Finite Group Databasebeta

← #11664570

Subgroups — #11664570

Order 768.

2134 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86242C211yesmaximal, characteristic
2#86098C211yesmaximal, characteristic, complemented
3C3×#3142C211yesmaximal, characteristic, complemented
4#83282C211yesmaximal, characteristic, complemented
5C2×#5067C211yesmaximal, characteristic, complemented
6#81740C212yesmaximal, complemented
7#81737C212yesmaximal, complemented
8#81715C212yesmaximal, complemented
9D3×#490C212yesmaximal, complemented
10#81638C218yesmaximal, complemented
11#78882C214yesmaximal, complemented
12#78876C214yesmaximal, complemented
13#78526C211yesmaximal, characteristic, complemented
14#78516C211yesmaximal, characteristic, complemented
15#60626—31nomaximal, Sylow
16C2×#1009C2212yes—
17C3×#532C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic, complemented
19#5830C2212yes—
20C2×#907C2212yes—
21#5725C2212yes—
22D3×#127C2212yes—
23#5690—22no—
24#5689C2214yescomplemented
25#5688C2212yescomplemented
26D3×#125C2212yescomplemented
27#5685—22no—
28#5674C2214yes—
29#5609—24no—
30#5553—22no—
31D4×D12—22no—
32#5511—24no—
33#5504C2212yes—
34#5500C2212yescomplemented
35C2×#874C2211yescharacteristic
36C4×#235C2211yescharacteristic
37#5470C2211yescharacteristic, complemented
38C3×#491C2212yescomplemented
39C3×#490C2212yescomplemented
40C3×#488C2214yescomplemented
41C3×#417C2214yescomplemented
42C3×C4×M16C2211yescharacteristic, complemented
43#5233C2214yescomplemented
44#5131C2214yescomplemented
45#5122C2212yescomplemented
46#5082C2211yescharacteristic, complemented
47#5082C2211yescharacteristic, complemented
48#5071C2211yescharacteristic, complemented
49#5071C2211yescharacteristic, complemented
50#5067C2212yescomplemented
51#5067C2212yescomplemented
52#5067C2214yescomplemented
53#5055C2214yescomplemented
54C2×#809C2214yescomplemented
55C2×#803C2211yescharacteristic, complemented
56#4804C2218yescomplemented
57#4803C2218yescomplemented
58#4800C2218yescomplemented
59#4794C2218yescomplemented
60#4770C2218yescomplemented
61D3×#102C2218yescomplemented
62#4764C2218yescomplemented
63#4760C2218yescomplemented
64#4713C2212yescomplemented
65#4713C2212yescomplemented
66#4707C2212yescomplemented
67D3×#97C2212yescomplemented
68#4694C2214yescomplemented
69#4689C2214yescomplemented
70#3504—31no—
71#3498—31no—
72#3142—32no—
73#3142D311yescharacteristic, complemented
74#3138—32no—
75#3138—31no—
76#3136—38no—
77#3136—34no—
78#3135—32no—
79#3135—31no—
80C2×#490—32no—
81C2×#490—31no—
82#2897—34no—
83#2875—31no—
84C2×C3×#41C2312yes—
85#1009—216no—
86C2×#239C2314yes—
87C2×D3×D4—24no—
88C2×D3×D4—24no—
89C2×D3×D4C2314yes—
90C2×#235—22no—
91C2×#235C2311yescharacteristic
92C22×D12—22no—
93C22×D12—22no—
94C22×D12C2312yes—
95C2×C3×M16C2312yes—
96C3×C4⋊Q8C2311yescharacteristic
97C3×#127C2311yescharacteristic
98C3×#125—22no—
99C3×#125C2312yes—
100C3×#125C2312yes—
101C3×#124C2312yes—
102C3×#121—22no—
103C3×C4×D4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C42C2311yescharacteristic
107#947C2314yes—
108#940—24no—
109#940C2314yes—
110#937—24no—
111#930—24no—
112C4×#67—24no—
113C4×#67D414yes—
114#926C2312yes—
115#926C2311yescharacteristic
116#924—22no—
117#924—24no—
118C22×C3⋊C8C2312yes—
119#907—28no—
120C2×ΓL(1,𝔽25)C2314yes—
121C2×D3×C8C2314yes—
122#897—24no—
123#896—24no—
124#896—24no—
125#895—24no—
126#893—22no—
127#893—24no—
128#892C2314yes—
129D3×C4⋊C4—24no—
130D3×C4⋊C4C2314yes—
131#890—22no—
132#890C2314yes—
133#889—24no—
134#889—24no—
135#888—22no—
136#888—24no—
137#888C2314yes—
138#887—22no—
139#887—24no—
140#884—28no—
141#882—28no—
142#874—22no—
143#874C2312yes—
144#874C2312yes—
145C4×D12D412yes—
146#872C2312yes—
147C42×D3C2312yes—
148#870C2311yescharacteristic
149#870C2311yescharacteristic
150#869C2312yes—
151C4×Dic6—22no—
152C4×Dic6D412yes—
153C3×#103C2318yes—
154C3×#102C2318yes—
155C3×#97C2312yes—
156C3×C4×C8C2312yes—
157#835C2314yes—
158#835C2314yes—
159#832C2314yes—
160#832C2314yes—
161#821C2318yes—
162#821C2318yes—
163#815C2314yes—
164C8×Dic3C2314yes—
165#809C2318yes—
166#809C2318yes—
167#803C2312yes—
168#803C2311yescharacteristic
169#803C2311yescharacteristic
170C2×H(5,𝔽2)—32no—
171C2×#131—32no—
172#557—62no—
173#554—64no—
174#554—62no—
175#550—62no—
176D42—62no—
177D42—64no—
178#544—34no—
179#544—32no—
180#535—32no—
181#535—34no—
182#533—32no—
183#533—31no—
184#532—32no—
185#532C2×D311yescharacteristic
186C2×#127—32no—
187C2×#127—31no—
188C2×#125—32no—
189C2×#125C2×D311yescharacteristic, complemented
190#520—32no—
191C4×#41—31no—
192#491—31no—
193#491—31no—
194#491C2×D312yescomplemented
195#490—31no—
196#490—31no—
197#490—32no—
198#490—34no—
199#490—32no—
200#490—38no—
201#490—32no—
202#490—32no—
203#490C2×D312yescomplemented
204#489—34no—
205#489—38no—
206#488—38no—
207#488C2×D314yescomplemented
208#486—38no—
209#486—34no—
210#418—34no—
211#418—38no—
212#417—38no—
213#417C2×D314yescomplemented
214#416—34no—
215#416—38no—
216C2×#102—34no—
217C2×#102—38no—
218#405—32no—
219#405—34no—
220C4×M16—32no—
221C4×M16C2×D311yescharacteristic, complemented
222C2×#97—31no—
223C2×#97—32no—
224C23×D3—22no—
225C23×D3—24no—
226C3×#41—28no—
227C2×C3×Q8C2412yes—
228C2×C3×D4—22no—
229C2×C3×D4—22no—
230C2×C3×D4C2412yes—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2411yescharacteristic
235C2×#67—24no—
236C2×#67—28no—
237C2×#67C2×D412yes—
238#239—216no—
239D3×D4—216no—
240D3×D4—416no—
241D3×D4—88no—
242#235—48no—
243#235#4118yes—
244C2×D12—28no—
245C2×D12—24no—
246C2×D12—22no—
247C2×D12—24no—
248C2×D12—24no—
249C2×D12—48no—
250C2×D12—84no—
251C2×D12C2×D411yescharacteristic
252C2×D12C2414yes—
253C2×C4×D3—24no—
254C2×C4×D3—24no—
255C2×C4×D3C2×D414yes—
256C2×C4×D3C2412yes—
257C2×Dic6—22no—
258C2×Dic6—24no—
259C2×Dic6C2×D411yescharacteristic
260C3×M16—24no—
261C2×C3×C8C2414yes—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—22no—
266C3×C4⋊C4C2414yes—
267C3×C22⋊C4—24no—
268C3×C42C2×D412yes—
269C3×C42C2411yescharacteristic
270C3×C42C2411yescharacteristic
271#217—22no—
272#217C2×D412yes—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—28no—
275(C2×D3)⋊C4C2×D414yes—
276C4⋊Dic3—22no—
277C4⋊Dic3—24no—
278C4⋊Dic3C2×D412yes—
279C4⋊Dic3C2412yes—
280C4⋊Dic3C2412yes—
281Dic3⋊C4—22no—
282Dic3⋊C4—22no—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4C2×D414yes—
287C4×Dic3—22no—
288C4×Dic3—24no—
289C4×Dic3C2×D414yes—
290C4×Dic3C2412yes—
291C2×C3⋊C8—24no—
292C2×C3⋊C8C2412yes—
293C2×C3⋊C8C2412yes—
294ΓL(1,𝔽25)—28no—
295D3×C8—28no—
296H(5,𝔽2)—616no—
297C2×#41—31no—
298C2×#41—34no—
299C2×#41—62no—
300C2×#41C22×D312yes—
301C22×D4—34no—
302C22×D4—32no—
303C22×D4—62no—
304C22×D4—64no—
305C22×D4—64no—
306C22×D4—62no—
307#131—68no—
308C2×M16—34no—
309C2×M16C22×D312yes—
310C22×C8—32no—
311C22×C8—34no—
312C4⋊Q8—34no—
313C4⋊Q8—32no—
314C4⋊Q8—64no—
315C4⋊Q8—62no—
316C4⋊Q8C22×D311yescharacteristic
317#127—34no—
318#127—32no—
319#127—32no—
320#127—64no—
321#127—62no—
322#127C22×D311yescharacteristic
323#125—22no—
324#125—34no—
325#125—31no—
326#125—31no—
327#125—64no—
328#125—64no—
329#125—62no—
330#125C22×D312yes—
331#125C22×D312yes—
332#124—34no—
333#124C22×D312yes—
334C22⋊Q8—64no—
335C22⋊Q8—64no—
336C22⋊Q8—64no—
337C22⋊Q8—64no—
338C22⋊Q8—62no—
339#121—22no—
340#121—68no—
341#121—64no—
342#121—64no—
343#120—68no—
344#120—64no—
345C4×Q8—32no—
346C4×Q8—64no—
347C4×Q8—62no—
348C4×Q8—62no—
349C4×D4—22no—
350C4×D4—34no—
351C4×D4—32no—
352C4×D4—64no—
353C4×D4—62no—
354C4×D4—64no—
355#117—31no—
356#117—32no—
357#117—32no—
358#117—34no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4—64no—
362C2×C4⋊C4C22×D312yes—
363C2×C42—32no—
364C2×C42C22×D311yescharacteristic
365#103—34no—
366#103—34no—
367#103C22×D318yes—
368#102—34no—
369#102—34no—
370#102—38no—
371#102—38no—
372#102—38no—
373#102—38no—
374#102C22×D318yes—
375#97—34no—
376#97—31no—
377#97—31no—
378#97—32no—
379#97C22×D312yes—
380C4×C8—34no—
381C4×C8C22×D312yes—
382C23×C3—22no—
383C23×C3C22×D411yescharacteristic
384C22×D3—24no—
385C22×D3—44no—
386C22×D3—48no—
387C22×D3—88no—
388C22×D3—84no—
389C22×D3C22×D412yes—
390C3×Q8—24no—
391C3×D4—24no—
392C3×D4—44no—
393C3×D4—44no—
394C2×C3×C4—24no—
395C2×C3×C4—22no—
396C2×C3×C4—42no—
397C2×C3×C4—44no—
398C2×C3×C4—82no—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C22×D412yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4114yes—
403C2×C3×C4C2511yescharacteristic
404#67—48no—
405#67—88no—
406#67#12418yes—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3C22×D414yes—
410C2×Dic3C22×D412yes—
411D12—28no—
412D12—48no—
413D12—48no—
414D12—44no—
415D12—84no—
416D12#12414yes—
417C4×D3—44no—
418C4×D3—48no—
419C4×D3—84no—
420C4×D3C2×#4118yes—
421Dic6—84no—
422Dic6—82no—
423Dic6#12414yes—
424C3×C8—24no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C24—64no—
428C24—62no—
429#41—28no—
430#41—38no—
431#41—616no—
432#41—128no—
433C2×Q8—31no—
434C2×Q8—64no—
435C2×Q8—62no—
436C2×Q8C23×D312yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—32no—
440C2×D4—31no—
441C2×D4—34no—
442C2×D4—64no—
443C2×D4—616no—
444C2×D4—64no—
445C2×D4—68no—
446C2×D4—64no—
447C2×D4—62no—
448C2×D4—64no—
449C2×D4—68no—
450C2×D4—128no—
451C2×D4—1216no—
452C2×D4—244no—
453C2×D4—248no—
454C2×D4C23×D312yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—32no—
458C22×C4—34no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4D3×D412yes—
462C22×C4C23×D311yescharacteristic
463M16—24no—
464M16—68no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—68no—
468C2×C8—64no—
469C2×C8C23×D314yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—34no—
475C4⋊C4—32no—
476C4⋊C4—32no—
477C4⋊C4—32no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4C23×D314yes—
486C22⋊C4—24no—
487C22⋊C4—32no—
488C22⋊C4—34no—
489C22⋊C4—68no—
490C22⋊C4—64no—
491C22⋊C4—62no—
492C42—34no—
493C42—32no—
494C42—64no—
495C42—62no—
496C42D3×D412yes—
497C42C23×D311yescharacteristic
498C42C23×D311yescharacteristic
499C22×C3—22no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3C2×#12412yes—
503C22×C3C23×D411yescharacteristic
504C2×D3—24no—
505C2×D3—48no—
506C2×D3—44no—
507C2×D3—168no—
508C2×D3—164no—
509C2×D3C2×#12414yes—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4C2×#12412yes—
515C3×C4C22×#4112yes—
516Dic3—24no—
517Dic3—42no—
518Dic3—42no—
519Dic3C2×#12414yes—
520C23—22no—
521C23—32no—
522C23—64no—
523C23—128no—
524C23—124no—
525C23—244no—
526C23—248no—
527C23C2×D3×D411yescharacteristic
528Q8—24no—
529Q8—34no—
530Q8—242no—
531Q8—244no—
532D4—24no—
533D4—38no—
534D4—34no—
535D4—44no—
536D4—44no—
537D4—68no—
538D4—128no—
539D4—128no—
540D4—128no—
541D4—124no—
542D4—244no—
543D4—248no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—32no—
547C2×C4—38no—
548C2×C4—34no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—82no—
554C2×C4—124no—
555C2×C4—128no—
556C2×C4—244no—
557C2×C4C2×D3×D412yes—
558C2×C4C2×D3×D412yes—
559C2×C4C2×D3×D411yescharacteristic
560C2×C4D3×#4114yes—
561C2×C4C24×D311yescharacteristic
562C8—24no—
563C8—62no—
564C8—62no—
565C2×C3—22no—
566C2×C3—82no—
567C2×C3C22×#12411yescharacteristic
568C2×C3#358512yes—
569D3—24no—
570D3—84no—
571C22—22no—
572C22—34no—
573C22—42no—
574C22—44no—
575C22—64no—
576C22—124no—
577C22—128no—
578C22—484no—
579C22—488no—
580C22D3×#12412yes—
581C22C22×D3×D411yescharacteristic
582C4—22no—
583C4—22no—
584C4—34no—
585C4—42no—
586C4—44no—
587C4—64no—
588C4—122no—
589C4—122no—
590C4D3×#12412yes—
591C4C2×D3×#4112yes—
592C3#6062611yescharacteristic, complemented, Sylow
593C2—22no—
594C2—64no—
595C2—82no—
596C2—244no—
597C2C2×D3×#12411yescharacteristic
598C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3142C2
socleC22×C3C23×D4
prime radical#86242C2
hypercenter#3142D3
hypocenterC3#60626
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #86098C2semidirect
3. C3×#3142C2semidirect
4. #83282C2semidirect
5. C2×#5067C2semidirect
6. #81740C2semidirect
7. #81737C2semidirect
8. #81715C2semidirect
9. D3×#490C2semidirect
10. #81638C2semidirect
11. #78882C2semidirect
12. #78876C2semidirect
13. #78526C2semidirect
14. #78516C2semidirect
18. C2×C3×#125C22semidirect
24. #5689C22semidirect
25. #5688C22semidirect
26. D3×#125C22semidirect
34. #5500C22semidirect
37. #5470C22semidirect
38. C3×#491C22semidirect
39. C3×#490C22semidirect
40. C3×#488C22semidirect
41. C3×#417C22semidirect
42. C3×C4×M16C22semidirect
43. #5233C22semidirect
44. #5131C22semidirect
45. #5122C22semidirect
46. #5082C22semidirect
47. #5082C22semidirect
48. #5071C22semidirect
49. #5071C22semidirect
50. #5067C22semidirect
51. #5067C22semidirect
52. #5067C22semidirect
53. #5055C22semidirect
54. C2×#809C22semidirect
55. C2×#803C22semidirect
56. #4804C22semidirect
57. #4803C22semidirect
58. #4800C22semidirect
59. #4794C22semidirect
60. #4770C22semidirect
61. D3×#102C22semidirect
62. #4764C22semidirect
63. #4760C22semidirect
64. #4713C22semidirect
65. #4713C22semidirect
66. #4707C22semidirect
67. D3×#97C22semidirect
68. #4694C22semidirect
69. #4689C22semidirect
73. #3142D3semidirect
189. C2×#125C2×D3semidirect
194. #491C2×D3semidirect
203. #490C2×D3semidirect
207. #488C2×D3semidirect
213. #417C2×D3semidirect
221. C4×M16C2×D3semidirect
592. C3#60626semidirect

Maximal subgroups

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

subgroupindex
#862422
#860982
C3×#31422
#832822
C2×#50672
#817402
#817372
#817152
D3×#4902
#816382
#788822
#788762
#785262
#785162
#606263