Finite Group Databasebeta

← #10778708

Subgroups — #10778708

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81564C211yesmaximal, characteristic, complemented
2#81070C211yesmaximal, characteristic, complemented
3#80336C211yesmaximal, characteristic
4#79425C211yesmaximal, characteristic, complemented
5C3×#1677C211yesmaximal, characteristic, complemented
6#71202C211yesmaximal, characteristic, complemented
7#71138C211yesmaximal, characteristic, complemented
8#71011C211yesmaximal, characteristic, complemented
9#70849C211yesmaximal, characteristic, complemented
10#68544C211yesmaximal, characteristic, complemented
11#68528C211yesmaximal, characteristic, complemented
12#68498C211yesmaximal, characteristic, complemented
13#68481C211yesmaximal, characteristic, complemented
14#68397C211yesmaximal, characteristic, complemented
15D3×#331C211yesmaximal, characteristic, complemented
16#19234—31nomaximal, Sylow
17C2×#901—21no—
18#5728C2211yescharacteristic
19D3×#127C2211yescharacteristic
20#5548C2211yescharacteristic
21C3×#486C2211yescharacteristic
22C3×#476C2211yescharacteristic, complemented
23C3×#459C2211yescharacteristic
24C3×C8×D4C2211yescharacteristic, complemented
25#5156—21no—
26#5151—21no—
27#5111—21no—
28#5088C2211yescharacteristic
29#5084C2211yescharacteristic, complemented
30#5072C2211yescharacteristic
31#5070C2211yescharacteristic
32#5019C2211yescharacteristic, complemented
33#5015C2211yescharacteristic
34#5012C2211yescharacteristic, complemented
35#4864—21no—
36#4854—21no—
37#4843C2211yescharacteristic
38#4841C2211yescharacteristic, complemented
39#4834C2211yescharacteristic, complemented
40#4833C2211yescharacteristic
41D3×C4⋊C8C2211yescharacteristic
42D3×C4⋊C8C2211yescharacteristic
43#4778—21no—
44#4773—21no—
45#4770—21no—
46#4770—21no—
47#4764—21no—
48#4764—21no—
49#4762—21no—
50#4760—21no—
51#4728—21no—
52#4727—21no—
53#4694C2211yescharacteristic
54#4688C2211yescharacteristic, complemented
55C4×ΓL(1,𝔽25)C2211yescharacteristic
56C3×#334C2211yescharacteristic, complemented
57C3×#333C2211yescharacteristic, complemented
58C3×#331C2211yescharacteristic
59#4546C2211yescharacteristic, complemented
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic, complemented
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4482C2211yescharacteristic, complemented
65#4480C2211yescharacteristic, complemented
66#4479C2211yescharacteristic, complemented
67#4458C2211yescharacteristic, complemented
68#3246—31no—
69#3205—31no—
70#3138—31no—
71#2935—31no—
72#1701—31no—
73#1701—31no—
74#1697—31no—
75#1697—31no—
76#1681—31no—
77#1681—31no—
78#1677—31no—
79#1677D311yescharacteristic, complemented
80#1565—31no—
81#1542—31no—
82C2×#331—31no—
83C2×D3×D4—21no—
84C2×D3×D4—21no—
85C2×D3×D4—21no—
86C2×#235—21no—
87C2×C3×D8—21no—
88C22×C3×C8—21no—
89C3×C4⋊Q8C2311yescharacteristic
90C3×#127C2311yescharacteristic
91C3×C4×D4C2311yescharacteristic
92#946—21no—
93Q8×Dic3—21no—
94#940—21no—
95#940—21no—
96#940—21no—
97#940D411yescharacteristic
98#938D411yescharacteristic
99C2×#214—21no—
100C2×#213—21no—
101C2×#213—21no—
102C4×#67—21no—
103C2×#208—21no—
104C2×D24—21no—
105C2×#204—21no—
106#901—44no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)D411yescharacteristic
109C2×ΓL(1,𝔽25)D411yescharacteristic
110C2×D3×C8—21no—
111C2×D3×C8—21no—
112C2×D3×C8—21no—
113#897—21no—
114#895D411yescharacteristic
115#892—21no—
116#890—21no—
117#888D411yescharacteristic
118#887—21no—
119#885—21no—
120#884—21no—
121#874C2311yescharacteristic
122C4×D12C2311yescharacteristic
123C42×D3C2311yescharacteristic
124#869C2311yescharacteristic
125C3×#106—21no—
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×C4⋊C8C2311yescharacteristic
129C3×#102—21no—
130C3×#102—21no—
131C3×#102—21no—
132C3×#102—21no—
133C3×C22⋊C8—21no—
134C3×C4×C8C2311yescharacteristic
135#832—21no—
136#832—21no—
137#832—21no—
138#830—21no—
139#821—21no—
140#821—21no—
141#821—21no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144(C2×D3)⋊C8—21no—
145#816—21no—
146#815D411yescharacteristic
147#815D411yescharacteristic
148Dic3⋊C8—21no—
149Dic3⋊C8—21no—
150C8×Dic3—21no—
151#810—21no—
152#809—21no—
153#809—21no—
154#809—21no—
155#809—21no—
156#808—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#131—61no—
162#533—31no—
163#532—31no—
164C2×#127—31no—
165#493—61no—
166#490—61no—
167#490—61no—
168#486—31no—
169#486—31no—
170#486—31no—
171#486—61no—
172#486C2×D311yescharacteristic
173#476—31no—
174#476C2×D311yescharacteristic, complemented
175#475—61no—
176#465—61no—
177#465—61no—
178#461—31no—
179#461—31no—
180#460—61no—
181#459—31no—
182#459—31no—
183#459—31no—
184#459C2×D311yescharacteristic
185#447—61no—
186#447—61no—
187#436—31no—
188#436—31no—
189#435—61no—
190#435—61no—
191C8×D4—31no—
192C8×D4C2×D311yescharacteristic, complemented
193#431—31no—
194C2×C4⋊C8—31no—
195C2×C4⋊C8—31no—
196#418—61no—
197#418—61no—
198#408—61no—
199C4×M16—31no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334C2×D311yescharacteristic, complemented
204#333—31no—
205#333C2×D311yescharacteristic, complemented
206#332—31no—
207#332—31no—
208#331—31no—
209#331—31no—
210#331—31no—
211#331C2×D311yescharacteristic
212C23×D3—41no—
213C2×C3×Q8—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C2×#67—21no—
220C2×#67—21no—
221C2×#67—21no—
222C2×#67—41no—
223C2×#67—41no—
224C2×#67—41no—
225D3×D4—42no—
226D3×D4—42no—
227D3×D4—81no—
228D3×D4—81no—
229D3×D4—82no—
230#235—42no—
231#235—42no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12C2×D411yescharacteristic
237C2×C4×D3—21no—
238C2×C4×D3C2×D411yescharacteristic
239C2×C4×D3C2×D411yescharacteristic
240C2×C4×D3#4111yescharacteristic
241C2×Dic6—21no—
242C2×Dic6—21no—
243C3×D8—81no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—42no—
248C2×C3×C8C2×D411yescharacteristic
249C2×C3×C8C2×D411yescharacteristic
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C22⋊C4—21no—
254C3×C42C2411yescharacteristic
255#217—21no—
256#214—81no—
257#213—81no—
258#213—81no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—41no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—41no—
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—21no—
267Dic3⋊C4—41no—
268C4×Dic3—41no—
269C4×Dic3C2×D411yescharacteristic
270C4×Dic3C2×D411yescharacteristic
271C4×Dic3#4111yescharacteristic
272#208—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D411yescharacteristic
277C2×C3⋊C8C2×D411yescharacteristic
278D24—81no—
279#204—81no—
280ΓL(1,𝔽25)—22no—
281ΓL(1,𝔽25)—22no—
282ΓL(1,𝔽25)—42no—
283D3×C8—42no—
284D3×C8—42no—
285D3×C8—81no—
286C2×#41—61no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—21no—
293C2×D8—61no—
294C2×D8—61no—
295C2×D8—61no—
296#131—124no—
297C2×M16—31no—
298C2×M16—31no—
299C2×M16—61no—
300C2×M16—61no—
301C22×C8—21no—
302C22×C8—61no—
303C22×C8—61no—
304C22×C8—61no—
305C4⋊Q8—31no—
306C4⋊Q8—31no—
307C4⋊Q8C22×D311yescharacteristic
308#127—31no—
309#127—31no—
310#127—61no—
311#127—61no—
312#127—61no—
313#127C22×D311yescharacteristic
314#125—61no—
315#124—61no—
316C22⋊Q8—61no—
317C22⋊Q8—61no—
318#121—31no—
319#121—31no—
320#121—61no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×D4—31no—
324C4×D4—61no—
325C4×D4C22×D311yescharacteristic
326#117—61no—
327C2×C42—31no—
328#106—21no—
329#106—61no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—61no—
334C4⋊C8—61no—
335C4⋊C8C22×D311yescharacteristic
336C4⋊C8C22×D311yescharacteristic
337C4⋊C8C22×D311yescharacteristic
338#103—61no—
339#103—61no—
340#102—21no—
341#102—21no—
342#102—21no—
343#102—21no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349#102—61no—
350#102—61no—
351#102—61no—
352#102—61no—
353#102—61no—
354C22⋊C8—21no—
355C22⋊C8—61no—
356C22⋊C8—61no—
357C22⋊C8—61no—
358C22⋊C8—61no—
359#97—31no—
360#97—31no—
361C4×C8—31no—
362C4×C8—61no—
363C4×C8C22×D311yescharacteristic
364C23×C3—21no—
365C23×C3—41no—
366C22×D3—21no—
367C22×D3—41no—
368C22×D3—82no—
369C22×D3—82no—
370C22×D3—82no—
371C22×D3#12111yescharacteristic
372C3×Q8—81no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—81no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387#67—42no—
388#67—42no—
389#67—81no—
390#67—82no—
391#67—82no—
392#67—82no—
393C2×Dic3—21no—
394C2×Dic3—41no—
395C2×Dic3#12111yescharacteristic
396C2×Dic3#12111yescharacteristic
397C2×Dic3#12111yescharacteristic
398D12—21no—
399D12—21no—
400D12—42no—
401D12—41no—
402D12—41no—
403D12—41no—
404D12—41no—
405D12—81no—
406C4×D3—21no—
407C4×D3—21no—
408C4×D3—41no—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3#13512yes—
412C4×D3AGL(1,ℤ/8)12yes—
413Dic6—42no—
414Dic6—41no—
415Dic6—41no—
416C3×C8—21no—
417C3×C8—21no—
418C3×C8—41no—
419C3×C8—41no—
420C3×C8—41no—
421C3⋊C8—21no—
422C3⋊C8—21no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—121no—
427#41—122no—
428#41—122no—
429C2×Q8—21no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—31no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—122no—
444C2×D4—122no—
445C2×D4—121no—
446C2×D4—121no—
447C2×D4—121no—
448C2×D4—241no—
449C2×D4—241no—
450C2×D4—242no—
451C2×D4D3×D411yescharacteristic
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457SD16—241no—
458SD16—241no—
459D8—81no—
460D8—241no—
461D8—241no—
462D8—241no—
463M16—62no—
464M16—62no—
465M16—122no—
466M16—122no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—31no—
471C2×C8—31no—
472C2×C8—42no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—122no—
477C2×C8—122no—
478C2×C8—241no—
479C2×C8D3×D411yescharacteristic
480C2×C8D3×D411yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—31no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—121no—
488C4⋊C4—121no—
489C4⋊C4D3×D411yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—61no—
492C22⋊C4—61no—
493C22⋊C4—61no—
494C22⋊C4—121no—
495C42—31no—
496C42—31no—
497C42—31no—
498C42—121no—
499C42C23×D311yescharacteristic
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12111yescharacteristic
507C2×D3—21no—
508C2×D3—21no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—164no—
516C2×D3#46312yes—
517C3×C4—21no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4C2×#13511yescharacteristic
521C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
522C3×C4#57611yescharacteristic
523C3×C4#57711yescharacteristic
524Dic3—21no—
525Dic3—21no—
526Dic3—41no—
527Dic3—81no—
528Dic3#46312yes—
529C23—21no—
530C23—31no—
531C23—41no—
532C23—61no—
533C23—121no—
534C23—242no—
535C23—242no—
536C23—242no—
537Q8—81no—
538Q8—122no—
539Q8—121no—
540Q8—121no—
541D4—21no—
542D4—21no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—61no—
548D4—61no—
549D4—81no—
550D4—122no—
551D4—122no—
552D4—122no—
553D4—121no—
554D4—121no—
555D4—121no—
556D4—121no—
557D4—241no—
558D4—242no—
559D4—241no—
560D4—242no—
561D4—242no—
562C2×C4—21no—
563C2×C4—31no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—31no—
567C2×C4—31no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4C2×D3×D411yescharacteristic
579C2×C4C2×D3×D411yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C8—21no—
582C8—21no—
583C8—41no—
584C8—41no—
585C8—41no—
586C8—61no—
587C8—61no—
588C8—121no—
589C8—121no—
590C8—121no—
591C2×C3—81no—
592C2×C3—161no—
593C2×C3C2×#46311yescharacteristic
594C2×C3#306211yescharacteristic
595C2×C3#306811yescharacteristic
596D3—22no—
597D3—81no—
598D3—161no—
599C22—32no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—61no—
604C22—61no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—121no—
609C22—121no—
610C22—121no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—484no—
615C22D3×#12111yescharacteristic
616C4—21no—
617C4—32no—
618C4—41no—
619C4—61no—
620C4—61no—
621C4—81no—
622C4—121no—
623C4—241no—
624C4#576211yescharacteristic
625C4D3×#13511yescharacteristic
626C4#576911yescharacteristic
627C4D3×AGL(1,ℤ/8)11yescharacteristic
628C3#1923411yescharacteristic, complemented, Sylow
629C2—62no—
630C2—81no—
631C2—161no—
632C2—241no—
633C2—481no—
634C2D3×#46311yescharacteristic
635C2#8066111yescharacteristic
636C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1677C2
socleC22×C3C2×#121
prime radical#80336C2
hypercenter#1677D3
hypocenterC3#19234
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1677

Complements

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

NH ≅ G/Ntype
1. #81564C2semidirect
2. #81070C2semidirect
4. #79425C2semidirect
5. C3×#1677C2semidirect
6. #71202C2semidirect
7. #71138C2semidirect
8. #71011C2semidirect
9. #70849C2semidirect
10. #68544C2semidirect
11. #68528C2semidirect
12. #68498C2semidirect
13. #68481C2semidirect
14. #68397C2semidirect
15. D3×#331C2semidirect
22. C3×#476C22semidirect
24. C3×C8×D4C22semidirect
29. #5084C22semidirect
32. #5019C22semidirect
34. #5012C22semidirect
38. #4841C22semidirect
39. #4834C22semidirect
54. #4688C22semidirect
56. C3×#334C22semidirect
57. C3×#333C22semidirect
59. #4546C22semidirect
61. #4486C22semidirect
64. #4482C22semidirect
65. #4480C22semidirect
66. #4479C22semidirect
67. #4458C22semidirect
79. #1677D3semidirect
174. #476C2×D3semidirect
192. C8×D4C2×D3semidirect
203. #334C2×D3semidirect
205. #333C2×D3semidirect
628. C3#19234semidirect

Maximal subgroups

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

subgroupindex
#815642
#810702
#803362
#794252
C3×#16772
#712022
#711382
#710112
#708492
#685442
#685282
#684982
#684812
#683972
D3×#3312
#192343