Finite Group Databasebeta

← #10765694

Subgroups — #10765694

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#79299C211yesmaximal, characteristic
2#79295C211yesmaximal, characteristic, complemented
3#79292C211yesmaximal, characteristic, complemented
4C3×#1453C211yesmaximal, characteristic, complemented
5#70749C211yesmaximal, characteristic, complemented
6#69854C211yesmaximal, characteristic, complemented
7#69852C211yesmaximal, characteristic, complemented
8#68089C211yesmaximal, characteristic, complemented
9#68087C211yesmaximal, characteristic, complemented
10#68085C211yesmaximal, characteristic, complemented
11#68083C211yesmaximal, characteristic, complemented
12#68081C211yesmaximal, characteristic, complemented
13#68079C211yesmaximal, characteristic, complemented
14#68055C211yesmaximal, characteristic, complemented
15#68055C211yesmaximal, characteristic, complemented
16#17955—31nomaximal, Sylow
17#5479C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#431C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×C4×M16C2211yescharacteristic
22#4924C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4847C411yescharacteristic
25#4845C411yescharacteristic
26#4844C411yescharacteristic
27#4840—21no—
28#4839—21no—
29#4837—21no—
30#4835—21no—
31#4829C411yescharacteristic
32#4737C411yescharacteristic
33#4734C411yescharacteristic
34#4733C411yescharacteristic
35#4730—21no—
36#4730—21no—
37#4729—21no—
38#4728—21no—
39#4723C411yescharacteristic
40C4×Dic12C2211yescharacteristic
41C4×D24C2211yescharacteristic
42C4×#204C2211yescharacteristic
43C4×#204C2211yescharacteristic
44#4688C2211yescharacteristic
45#4688C2211yescharacteristic
46#4688C2211yescharacteristic
47C8×D12C2211yescharacteristic
48#4679C2211yescharacteristic
49#4679C2211yescharacteristic
50#4679C2211yescharacteristic
51C8×Dic6C2211yescharacteristic
52C3×#322C2211yescharacteristic
53C3×#322C2211yescharacteristic
54C3×#322C2211yescharacteristic
55C3×C82C2211yescharacteristic
56#4459C2211yescharacteristic
57#4459C2211yescharacteristic
58#4459C2211yescharacteristic
59#4459C2211yescharacteristic
60#4457C2211yescharacteristic
61#4457C2211yescharacteristic
62#4457C2211yescharacteristic
63#4457C2211yescharacteristic
64#4456C2211yescharacteristic
65#4456C2211yescharacteristic
66#4456C2211yescharacteristic
67#4456C2211yescharacteristic
68#3102—31no—
69#2993—31no—
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#1453D311yescharacteristic, complemented
83C2×#235C2×C411yescharacteristic
84C2×C3×M16—21no—
85C2×C3×C42C2311yescharacteristic
86#930C2×C411yescharacteristic
87#929—21no—
88C4×#67—21no—
89C4×#67—21no—
90C4×#67—21no—
91#926C2×C411yescharacteristic
92#924C2×C411yescharacteristic
93C2×#208—21no—
94C22×C3⋊C8—21no—
95C2×Dic12C2×C411yescharacteristic
96C2×D24C2×C411yescharacteristic
97C2×#204C2×C411yescharacteristic
98C2×#204C2×C411yescharacteristic
99C2×ΓL(1,𝔽25)—21no—
100C2×ΓL(1,𝔽25)—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×D3×C8—21no—
103#876—21no—
104#875C2×C411yescharacteristic
105C4×D12C2×C411yescharacteristic
106C4×D12C2311yescharacteristic
107C4×D12C2311yescharacteristic
108#872—21no—
109#872—21no—
110C42×D3—21no—
111#870C2×C411yescharacteristic
112C4×Dic6C2×C411yescharacteristic
113C4×Dic6C2311yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×C4⋊C8C2×C411yescharacteristic
116C3×C4⋊C8C2×C411yescharacteristic
117C3×C4⋊C8D411yescharacteristic
118C3×C4⋊C8D411yescharacteristic
119C3×C22⋊C8C2×C411yescharacteristic
120C3×C22⋊C8C2×C411yescharacteristic
121C3×C22⋊C8D411yescharacteristic
122C3×C22⋊C8D411yescharacteristic
123C3×#97—21no—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#821C2×C411yescharacteristic
131#821C2×C411yescharacteristic
132#821C2×C411yescharacteristic
133#821C2×C411yescharacteristic
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138#818C2×C411yescharacteristic
139#818C2×C411yescharacteristic
140#817C2×C411yescharacteristic
141#817C2×C411yescharacteristic
142#816C2×C411yescharacteristic
143#816C2×C411yescharacteristic
144#816C2×C411yescharacteristic
145#816C2×C411yescharacteristic
146#815—21no—
147#815—21no—
148#815—21no—
149Dic3⋊C8—21no—
150Dic3⋊C8—21no—
151Dic3⋊C8—21no—
152Dic3⋊C8—21no—
153C8×Dic3—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#529—31no—
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#431C2×D311yescharacteristic
189#431C2×D311yescharacteristic
190#424—61no—
191#423—31no—
192#423—61no—
193C2×C4⋊C8—31no—
194C4×M16C2×D311yescharacteristic
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#322C2×D311yescharacteristic
208#322C2×D311yescharacteristic
209#322C2×D311yescharacteristic
210C82C2×D311yescharacteristic
211C22×C3×C4C22×C411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C2×#67—21no—
216#235—41no—
217#235—41no—
218C2×D12C22×C411yescharacteristic
219C2×D12C22×C411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×Dic6C22×C411yescharacteristic
225C2×Dic6C22×C411yescharacteristic
226C3×M16—42no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8C22×C411yescharacteristic
230C2×C3×C8C22×C411yescharacteristic
231C2×C3×C8C22×C411yescharacteristic
232C2×C3×C8C22×C411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C42C22×C411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42#4111yescharacteristic
240C3×C42C2411yescharacteristic
241#217—21no—
242#217—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249C4⋊Dic3C22×C411yescharacteristic
250C4⋊Dic3C22×C411yescharacteristic
251C4⋊Dic3C22×C411yescharacteristic
252C4⋊Dic3C22×C411yescharacteristic
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263#208—81no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—81no—
269Dic12—41no—
270D24—41no—
271#204—41no—
272#204—41no—
273ΓL(1,𝔽25)—81no—
274ΓL(1,𝔽25)—81no—
275ΓL(1,𝔽25)—81no—
276D3×C8—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—61no—
288C22×C8—61no—
289#126—61no—
290#125—31no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294#121—31no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—31no—
298C4×D4—31no—
299C4×D4—31no—
300C4×D4—31no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4—61no—
304#117—31no—
305#117—61no—
306#117—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—31no—
310#107—31no—
311#106—31no—
312#106—31no—
313C4⋊C8—31no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8C4×D311yescharacteristic
324C4⋊C8C4×D311yescharacteristic
325C4⋊C8D1211yescharacteristic
326C4⋊C8D1211yescharacteristic
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—61no—
336C22⋊C8—61no—
337C22⋊C8—61no—
338C22⋊C8—61no—
339C22⋊C8C4×D311yescharacteristic
340C22⋊C8C4×D311yescharacteristic
341C22⋊C8D1211yescharacteristic
342C22⋊C8D1211yescharacteristic
343#97—21no—
344#97—61no—
345#97—61no—
346#97—61no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C4×C8C22×D311yescharacteristic
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3C4×D411yescharacteristic
355C22×D3—21no—
356C22×D3—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4C4×D411yescharacteristic
363C2×C3×C4C4×D411yescharacteristic
364C2×C3×C4C4×D411yescharacteristic
365C2×C3×C4C23×C411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368#67—42no—
369#67—81no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376D12—21no—
377D12—21no—
378D12—21no—
379D12—21no—
380C4×D3—42no—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—81no—
386Dic6—21no—
387Dic6—21no—
388Dic6—21no—
389Dic6—21no—
390C3×C8—22no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—21no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404#41—121no—
405#41—121no—
406C2×Q8—31no—
407C2×Q8—31no—
408C2×D4—31no—
409C2×D4—31no—
410C2×D4—61no—
411C2×D4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4C2×C4×D311yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4#23511yescharacteristic
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—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—241no—
435C2×C8—241no—
436C2×C8C2×C4×D311yescharacteristic
437C2×C8C2×C4×D311yescharacteristic
438C2×C8C2×C4×D311yescharacteristic
439C2×C8C2×C4×D311yescharacteristic
440C2×C8C2×D1211yescharacteristic
441C2×C8C2×D1211yescharacteristic
442C2×C8C2×D1211yescharacteristic
443C2×C8C2×D1211yescharacteristic
444C4⋊C4—31no—
445C4⋊C4—31no—
446C4⋊C4—31no—
447C4⋊C4—31no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C42—61no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42C2×C4×D311yescharacteristic
467C42C2×D1211yescharacteristic
468C42#23511yescharacteristic
469C42C23×D311yescharacteristic
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3C2×C4×D411yescharacteristic
474C2×D3—41no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4#56811yescharacteristic
485C3×C4#56811yescharacteristic
486C3×C4#57711yescharacteristic
487C3×C4#57711yescharacteristic
488Dic3—41no—
489Dic3—41no—
490Dic3—41no—
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494C23—61no—
495C23—61no—
496C23C4×D1211yescharacteristic
497Q8—61no—
498Q8—61no—
499Q8—61no—
500Q8—61no—
501D4—61no—
502D4—61no—
503D4—61no—
504D4—61no—
505D4—122no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—122no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—241no—
524C2×C4C4×D1211yescharacteristic
525C2×C4C4×D1211yescharacteristic
526C2×C4C4×D1211yescharacteristic
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C22×D1211yescharacteristic
529C2×C4C2×#23511yescharacteristic
530C8—22no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—121no—
541C8—121no—
542C8—121no—
543C8—121no—
544C2×C3—41no—
545C2×C3#293011yescharacteristic
546C2×C3#295211yescharacteristic
547C2×C3#295611yescharacteristic
548D3—81no—
549D3—81no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22C2×C4×D1211yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—121no—
565C4—121no—
566C4—121no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4#575011yescharacteristic
571C4#575011yescharacteristic
572C4#575111yescharacteristic
573C4#575111yescharacteristic
574C3#1795511yescharacteristic, complemented, Sylow
575C2—41no—
576C2—241no—
577C2—241no—
578C2#7854111yescharacteristic
579C2#7855411yescharacteristic
580C2#7857411yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1453C2
socleC22×C3C2×C4×D4
prime radical#4734C4
cosocleC3×C42C24
hypercenter#1453D3
hypocenterC3#17955
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #79295C2semidirect
3. #79292C2semidirect
4. C3×#1453C2semidirect
5. #70749C2semidirect
6. #69854C2semidirect
7. #69852C2semidirect
8. #68089C2semidirect
9. #68087C2semidirect
10. #68085C2semidirect
11. #68083C2semidirect
12. #68081C2semidirect
13. #68079C2semidirect
14. #68055C2semidirect
15. #68055C2semidirect
82. #1453D3semidirect
574. C3#17955semidirect

Maximal subgroups

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

subgroupindex
#792992
#792952
#792922
C3×#14532
#707492
#698542
#698522
#680892
#680872
#680852
#680832
#680812
#680792
#680552
#179553