Finite Group Databasebeta

← #10765281

Subgroups — #10765281

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79301C211yesmaximal, characteristic
2#79291C211yesmaximal, characteristic, complemented
3#79290C211yesmaximal, characteristic, complemented
4C3×#1454C211yesmaximal, characteristic, complemented
5#70763C211yesmaximal, characteristic, complemented
6#70758C211yesmaximal, characteristic, complemented
7#70752C211yesmaximal, characteristic, complemented
8#68088C211yesmaximal, characteristic, complemented
9#68085C211yesmaximal, characteristic, complemented
10#68081C211yesmaximal, characteristic, complemented
11#68078C211yesmaximal, characteristic, complemented
12#68077C211yesmaximal, characteristic, complemented
13#68076C211yesmaximal, characteristic, complemented
14C8×#204C211yesmaximal, characteristic, complemented
15C8×D24C211yesmaximal, characteristic, complemented
16#17920—31nomaximal, Sylow
17#5477C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#431C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#4926C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4850C411yescharacteristic
25#4848C411yescharacteristic
26#4843C411yescharacteristic
27#4842C411yescharacteristic
28#4836—21no—
29#4836—21no—
30#4833—21no—
31D3×C4⋊C8—21no—
32#4737C411yescharacteristic
33#4737C411yescharacteristic
34#4734C411yescharacteristic
35#4734C411yescharacteristic
36#4727—21no—
37#4727—21no—
38#67⋊C8—21no—
39#67⋊C8—21no—
40C4×D24C2211yescharacteristic
41C4×D24C2211yescharacteristic
42C4×#204C2211yescharacteristic
43C4×#204C2211yescharacteristic
44#4688C2211yescharacteristic
45#4688C2211yescharacteristic
46C8×D12C2211yescharacteristic
47C8×D12C2211yescharacteristic
48C8×D12C2211yescharacteristic
49C8×D12C2211yescharacteristic
50C8×Dic6C2211yescharacteristic
51C8×Dic6C2211yescharacteristic
52C3×#322C2211yescharacteristic
53C3×#322C2211yescharacteristic
54C3×#322C2211yescharacteristic
55C3×C82C2211yescharacteristic
56#4459C2211yescharacteristic
57#4459C2211yescharacteristic
58#4459C2211yescharacteristic
59#4459C2211yescharacteristic
60#4459C2211yescharacteristic
61#4459C2211yescharacteristic
62#4458C2211yescharacteristic
63#4458C2211yescharacteristic
64#4457C2211yescharacteristic
65#4457C2211yescharacteristic
66#4456C2211yescharacteristic
67#4456C2211yescharacteristic
68#3106—31no—
69#2988—31no—
70#2981—31no—
71#1588—31no—
72#1586—31no—
73#1583—31no—
74#1581—31no—
75#1580—31no—
76SD16⋊C8—31no—
77C8×SD16—31no—
78C8×D8—31no—
79#1573—31no—
80#1488—31no—
81#1480—31no—
82#1454D311yescharacteristic, complemented
83C2×#235—21no—
84C2×#235C2×C411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86#930—21no—
87#930C2×C411yescharacteristic
88#930C2×C411yescharacteristic
89C4×#67—21no—
90C4×#67—21no—
91#926C2×C411yescharacteristic
92C22×C3⋊C8—21no—
93C2×D24C2×C411yescharacteristic
94C2×D24C2×C411yescharacteristic
95C2×#204C2×C411yescharacteristic
96C2×#204C2×C411yescharacteristic
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101C2×D3×C8—21no—
102C2×D3×C8—21no—
103#875—21no—
104#874C2×C411yescharacteristic
105C4×D12—21no—
106C4×D12C2×C411yescharacteristic
107C4×D12C2311yescharacteristic
108C4×D12C2311yescharacteristic
109C4×D12C2311yescharacteristic
110#872—21no—
111C42×D3—21no—
112#869C2×C411yescharacteristic
113C4×Dic6C2×C411yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2×C411yescharacteristic
117C3×C4⋊C8C2×C411yescharacteristic
118C3×C4⋊C8D411yescharacteristic
119C3×C4⋊C8D411yescharacteristic
120C3×C22⋊C8—21no—
121C3×C22⋊C8C2×C411yescharacteristic
122C3×C22⋊C8C2×C411yescharacteristic
123C3×C22⋊C8D411yescharacteristic
124C3×C22⋊C8D411yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131#830—21no—
132#821C2×C411yescharacteristic
133#821C2×C411yescharacteristic
134#821C2×C411yescharacteristic
135#821C2×C411yescharacteristic
136#821C2×C411yescharacteristic
137#821C2×C411yescharacteristic
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144#818C2×C411yescharacteristic
145#818C2×C411yescharacteristic
146#817C2×C411yescharacteristic
147#817C2×C411yescharacteristic
148#816C2×C411yescharacteristic
149#816C2×C411yescharacteristic
150Dic3⋊C8—21no—
151Dic3⋊C8—21no—
152C8×Dic3—21no—
153C8×Dic3—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160#532—31no—
161C4×#41—31no—
162#482—31no—
163#482—31no—
164#477—31no—
165#476—31no—
166#460—31no—
167#459—31no—
168#449—31no—
169#449—31no—
170C8×Q8—31no—
171C8×Q8—31no—
172C4×SD16—31no—
173C4×SD16—31no—
174C4×D8—31no—
175C4×D8—31no—
176#436—31no—
177#436—31no—
178#435—61no—
179#435—61no—
180C8×D4—31no—
181C8×D4—31no—
182C8×D4—31no—
183C8×D4—31no—
184C8×D4—61no—
185C8×D4—61no—
186#432—31no—
187#431—61no—
188#431C2×D311yescharacteristic
189#431C2×D311yescharacteristic
190#431C2×D311yescharacteristic
191C2×C4⋊C8—31no—
192C2×C4⋊C8—61no—
193#408—61no—
194#408—61no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201D4⋊C8—31no—
202D4⋊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—
216C2×#67—21no—
217#235—22no—
218#235—22no—
219#235—81no—
220#235—81no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12C22×C411yescharacteristic
224C2×D12C22×C411yescharacteristic
225C2×D12C22×C411yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×Dic6—21no—
232C2×Dic6C22×C411yescharacteristic
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C22×C411yescharacteristic
236C2×C3×C8C22×C411yescharacteristic
237C2×C3×C8C22×C411yescharacteristic
238C2×C3×C8C22×C411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C3×C42C22×C411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42#4111yescharacteristic
246C3×C42C2411yescharacteristic
247#217—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C22×C411yescharacteristic
256C4⋊Dic3C22×C411yescharacteristic
257C4⋊Dic3C22×C411yescharacteristic
258C4⋊Dic3C22×C411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—81no—
269D24—22no—
270D24—22no—
271#204—22no—
272#204—22no—
273ΓL(1,𝔽25)—81no—
274ΓL(1,𝔽25)—81no—
275D3×C8—81no—
276D3×C8—81no—
277D3×C8—81no—
278D3×C8—81no—
279C2×#41—31no—
280C2×#41—61no—
281C2×SD16—31no—
282C2×SD16—31no—
283C2×D8—31no—
284C2×D8—31no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—61no—
288C22×C8—61no—
289C22×C8—61no—
290C22×C8—61no—
291C22×C8—61no—
292C4⋊Q8—31no—
293#127—31no—
294#124—61no—
295#121—31no—
296#121—31no—
297#121—61no—
298C4×Q8—31no—
299C4×Q8—31no—
300C4×D4—31no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—31no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4—61no—
307#117—31no—
308#117—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—31no—
312#107—31no—
313#106—31no—
314#106—31no—
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⋊C8C4×D311yescharacteristic
324C4⋊C8C4×D311yescharacteristic
325C4⋊C8D1211yescharacteristic
326C4⋊C8D1211yescharacteristic
327#103—31no—
328#103—31no—
329#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102—31no—
335C22⋊C8—21no—
336C22⋊C8—61no—
337C22⋊C8—61no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8C4×D311yescharacteristic
344C22⋊C8C4×D311yescharacteristic
345C22⋊C8D1211yescharacteristic
346C22⋊C8D1211yescharacteristic
347#97—61no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3C4×D411yescharacteristic
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4C4×D411yescharacteristic
366C2×C3×C4C4×D411yescharacteristic
367C2×C3×C4C4×D411yescharacteristic
368C2×C3×C4C23×C411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371#67—24no—
372#67—81no—
373#67—81no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379D12—21no—
380D12—21no—
381D12—21no—
382D12—21no—
383D12—21no—
384D12—81no—
385D12—81no—
386D12#13112yes—
387C4×D3—24no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—81no—
395Dic6—21no—
396Dic6—81no—
397Dic6#13112yes—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—21no—
403C3×C8—21no—
404C3×C8—21no—
405C3×C8—21no—
406C3×C8—41no—
407C3×C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—62no—
413#41—62no—
414#41—241no—
415#41—241no—
416C2×Q8—31no—
417C2×Q8—61no—
418C2×D4—31no—
419C2×D4—31no—
420C2×D4—31no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4#23511yescharacteristic
434SD16—62no—
435SD16—62no—
436D8—62no—
437D8—62no—
438M16—241no—
439M16—241no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—241no—
447C2×C8—241no—
448C2×C8—241no—
449C2×C8—241no—
450C2×C8—241no—
451C2×C8C2×C4×D311yescharacteristic
452C2×C8C2×C4×D311yescharacteristic
453C2×C8C2×C4×D311yescharacteristic
454C2×C8C2×C4×D311yescharacteristic
455C2×C8C2×D1211yescharacteristic
456C2×C8C2×D1211yescharacteristic
457C2×C8C2×D1211yescharacteristic
458C2×C8C2×D1211yescharacteristic
459C4⋊C4—31no—
460C4⋊C4—31no—
461C4⋊C4—31no—
462C4⋊C4—31no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42C2×C4×D311yescharacteristic
477C42C2×D1211yescharacteristic
478C42#23511yescharacteristic
479C42C23×D311yescharacteristic
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3C2×C4×D411yescharacteristic
484C2×D3—22no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4C2×#13111yescharacteristic
498C3×C4C2×#13111yescharacteristic
499C3×C4#57611yescharacteristic
500C3×C4#57711yescharacteristic
501Dic3—22no—
502Dic3—41no—
503Dic3—41no—
504Dic3—41no—
505Dic3—41no—
506C23—61no—
507C23—61no—
508C23—61no—
509C23C4×D1211yescharacteristic
510Q8—32no—
511Q8—61no—
512Q8—241no—
513D4—32no—
514D4—64no—
515D4—61no—
516D4—61no—
517D4—61no—
518D4—61no—
519D4—61no—
520D4—241no—
521D4—241no—
522D4—241no—
523D4—241no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—64no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—241no—
542C2×C4C4×D1211yescharacteristic
543C2×C4C4×D1211yescharacteristic
544C2×C4C4×D1211yescharacteristic
545C2×C4C22×C4×D311yescharacteristic
546C2×C4C22×D1211yescharacteristic
547C2×C4C2×#23511yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—121no—
559C8—121no—
560C8—121no—
561C8—121no—
562C2×C3—41no—
563C2×C3#292911yescharacteristic
564C2×C3#294611yescharacteristic
565C2×C3#295711yescharacteristic
566D3—42no—
567D3—81no—
568D3—81no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—62no—
573C22—121no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22C2×C4×D1211yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—62no—
587C4—121no—
588C4—121no—
589C4—121no—
590C4—121no—
591C4D3×#13111yescharacteristic
592C4D3×#13111yescharacteristic
593C4#575111yescharacteristic
594C4#575211yescharacteristic
595C3#1792011yescharacteristic, complemented, Sylow
596C2—41no—
597C2—122no—
598C2—241no—
599C2—241no—
600C2#7855311yescharacteristic
601C2#7856011yescharacteristic
602C2#7858011yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1454C2
socleC22×C3C2×C4×D4
prime radical#79301C2
cosocleC3×C42C24
hypercenter#1454D3
hypocenterC3#17920
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #79291C2semidirect
3. #79290C2semidirect
4. C3×#1454C2semidirect
5. #70763C2semidirect
6. #70758C2semidirect
7. #70752C2semidirect
8. #68088C2semidirect
9. #68085C2semidirect
10. #68081C2semidirect
11. #68078C2semidirect
12. #68077C2semidirect
13. #68076C2semidirect
14. C8×#204C2semidirect
15. C8×D24C2semidirect
82. #1454D3semidirect
595. C3#17920semidirect

Maximal subgroups

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

subgroupindex
#793012
#792912
#792902
C3×#14542
#707632
#707582
#707522
#680882
#680852
#680812
#680782
#680772
#680762
C8×#2042
C8×D242
#179203