximal, characteristic, complemented
11#70844C211yesmaximal, characteristic, complemented
12#70842C211yesmaximal, characteristic, complemented
13#70743C211yesmaximal, characteristic, complemented
14#70497C211yesmaximal, characteristic, complemented
15#70493C211yesmaximal, characteristic, complemented
16#17955—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#446C2211yescharacteristic
20C3×C8×Q8C2211yescharacteristic
21C3×#436—21no—
22C3×#436C2211yescharacteristic
23C3×#435—21no—
24C3×C8×D4C2211yescharacteristic
25C3×#433—21no—
26C3×#431C2211yescharacteristic
27C3×#424—21no—
28C3×#423C2211yescharacteristic
29#5066C411yescharacteristic
30#5063C411yescharacteristic
31#5057C411yescharacteristic
32#5054C411yescharacteristic
33#5047C411yescharacteristic
34#5045C411yescharacteristic
35#5037C411yescharacteristic
36#5035C411yescharacteristic
37C4×#216C2211yescharacteristic
38C4×#215C2211yescharacteristic
39C4×#214C2211yescharacteristic
40C4×#213C2211yescharacteristic
41#4925C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4837—21no—
45#4835—21no—
46#4730—21no—
47#4730—21no—
48#4688C2211yescharacteristic
49#4688C2211yescharacteristic
50#4679C2211yescharacteristic
51#4679C2211yescharacteristic
52#4535C2211yescharacteristic
53#4535C2211yescharacteristic
54D4⋊(C3⋊C8)C2211yescharacteristic
55D4⋊(C3⋊C8)C2211yescharacteristic
56#4484C2211yescharacteristic
57#4484C2211yescharacteristic
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4480C2211yescharacteristic
62#4480C2211yescharacteristic
63#4480C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3102—31no—
69#2993D311yescharacteristic, complemented
70#2989—31no—
71#1594—31no—
72#1593—31no—
73#1590—31no—
74#1590—31no—
75#1587—31no—
76#1586—31no—
77#1583—31no—
78#1582—31no—
79#1570—31no—
80#1489—31no—
81#1483—31no—
82#1453—31no—
83C2×#235C2×C411yescharacteristic
84C2×C3×M16—21no—
85C2×C3×M16—21no—
86C22×C3×C8—21no—
87C3×#126—21no—
88C3×#125C2×C411yescharacteristic
89C3×#124C2×C411yescharacteristic
90C3×#123—21no—
91C3×C22⋊Q8C2×C411yescharacteristic
92C3×#121C2×C411yescharacteristic
93C3×C4×Q8C2311yescharacteristic
94C3×C4×D4—21no—
95C3×C4×D4C2311yescharacteristic
96C3×#117—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#216C2×C411yescharacteristic
99C2×#215C2×C411yescharacteristic
100C2×#214C2×C411yescharacteristic
101C2×#213C2×C411yescharacteristic
102C4×#67—21no—
103C4×#67—21no—
104#926C2×C411yescharacteristic
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⋊C8—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8D411yescharacteristic
119C3×C4⋊C8D411yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8D411yescharacteristic
125C3×C22⋊C8D411yescharacteristic
126C3×#97—21no—
127C3×C4×C8—21no—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#835C2×C411yescharacteristic
131#835C2×C411yescharacteristic
132#832C2×C411yescharacteristic
133#832C2×C411yescharacteristic
134#830C2×C411yescharacteristic
135#830C2×C411yescharacteristic
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138#815—21no—
139#815—21no—
140Dic3⋊C8—21no—
141Dic3⋊C8—21no—
142#810C2×C411yescharacteristic
143#810C2×C411yescharacteristic
144#809C2×C411yescharacteristic
145#809C2×C411yescharacteristic
146#808C2×C411yescharacteristic
147#808C2×C411yescharacteristic
148#807C2×C411yescharacteristic
149#807C2×C411yescharacteristic
150C4⋊(C3⋊C8)—21no—
151C4⋊(C3⋊C8)C2×C411yescharacteristic
152C4⋊(C3⋊C8)C2×C411yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803—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#446C2×D311yescharacteristic
173C8×Q8C2×D311yescharacteristic
174C4×Q16—31no—
175C4×SD16—31no—
176C4×SD16—31no—
177C4×D8—31no—
178#436—21no—
179#436—31no—
180#436—31no—
181#436C2×D311yescharacteristic
182#435—21no—
183#435—61no—
184#435—61no—
185C8×D4C2×D311yescharacteristic
186#433—21no—
187#432—61no—
188#431—31no—
189#431C2×D311yescharacteristic
190#424—21no—
191#423—61no—
192#423C2×D311yescharacteristic
193C2×C4⋊C8—31no—
194C4×M16—31no—
195#334—31no—
196#334—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8—31no—
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—81no—
227C3×M16—81no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—81no—
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C22×C411yescharacteristic
240C3×C4⋊C4C22×C411yescharacteristic
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C42—21no—
247C3×C42C22×C411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42#4111yescharacteristic
250C3×C42C2411yescharacteristic
251#217—21no—
252#216—41no—
253#215—41no—
254#214—41no—
255#213—41no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258C4⋊Dic3C22×C411yescharacteristic
259C4⋊Dic3C22×C411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265#208—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—81no—
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×C3⋊C8C22×C411yescharacteristic
274C2×C3⋊C8C22×C411yescharacteristic
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—21no—
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—21no—
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8#6711yescharacteristic
324C4⋊C8#6711yescharacteristic
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—21no—
336C22⋊C8—21no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8#6711yescharacteristic
342C22⋊C8#6711yescharacteristic
343#97—21no—
344#97—61no—
345#97—61no—
346#97—61no—
347C4×C8—21no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
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—21no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—22no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—41no—
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×C4#23511yescharacteristic
418C22×C4C2×#6711yescharacteristic
419Q16—121no—
420SD16—121no—
421SD16—121no—
422D8—121no—
423M16—81no—
424M16—81no—
425M16—122no—
426M16—241no—
427M16—241no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—81no—
439C2×C8—241no—
440C2×C8C2×#6711yescharacteristic
441C2×C8C2×#6711yescharacteristic
442C2×C8C2×#6711yescharacteristic
443C2×C8C2×#6711yescharacteristic
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
467C42#23511yescharacteristic
468C42C2×#6711yescharacteristic
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#57711yescharacteristic
490C3×C4#57711yescharacteristic
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494C23—21no—
495C23—61no—
496C23C4×#6711yescharacteristic
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×C4C4×#6711yescharacteristic
525C2×C4C4×#6711yescharacteristic
526C2×C4C4×#6711yescharacteristic
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×#23511yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—62no—
537C8—61no—
538C8—61no—
539C8—61no—
540C8—61no—
541C8—121no—
542C8—121no—
543C8—121no—
544C2×C3—41no—
545C2×C3—81no—
546C2×C3#293011yescharacteristic
547C2×C3#295211yescharacteristic
548C2×C3#295611yescharacteristic
549D3—81no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22C2×C4×#6711yescharacteristic
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#583611yescharacteristic
573C4#583711yescharacteristic
574C3#1795511yescharacteristic, complemented, Sylow
575C2—41no—
576C2—81no—
577C2—241no—
578C2#8255811yescharacteristic
579C2#8379511yescharacteristic
580C2#8381111yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2993C2
socleC22×C3C2×C4×D4
prime radical#5037C4
cosocleC3×C42C24
hypercenter#2993D3
hypocenterC3#17955
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
1. C3×#2993C2semidirect
3. #79292C2semidirect
4. #71751C2semidirect
5. #70952C2semidirect
6. #70950C2semidirect
7. #70935C2semidirect
8. #70933C2semidirect
9. #70866C2semidirect
10. #70861C2semidirect
11. #70844C2semidirect
12. #70842C2semidirect
13. #70743C2semidirect
14. #70497C2semidirect
15. #70493C2semidirect
69. #2993D3semidirect
574. C3#17955semidirect

Maximal subgroups

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

subgroupindex
C3×#29932
#831402
#792922
#717512
#709522
#709502
#709352
#709332
#708662
#708612
#708442
#708422
#707432
#704972
#704932
#179553