Finite Group Databasebeta

← #10765602

Subgroups — #10765602

Order 768.

608 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#79593C211yesmaximal, characteristic
2#79523C211yesmaximal, characteristic, complemented
3#79431C211yesmaximal, characteristic, complemented
4C3×#1586C211yesmaximal, characteristic, complemented
5#70881C211yesmaximal, characteristic, complemented
6#70862C211yesmaximal, characteristic, complemented
7#70842C211yesmaximal, characteristic, complemented
8#68534C211yesmaximal, characteristic, complemented
9#68523C211yesmaximal, characteristic, complemented
10#68173C211yesmaximal, characteristic, complemented
11#68168C211yesmaximal, characteristic, complemented
12#68163C211yesmaximal, characteristic, complemented
13#68162C211yesmaximal, characteristic, complemented
14#68080C211yesmaximal, characteristic, complemented
15#68074C211yesmaximal, characteristic, complemented
16#17948—31nomaximal, Sylow
17#5548C2211yescharacteristic
18C4×#237C2211yescharacteristic
19C3×C4×D8C2211yescharacteristic
20C3×#436C2211yescharacteristic
21C3×C8×D4C2211yescharacteristic
22#5158C411yescharacteristic
23#5150C411yescharacteristic
24#5127—21no—
25#5118—21no—
26#5111—21no—
27#5110—21no—
28C4×#214C2211yescharacteristic
29C4×#213C2211yescharacteristic
30#5012C2211yescharacteristic
31#5012C2211yescharacteristic
32#4885C411yescharacteristic
33#4875C411yescharacteristic
34#4838C2211yescharacteristic
35#4835C2211yescharacteristic
36D3×C4⋊C8C2211yescharacteristic
37#4831C2211yescharacteristic
38#4781C411yescharacteristic
39#4779C411yescharacteristic
40#4762C411yescharacteristic
41#4761C411yescharacteristic
42#4728—21no—
43#4728—21no—
44#4727—21no—
45#4719—21no—
46C4×#204C2211yescharacteristic
47C8×D12C2211yescharacteristic
48#4685C2211yescharacteristic
49#4685C2211yescharacteristic
50#4685C2211yescharacteristic
51#4679C2211yescharacteristic
52C3×#334C2211yescharacteristic
53C3×D4⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56D4⋊(C3⋊C8)C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4479C2211yescharacteristic
62#4459C2211yescharacteristic
63#4458C2211yescharacteristic
64#4456C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3106—31no—
69#2989—31no—
70#2981—31no—
71#1594—31no—
72#1592—31no—
73#1590—31no—
74#1589—31no—
75#1586D311yescharacteristic, complemented
76#1585—31no—
77#1582—31no—
78#1581—31no—
79#1573—31no—
80#1488—31no—
81#1480—31no—
82#1454—31no—
83C2×#237C2×C411yescharacteristic
84C2×#235—21no—
85C2×C3×D8C2×C411yescharacteristic
86C2×C3×M16—21no—
87C22×C3×C8—21no—
88C3×C4×D4C2311yescharacteristic
89C3×C4×D4C2311yescharacteristic
90#940C2×C411yescharacteristic
91#938C2×C411yescharacteristic
92D4×Dic3C2×C411yescharacteristic
93C2×#214C2×C411yescharacteristic
94C2×#213C2×C411yescharacteristic
95C4×#67—21no—
96C4×#67—21no—
97C2×C4×Dic3—21no—
98C2×#208—21no—
99C2×#208—21no—
100C2×#204C2×C411yescharacteristic
101C2×ΓL(1,𝔽25)—21no—
102C2×D3×C8—21no—
103C2×D3×C8—21no—
104#895C2×C411yescharacteristic
105#892C2×C411yescharacteristic
106#888C2×C411yescharacteristic
107#887C2×C411yescharacteristic
108#885—21no—
109#884—21no—
110#67⋊C4—21no—
111#877—21no—
112C4×D12C2311yescharacteristic
113C42×D3C2311yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×#107C2×C411yescharacteristic
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×#102C2×C411yescharacteristic
119C3×#102C2×C411yescharacteristic
120C3×C22⋊C8—21no—
121C3×C22⋊C8—21no—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124C3×C4×C8C2311yescharacteristic
125#832C2×C411yescharacteristic
126#832C2×C411yescharacteristic
127#830—21no—
128#830—21no—
129#821C2×C411yescharacteristic
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8C2×C411yescharacteristic
135(C2×D3)⋊C8C2×C411yescharacteristic
136(C2×D3)⋊C8D411yescharacteristic
137(C2×D3)⋊C8D411yescharacteristic
138#817C2×C411yescharacteristic
139#816C2×C411yescharacteristic
140#815—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8—21no—
147Dic3⋊C8C2×C411yescharacteristic
148Dic3⋊C8C2×C411yescharacteristic
149Dic3⋊C8D411yescharacteristic
150Dic3⋊C8D411yescharacteristic
151#810C2×C411yescharacteristic
152#809C2×C411yescharacteristic
153#808C2×C411yescharacteristic
154#807C2×C411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
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—
170#446—31no—
171#446—31no—
172C4×SD16—31no—
173C4×SD16—31no—
174C4×D8—31no—
175C4×D8C2×D311yescharacteristic
176#436—31no—
177#436—31no—
178#436—31no—
179#436C2×D311yescharacteristic
180#435—61no—
181#435—61no—
182#435—61no—
183#435—61no—
184C8×D4—31no—
185C8×D4C2×D311yescharacteristic
186#432—31no—
187#432—61no—
188#431—31no—
189#431—31no—
190#431—31no—
191#423—61no—
192C2×C4⋊C8—31no—
193#408—61no—
194#408—61no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334C2×D311yescharacteristic
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322C2×D311yescharacteristic
210C82—31no—
211C2×C3×D4C22×C411yescharacteristic
212C2×C3×D4C22×C411yescharacteristic
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C2×#67—21no—
216C2×#67—21no—
217C2×#67—21no—
218C2×#67—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221#237—41no—
222#237—41no—
223#235—81no—
224#235—81no—
225C2×D12—21no—
226C2×D12C22×C411yescharacteristic
227C2×C4×D3—21no—
228C2×C4×D3C22×C411yescharacteristic
229C2×C4×D3C2×D411yescharacteristic
230C2×C4×D3#4111yescharacteristic
231C2×Dic6—21no—
232C2×Dic6C22×C411yescharacteristic
233C3×D8—41no—
234C3×M16—81no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—81no—
239C2×C3×C8C22×C411yescharacteristic
240C2×C3×C8C22×C411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C3×C4⋊C4C22×C411yescharacteristic
244C3×C4⋊C4C22×C411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C42C2411yescharacteristic
248#217—21no—
249#217—21no—
250#214—41no—
251#213—41no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255C4⋊Dic3C22×C411yescharacteristic
256C4⋊Dic3C22×C411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3#4111yescharacteristic
266#208—81no—
267#208—81no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275#204—41no—
276ΓL(1,𝔽25)—81no—
277D3×C8—81no—
278D3×C8—81no—
279C2×#41—31no—
280C2×#41—61no—
281C2×SD16—31no—
282C2×SD16—31no—
283C2×D8—31no—
284C2×D8C4×D311yescharacteristic
285C2×M16—21no—
286C2×M16—61no—
287C2×M16—61no—
288C2×M16—61no—
289C22×C8—21no—
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—61no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—31no—
308#117—61no—
309C2×C42—31no—
310C2×C42—61no—
311#107—31no—
312#107C4×D311yescharacteristic
313#106—31no—
314#106—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
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#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102C4×D311yescharacteristic
334#102C4×D311yescharacteristic
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—31no—
340C22⋊C8—31no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347#97—61no—
348#97—61no—
349#97—61no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—31no—
353C4×C8C22×D311yescharacteristic
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C23×C3—21no—
358C22×D3—21no—
359C22×D3C4×D411yescharacteristic
360C3×D4—21no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4C23×C411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373#67—42no—
374#67—81no—
375#67—81no—
376#67—81no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—42no—
381C2×Dic3—81no—
382C2×Dic3C4×D411yescharacteristic
383C2×Dic3C4×D411yescharacteristic
384C2×Dic3C4×D411yescharacteristic
385D12—21no—
386D12—21no—
387D12—81no—
388C4×D3—21no—
389C4×D3—21no—
390C4×D3—21no—
391C4×D3—21no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—41no—
395Dic6—21no—
396Dic6—21no—
397Dic6—81no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—121no—
413#41—121no—
414#41—241no—
415#41—241no—
416C2×Q8—31no—
417C2×Q8—61no—
418C2×D4—31no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4C2×C4×D311yescharacteristic
425C2×D4C2×C4×D311yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434SD16—121no—
435SD16—121no—
436D8—41no—
437D8—121no—
438M16—81no—
439M16—241no—
440M16—241no—
441M16—241no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—31no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—81no—
453C2×C8—241no—
454C2×C8—241no—
455C2×C8C2×C4×D311yescharacteristic
456C2×C8C2×C4×D311yescharacteristic
457C2×C8D3×D411yescharacteristic
458C2×C8D3×D411yescharacteristic
459C4⋊C4—31no—
460C4⋊C4—31no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4C2×C4×D311yescharacteristic
465C4⋊C4C2×C4×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C42—31no—
474C42—31no—
475C42—31no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42C23×D311yescharacteristic
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C4×D411yescharacteristic
487C2×D3—21no—
488C2×D3—21no—
489C2×D3—21no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C3×C4—21no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4#56811yescharacteristic
497C3×C4#56811yescharacteristic
498C3×C4#57611yescharacteristic
499C3×C4#57711yescharacteristic
500Dic3—21no—
501Dic3—21no—
502Dic3—21no—
503Dic3—41no—
504Dic3—41no—
505Dic3—41no—
506C23—21no—
507C23—21no—
508C23—31no—
509C23—61no—
510Q8—61no—
511Q8—61no—
512Q8—241no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—21no—
517D4—61no—
518D4—61no—
519D4—122no—
520D4—241no—
521D4—241no—
522D4—241no—
523D4—241no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—31no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—241no—
545C2×C4C22×C4×D311yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—61no—
556C8—61no—
557C8—61no—
558C8—61no—
559C8—121no—
560C8—121no—
561C8—121no—
562C2×C3—81no—
563C2×C3—81no—
564C2×C3#293011yescharacteristic
565C2×C3#294611yescharacteristic
566C2×C3#295711yescharacteristic
567D3—41no—
568D3—81no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—61no—
576C22—61no—
577C22—61no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22C4×D3×D411yescharacteristic
582C4—21no—
583C4—41no—
584C4—41no—
585C4—61no—
586C4—61no—
587C4—61no—
588C4—121no—
589C4—121no—
590C4—121no—
591C4#574511yescharacteristic
592C4#575011yescharacteristic
593C4#577411yescharacteristic
594C4#577511yescharacteristic
595C3#1794811yescharacteristic, complemented, Sylow
596C2—81no—
597C2—81no—
598C2—121no—
599C2—241no—
600C2#7947911yescharacteristic
601C2#7986611yescharacteristic
602C2#7989011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1586C2
socleC22×C3C2×C4×D4
prime radical#79593C2
cosocleC3×C42C24
hypercenter#1586D3
hypocenterC3#17948
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #79523C2semidirect
3. #79431C2semidirect
4. C3×#1586C2semidirect
5. #70881C2semidirect
6. #70862C2semidirect
7. #70842C2semidirect
8. #68534C2semidirect
9. #68523C2semidirect
10. #68173C2semidirect
11. #68168C2semidirect
12. #68163C2semidirect
13. #68162C2semidirect
14. #68080C2semidirect
15. #68074C2semidirect
75. #1586D3semidirect
595. C3#17948semidirect

Maximal subgroups

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

subgroupindex
#795932
#795232
#794312
C3×#15862
#708812
#708622
#708422
#685342
#685232
#681732
#681682
#681632
#681622
#680802
#680742
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