Finite Group Databasebeta

← #10951987

Subgroups — #10951987

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3290C211yesmaximal, characteristic, complemented
2#84017C211yesmaximal, characteristic, complemented
3#84007C211yesmaximal, characteristic, complemented
4#79891C211yesmaximal, characteristic, complemented
5#72124C211yesmaximal, characteristic, complemented
6#72123C211yesmaximal, characteristic, complemented
7#71325C211yesmaximal, characteristic, complemented
8#71320C211yesmaximal, characteristic, complemented
9#71098C211yesmaximal, characteristic, complemented
10#71096C211yesmaximal, characteristic, complemented
11#71095C211yesmaximal, characteristic, complemented
12#71092C211yesmaximal, characteristic, complemented
13#70925C211yesmaximal, characteristic, complemented
14#70923C211yesmaximal, characteristic, complemented
15#70921C211yesmaximal, characteristic, complemented
16#33859—31nomaximal, Sylow
17C3×#578C2211yescharacteristic, complemented
18C3×#577—21no—
19C3×#576C2211yescharacteristic
20#5770—21no—
21#5753C2211yescharacteristic, complemented
22#5752C2211yescharacteristic
23C3×#495C2211yescharacteristic, complemented
24C3×#473C2211yescharacteristic
25C3×#472C2211yescharacteristic, complemented
26C3×#471—21no—
27C3×#455—21no—
28C3×#454—21no—
29C3×#443C2211yescharacteristic
30C3×#443C2211yescharacteristic, complemented
31#5195—21no—
32#5195C2211yescharacteristic, complemented
33#5194—21no—
34#5194—21no—
35#5194C2211yescharacteristic
36#5193—21no—
37#5193C2211yescharacteristic
38C2×#829—21no—
39C2×#829C2211yescharacteristic, complemented
40#5179—21no—
41#5179C2211yescharacteristic, complemented
42C2×#828C2211yescharacteristic, complemented
43C2×#827C2211yescharacteristic, complemented
44#5147—21no—
45#5147C2211yescharacteristic
46C2×#826—21no—
47C2×#826C2211yescharacteristic
48#5143C2211yescharacteristic
49#5142C2211yescharacteristic, complemented
50#5140C2211yescharacteristic
51#5140C2211yescharacteristic, complemented
52#4898—21no—
53#4827—21no—
54#4825—21no—
55#4700C2211yescharacteristic, complemented
56#4696C2211yescharacteristic, complemented
57#4567C2211yescharacteristic, complemented
58#4565C2211yescharacteristic, complemented
59#4563C2211yescharacteristic, complemented
60#4561C2211yescharacteristic
61#4561C2211yescharacteristic, complemented
62#4497C2211yescharacteristic, complemented
63#4496C2211yescharacteristic, complemented
64#4495C2211yescharacteristic, complemented
65#4493C2211yescharacteristic, complemented
66#4463C2211yescharacteristic, complemented
67C4×C3⋊C16C2211yescharacteristic, complemented
68#3417—31no—
69#3415—31no—
70#3290—31no—
71#3290D311yescharacteristic, complemented
72#2249—31no—
73#2222—31no—
74#2221—31no—
75#2212—31no—
76#2212—31no—
77#2194—31no—
78#2194—31no—
79#2193—31no—
80#2193—31no—
81#2178—31no—
82#2170—31no—
83C3×2−1+4—21no—
84C3×H(5,𝔽2)—21no—
85#1010—21no—
86#1009—21no—
87C3×#137—21no—
88C3×#137—21no—
89C3×#137—41no—
90C3×#137—41no—
91C3×AGL(1,ℤ/8)—21no—
92C3×AGL(1,ℤ/8)—21no—
93C3×AGL(1,ℤ/8)—41no—
94C3×AGL(1,ℤ/8)—41no—
95C3×#135—21no—
96C3×#135—21no—
97C3×#135—41no—
98C3×#135D411yescharacteristic
99C3×#135C2311yescharacteristic
100C2×C3×Q16—21no—
101C2×C3×Q16C2311yescharacteristic
102C2×C3×SD16—21no—
103C2×C3×SD16—41no—
104C2×C3×D8—21no—
105C2×C3×D8C2311yescharacteristic
106C3×#131—21no—
107C3×#131C2311yescharacteristic
108C3×#131C2311yescharacteristic
109C3×#124—21no—
110#951—21no—
111#949—21no—
112#918—41no—
113#916—41no—
114D3×SD16—41no—
115#911—41no—
116#909—21no—
117#909—41no—
118#908—21no—
119#908—41no—
120C2×Dic12—21no—
121C2×Dic12C2311yescharacteristic
122#904—21no—
123#904—21no—
124#904C2311yescharacteristic
125C2×D24—21no—
126C2×D24C2311yescharacteristic
127C2×#204—21no—
128#901—21no—
129#875—21no—
130C3×#108D411yescharacteristic
131C3×#108C2311yescharacteristic
132C3×#104—21no—
133C3×#104D411yescharacteristic
134C3×#104D411yescharacteristic
135C3×#101—21no—
136C3×#100—21no—
137C3×C4×C8C2311yescharacteristic
138#829—21no—
139#829—21no—
140#829—21no—
141#829—21no—
142#829D411yescharacteristic
143#828—21no—
144#828—21no—
145#828—41no—
146#828D411yescharacteristic
147#827—21no—
148#827—21no—
149#827—41no—
150#827D411yescharacteristic
151#826—21no—
152#826—21no—
153#826—21no—
154#826—21no—
155#826D411yescharacteristic
156#824—21no—
157#823—21no—
158#819C2311yescharacteristic
159#812—21no—
160#812—21no—
161#812C2311yescharacteristic
162#812C2311yescharacteristic
163C2×C3⋊C16—21no—
164C2×C3⋊C16—21no—
165C2×C3⋊C16C2311yescharacteristic
166C2×C3⋊C16C2311yescharacteristic
167#805—21no—
168#578—31no—
169#578C2×D311yescharacteristic, complemented
170#577—21no—
171#577—61no—
172#576—31no—
173#576C2×D311yescharacteristic
174#510—31no—
175#510—31no—
176#510—61no—
177#510—61no—
178#509—31no—
179#509—31no—
180#509—61no—
181#509—61no—
182#508—31no—
183#508—61no—
184#508—61no—
185C2×Q32—31no—
186C2×Q32—61no—
187C2×SD32—31no—
188C2×SD32—31no—
189C2×D16—31no—
190C2×D16—61no—
191#504—31no—
192#504—31no—
193#495—31no—
194#495C2×D311yescharacteristic, complemented
195#473—31no—
196#473C2×D311yescharacteristic
197#472—31no—
198#472C2×D311yescharacteristic, complemented
199#471—21no—
200#471—61no—
201#455—21no—
202#455—61no—
203#454—21no—
204#454—61no—
205#443—31no—
206#443C2×D311yescharacteristic
207#443C2×D311yescharacteristic, complemented
208#368—31no—
209#364—31no—
210#362—31no—
211#362—31no—
212#361—31no—
213#361—31no—
214#359—31no—
215#359—31no—
216#350—31no—
217#350—31no—
218C4×C16—31no—
219C3×#41—21no—
220C3×#41—41no—
221C3×#41—41no—
222C3×#41—41no—
223C3×#41—41no—
224C3×#41—81no—
225C3×#41C2×D411yescharacteristic
226C3×#41C2×D411yescharacteristic
227C2×C3×Q8—21no—
228C2×C3×Q8—41no—
229C2×C3×Q8—41no—
230C2×C3×D4—21no—
231C2×C3×D4—41no—
232C2×C3×D4—41no—
233C2×C3×D4—81no—
234#239—41no—
235D3×Q8—41no—
236#237—41no—
237#237—81no—
238D3×D4—41no—
239D3×D4—81no—
240#235—21no—
241#235—41no—
242#235—41no—
243C2×D12—21no—
244C2×D12—41no—
245C2×Dic6—21no—
246C2×Dic6—41no—
247C3×Q16—21no—
248C3×Q16—21no—
249C3×Q16—41no—
250C3×Q16—41no—
251C3×Q16D811yescharacteristic
252C3×Q16C2×D411yescharacteristic
253C3×SD16—21no—
254C3×SD16—41no—
255C3×SD16—41no—
256C3×SD16—41no—
257C3×SD16—41no—
258C3×SD16—41no—
259C3×SD16—41no—
260C3×SD16D811yescharacteristic
261C3×SD16D811yescharacteristic
262C3×D8—21no—
263C3×D8—21no—
264C3×D8—41no—
265C3×D8—41no—
266C3×D8D811yescharacteristic
267C3×D8C2×D411yescharacteristic
268C3×M16—21no—
269C3×M16—21no—
270C3×M16—21no—
271C3×M16—41no—
272C3×M16C2×D411yescharacteristic
273C3×M16C2×D411yescharacteristic
274C2×C3×C8—21no—
275C2×C3×C8—21no—
276C2×C3×C8—41no—
277C2×C3×C8C2×D411yescharacteristic
278C2×C3×C8C2411yescharacteristic
279C3×C22⋊C4—81no—
280C3×C42C2×D411yescharacteristic
281#216—41no—
282#215—41no—
283#214—41no—
284#213—41no—
285(C2×D3)⋊C4—81no—
286#208—21no—
287Dic12—21no—
288Dic12—21no—
289Dic12—41no—
290Dic12C2×D411yescharacteristic
291D24—21no—
292D24—21no—
293D24—41no—
294D24C2×D411yescharacteristic
295#204—21no—
296#204—41no—
297#204—41no—
298#204—41no—
299#204—41no—
300ΓL(1,𝔽25)—41no—
301D3×C8—41no—
302C3⋊C16—21no—
303C3⋊C16—21no—
304C3⋊C16—21no—
305C3⋊C16C2×D411yescharacteristic
306C3⋊C16C2×D411yescharacteristic
3072−1+4—21no—
3082−1+4—61no—
309H(5,𝔽2)—21no—
310H(5,𝔽2)—61no—
311#137—21no—
312#137—21no—
313#137—41no—
314#137—41no—
315#137—61no—
316#137—61no—
317#137—121no—
318#137—121no—
319AGL(1,ℤ/8)—21no—
320AGL(1,ℤ/8)—21no—
321AGL(1,ℤ/8)—41no—
322AGL(1,ℤ/8)—41no—
323AGL(1,ℤ/8)—61no—
324AGL(1,ℤ/8)—61no—
325AGL(1,ℤ/8)—121no—
326AGL(1,ℤ/8)—121no—
327#135—21no—
328#135—21no—
329#135—31no—
330#135—41no—
331#135—61no—
332#135—61no—
333#135—121no—
334#135#6711yescharacteristic
335#135C22×D311yescharacteristic
336C2×Q16—21no—
337C2×Q16—31no—
338C2×Q16—61no—
339C2×Q16C22×D311yescharacteristic
340C2×SD16—21no—
341C2×SD16—41no—
342C2×SD16—61no—
343C2×SD16—121no—
344C2×D8—21no—
345C2×D8—31no—
346C2×D8—61no—
347C2×D8C22×D311yescharacteristic
348#131—21no—
349#131—61no—
350#131C22×D311yescharacteristic
351#131C22×D311yescharacteristic
352#124—21no—
353#124—61no—
354Q32—31no—
355Q32—61no—
356Q32—61no—
357Q32—61no—
358Q32—61no—
359SD32—31no—
360SD32—31no—
361SD32—61no—
362SD32—61no—
363SD32—61no—
364SD32—61no—
365SD32—121no—
366SD32—121no—
367D16—31no—
368D16—61no—
369D16—61no—
370D16—61no—
371D16—61no—
372M32—31no—
373M32—31no—
374M32—61no—
375M32—61no—
376C2×C16—31no—
377C2×C16—31no—
378C2×C16—61no—
379C2×C16—61no—
380#108—31no—
381#108#6711yescharacteristic
382#108C22×D311yescharacteristic
383#104—21no—
384#104—61no—
385#104#6711yescharacteristic
386#104#6711yescharacteristic
387#101—21no—
388#101—61no—
389#100—21no—
390#100—61no—
391C4×C8C22×D311yescharacteristic
392C23×C3—41no—
393C23×C3—81no—
394C22×D3—41no—
395C22×D3—81no—
396C3×Q8—21no—
397C3×Q8—41no—
398C3×Q8—41no—
399C3×Q8—81no—
400C3×Q8C2×D811yescharacteristic
401C3×Q8C2×D811yescharacteristic
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—21no—
405C3×D4—41no—
406C3×D4—41no—
407C3×D4—41no—
408C3×D4—81no—
409C3×D4—81no—
410C3×D4—81no—
411C3×D4C2×D811yescharacteristic
412C3×D4C2×D811yescharacteristic
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—81no—
419C2×C3×C4—81no—
420C2×C3×C4—81no—
421C2×C3×C4C22×D411yescharacteristic
422#67—41no—
423#67—81no—
424#67—81no—
425C2×Dic3—41no—
426C2×Dic3—81no—
427D12—21no—
428D12—41no—
429D12—41no—
430D12—81no—
431C4×D3—41no—
432C4×D3—81no—
433C4×D3—81no—
434Dic6—21no—
435Dic6—41no—
436Dic6—41no—
437Dic6—81no—
438C3×C8—21no—
439C3×C8—21no—
440C3×C8—41no—
441C3×C8C2×D811yescharacteristic
442C3×C8C2×D811yescharacteristic
443C3×C8C22×D411yescharacteristic
444C3×C8H(5,𝔽2)11yescharacteristic
445C3⋊C8—41no—
446#41—21no—
447#41—41no—
448#41—41no—
449#41—41no—
450#41—41no—
451#41—61no—
452#41—81no—
453#41—121no—
454#41—121no—
455#41—121no—
456#41—121no—
457#41—241no—
458#41C2×#6711yescharacteristic
459#41C2×#6711yescharacteristic
460C2×Q8—21no—
461C2×Q8—41no—
462C2×Q8—41no—
463C2×Q8—61no—
464C2×Q8—121no—
465C2×Q8—121no—
466C2×D4—21no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—61no—
470C2×D4—81no—
471C2×D4—121no—
472C2×D4—121no—
473C2×D4—241no—
474Q16—21no—
475Q16—21no—
476Q16—31no—
477Q16—41no—
478Q16—41no—
479Q16—61no—
480Q16—61no—
481Q16—121no—
482Q16—121no—
483Q16#21311yescharacteristic
484Q16D3×D411yescharacteristic
485SD16—21no—
486SD16—41no—
487SD16—41no—
488SD16—41no—
489SD16—41no—
490SD16—41no—
491SD16—41no—
492SD16—61no—
493SD16—121no—
494SD16—121no—
495SD16—121no—
496SD16—121no—
497SD16—121no—
498SD16—121no—
499SD16#21311yescharacteristic
500SD16#21311yescharacteristic
501D8—21no—
502D8—21no—
503D8—31no—
504D8—41no—
505D8—41no—
506D8—61no—
507D8—61no—
508D8—121no—
509D8—121no—
510D8#21311yescharacteristic
511D8D3×D411yescharacteristic
512M16—21no—
513M16—21no—
514M16—21no—
515M16—41no—
516M16—61no—
517M16—121no—
518M16C2×#6711yescharacteristic
519M16C2×#6711yescharacteristic
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—41no—
523C2×C8—121no—
524C2×C8C2×#6711yescharacteristic
525C2×C8C23×D311yescharacteristic
526C22⋊C4—81no—
527C22⋊C4—241no—
528C42C2×#6711yescharacteristic
529C16—31no—
530C16—31no—
531C16—61no—
532C16—61no—
533C16—61no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—41no—
537C22×C3—81no—
538C22×C3—161no—
539C22×C3—321no—
540C22×C3#57611yescharacteristic
541C2×D3—41no—
542C2×D3—81no—
543C2×D3—161no—
544C2×D3—321no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—81no—
550C3×C4D4211yescharacteristic
551C3×C4C22×D811yescharacteristic
552Dic3—41no—
553Dic3—81no—
554C23—41no—
555C23—81no—
556C23—121no—
557C23—241no—
558Q8—21no—
559Q8—41no—
560Q8—41no—
561Q8—61no—
562Q8—81no—
563Q8—121no—
564Q8—121no—
565Q8—241no—
566Q8C2×#21311yescharacteristic
567Q8C2×#21311yescharacteristic
568D4—21no—
569D4—21no—
570D4—21no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—61no—
575D4—81no—
576D4—81no—
577D4—81no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—241no—
582D4—241no—
583D4—241no—
584D4C2×#21311yescharacteristic
585D4C2×#21311yescharacteristic
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—241no—
597C2×C4—241no—
598C2×C4—241no—
599C2×C4C22×#6711yescharacteristic
600C8—21no—
601C8—21no—
602C8—41no—
603C8—121no—
604C8C2×#21311yescharacteristic
605C8C2×#21311yescharacteristic
606C8C2×D3×D411yescharacteristic
607C8#100411yescharacteristic
608C2×C3—21no—
609C2×C3—41no—
610C2×C3—41no—
611C2×C3—81no—
612C2×C3—161no—
613C2×C3D4×D811yescharacteristic
614D3—81no—
615D3—161no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—81no—
620C22—121no—
621C22—161no—
622C22—241no—
623C22—321no—
624C22—481no—
625C22—961no—
626C22#583511yescharacteristic
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—121no—
633C4—241no—
634C4C22×#21311yescharacteristic
635C4D4×#6711yescharacteristic
636C3#3385911yescharacteristic, complemented, Sylow
637C2—21no—
638C2—41no—
639C2—41no—
640C2—81no—
641C2—161no—
642C2—241no—
643C2—481no—
644C2D4×#21311yescharacteristic
subgroupG/H
centerC2D4×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3290C2
socleC2×C3D4×D8
hypercenter#3290D3
hypocenterC3#33859
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #3290

Complements

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

NH ≅ G/Ntype
1. C3×#3290C2semidirect
2. #84017C2semidirect
3. #84007C2semidirect
4. #79891C2semidirect
5. #72124C2semidirect
6. #72123C2semidirect
7. #71325C2semidirect
8. #71320C2semidirect
9. #71098C2semidirect
10. #71096C2semidirect
11. #71095C2semidirect
12. #71092C2semidirect
13. #70925C2semidirect
14. #70923C2semidirect
15. #70921C2semidirect
17. C3×#578C22semidirect
21. #5753C22semidirect
23. C3×#495C22semidirect
25. C3×#472C22semidirect
30. C3×#443C22semidirect
32. #5195C22semidirect
39. C2×#829C22semidirect
41. #5179C22semidirect
42. C2×#828C22semidirect
43. C2×#827C22semidirect
49. #5142C22semidirect
51. #5140C22semidirect
55. #4700C22semidirect
56. #4696C22semidirect
57. #4567C22semidirect
58. #4565C22semidirect
59. #4563C22semidirect
61. #4561C22semidirect
62. #4497C22semidirect
63. #4496C22semidirect
64. #4495C22semidirect
65. #4493C22semidirect
66. #4463C22semidirect
67. C4×C3⋊C16C22semidirect
71. #3290D3semidirect
169. #578C2×D3semidirect
194. #495C2×D3semidirect
198. #472C2×D3semidirect
207. #443C2×D3semidirect
636. C3#33859semidirect

Maximal subgroups

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

subgroupindex
C3×#32902
#840172
#840072
#798912
#721242
#721232
#713252
#713202
#710982
#710962
#710952
#710922
#709252
#709232
#709212
#338593