Finite Group Databasebeta

← #11664564

Subgroups — #11664564

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#86195C211yesmaximal, characteristic
2#86098C211yesmaximal, characteristic, complemented
3C2×C3×#490C211yesmaximal, characteristic, complemented
4#83159C211yesmaximal, characteristic, complemented
5#83012C211yesmaximal, characteristic, complemented
6#81716C212yesmaximal, complemented
7#81715C212yesmaximal, complemented
8#81714C218yesmaximal, complemented
9#81713C212yesmaximal, complemented
10D3×#490C212yesmaximal, complemented
11#78742C214yesmaximal, complemented
12#78737C214yesmaximal, complemented
13#78497C211yesmaximal, characteristic, complemented
14#78486C211yesmaximal, characteristic, complemented
15#60626—31nomaximal, Sylow
16C2×#1004C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic, complemented
19#5805C2212yes—
20D4×#67—24no—
21C2×#901C2212yes—
22#5718—22no—
23#5717—22no—
24#5716C2212yes—
25#5715C2214yes—
26D3×#127C2212yes—
27#5690—22no—
28#5689C2214yescomplemented
29#5688C2212yescomplemented
30D3×#125C2212yescomplemented
31#5685—22no—
32#5511—24no—
33#5504C2212yes—
34#5500C2212yescomplemented
35#5477C2211yescharacteristic
36C4×#235C2211yescharacteristic
37#5470C2211yescharacteristic, complemented
38C3×#490C2212yescomplemented
39C3×#490C2214yescomplemented
40C3×#490C2212yescomplemented
41C2×C3×#102C2214yescomplemented
42C2×C3×#97C2211yescharacteristic, complemented
43#5215C2214yescomplemented
44#5113C2214yescomplemented
45#5101C2212yescomplemented
46#5071C2212yescomplemented
47#5071C2212yescomplemented
48#5070C2214yescomplemented
49#5069C2212yescomplemented
50#5068C2214yescomplemented
51#5067C2212yescomplemented
52#4966C2214yescomplemented
53C4×#208C2211yescharacteristic, complemented
54#4772C2218yescomplemented
55#4771C2218yescomplemented
56#4770C2218yescomplemented
57D3×#102C2218yescomplemented
58#4768C2218yescomplemented
59#4765C2218yescomplemented
60#4764C2218yescomplemented
61#4760C2218yescomplemented
62#4714C2214yescomplemented
63#4713C2211yescharacteristic, complemented
64#4713C2211yescharacteristic, complemented
65#4708C2214yescomplemented
66#4707C2212yescomplemented
67D3×#97C2212yescomplemented
68#4703C2211yescharacteristic, complemented
69#4703C2211yescharacteristic, complemented
70#3504—31no—
71#3498—31no—
72#3142—32no—
73#3142—31no—
74#3138—32no—
75#3138—31no—
76#3136—38no—
77#3136—34no—
78#3135—32no—
79#3135—31no—
80C2×#490—32no—
81C2×#490D311yescharacteristic, complemented
82#2897—34no—
83#2875—31no—
84C22×C3×D4—22no—
85C22×C3×D4C2312yes—
86C22×#67—24no—
87#1004—216no—
88C2×#237—22no—
89C2×#237C2314yes—
90C2×D3×D4—22no—
91C2×D3×D4—24no—
92C2×D3×D4C2314yes—
93C2×#235C2311yescharacteristic
94C2×#235C2312yes—
95C22×C3×C8C2312yes—
96C3×#127—22no—
97C3×#127C2312yes—
98C3×#127C2312yes—
99C3×#125—22no—
100C3×#125C2312yes—
101C3×#125C2312yes—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C42C2311yescharacteristic
105#953—24no—
106#940—24no—
107#940C2314yes—
108#939—28no—
109#938—24no—
110#938—24no—
111#937—28no—
112#936C2314yes—
113D4×Dic3—24no—
114#930—22no—
115C4×#67—24no—
116C4×#67D414yes—
117#926C2312yes—
118#926C2311yescharacteristic
119#924—22no—
120#924—24no—
121C2×#208C2312yes—
122#901—28no—
123C2×ΓL(1,𝔽25)C2314yes—
124C2×D3×C8C2314yes—
125#897—24no—
126#896—24no—
127#896—24no—
128#893—22no—
129#892C2314yes—
130D3×C4⋊C4—24no—
131D3×C4⋊C4C2314yes—
132#890—22no—
133#890C2314yes—
134#889—24no—
135#889—24no—
136#888—22no—
137#888—24no—
138#888C2314yes—
139#887—22no—
140#887—24no—
141#875C2312yes—
142#874C2311yescharacteristic
143C4×D12—22no—
144C4×D12D412yes—
145#872C2312yes—
146C42×D3C2312yes—
147#870C2311yescharacteristic
148#870C2311yescharacteristic
149#869C2311yescharacteristic
150#869C2312yes—
151C4×Dic6—22no—
152C4×Dic6D412yes—
153C3×#102C2318yes—
154C3×#102C2318yes—
155C3×#97C2311yescharacteristic
156C3×#97C2311yescharacteristic
157C3×#97C2312yes—
158#832C2318yes—
159#832C2318yes—
160#821C2314yes—
161#821C2314yes—
162#816C2314yes—
163#816C2314yes—
164#815C2314yes—
165C8×Dic3C2314yes—
166#810C2318yes—
167#809C2318yes—
168#803C2312yes—
169C4×C3⋊C8C2312yes—
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#532—31no—
186C2×#127—32no—
187C2×#127C2×D311yescharacteristic
188C2×#125—32no—
189C2×#125C2×D311yescharacteristic, complemented
190#520—32no—
191C4×#41—31no—
192#491—32no—
193#491—31no—
194#491—31no—
195#490—32no—
196#490—32no—
197#490—32no—
198#490—38no—
199#490—31no—
200#490—31no—
201#490C2×D312yescomplemented
202#490C2×D314yescomplemented
203#490C2×D312yescomplemented
204#489—38no—
205#489—34no—
206#488—34no—
207#488—38no—
208#486—34no—
209#486—38no—
210#418—34no—
211#418—38no—
212#417—34no—
213#417—38no—
214#416—34no—
215#416—38no—
216C2×#102—38no—
217C2×#102C2×D314yescomplemented
218#405—32no—
219#405—34no—
220C4×M16—31no—
221C4×M16—32no—
222C2×#97—32no—
223C2×#97C2×D311yescharacteristic, complemented
224C24×C3—22no—
225C23×D3—24no—
226C2×C3×D4—24no—
227C2×C3×D4—22no—
228C2×C3×D4—24no—
229C2×C3×D4—28no—
230C2×C3×D4—84no—
231C2×C3×D4C2414yes—
232C22×C3×C4—22no—
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2411yescharacteristic
235C2×#67—22no—
236C2×#67—24no—
237C2×#67—24no—
238C2×#67—28no—
239C2×#67—416no—
240C2×#67C2×D412yes—
241C22×Dic3—24no—
242#237—216no—
243#237—48no—
244D3×D4—216no—
245D3×D4—48no—
246D3×D4—88no—
247#235—28no—
248#235#4118yes—
249C2×D12—22no—
250C2×D12C2×D411yescharacteristic
251C2×D12C2412yes—
252C2×C4×D3—22no—
253C2×C4×D3—24no—
254C2×C4×D3C2×D414yes—
255C2×C4×D3C2412yes—
256C2×Dic6—22no—
257C2×Dic6—24no—
258C2×Dic6C2×D411yescharacteristic
259C2×Dic6C2412yes—
260C2×C3×C8—24no—
261C2×C3×C8C2412yes—
262C2×C3×C8C2412yes—
263C3×C4⋊C4—22no—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—24no—
266C3×C4⋊C4C2414yes—
267C3×C42C2×D412yes—
268C3×C42C2411yescharacteristic
269C3×C42C2411yescharacteristic
270#217—22no—
271#217—28no—
272#217C2×D412yes—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4C2×D414yes—
276C4⋊Dic3—22no—
277C4⋊Dic3—22no—
278C4⋊Dic3—24no—
279C4⋊Dic3C2×D412yes—
280C4⋊Dic3C2412yes—
281C4⋊Dic3C2412yes—
282Dic3⋊C4—22no—
283Dic3⋊C4—22no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4C2×D414yes—
288C4×Dic3—22no—
289C4×Dic3—24no—
290C4×Dic3C2×D414yes—
291C4×Dic3C2412yes—
292#208—24no—
293C2×C3⋊C8C2414yes—
294ΓL(1,𝔽25)—28no—
295D3×C8—28no—
296H(5,𝔽2)—616no—
297C2×#41—31no—
298C2×#41—34no—
299C2×#41—32no—
300C2×#41—62no—
301C22×D4—22no—
302C22×D4—34no—
303C22×D4—64no—
304C22×D4—62no—
305C22×D4—64no—
306C22×D4C22×D312yes—
307#131—68no—
308C2×M16—32no—
309C2×M16—34no—
310C22×C8—34no—
311C22×C8C22×D312yes—
312C4⋊Q8—34no—
313C4⋊Q8—32no—
314C4⋊Q8—31no—
315C4⋊Q8—64no—
316C4⋊Q8—62no—
317#127—22no—
318#127—34no—
319#127—31no—
320#127—64no—
321#127C22×D312yes—
322#127C22×D312yes—
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#124—32no—
334C22⋊Q8—64no—
335C22⋊Q8—64no—
336C22⋊Q8—64no—
337C22⋊Q8—64no—
338C22⋊Q8—62no—
339#121—64no—
340#121—68no—
341#121—64no—
342#121—62no—
343#120—68no—
344#120—64no—
345C4×Q8—32no—
346C4×Q8—64no—
347C4×Q8—62no—
348C4×Q8—62no—
349C4×D4—34no—
350C4×D4—32no—
351C4×D4—64no—
352C4×D4—62no—
353C4×D4—64no—
354C4×D4—62no—
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#103—38no—
368#102—38no—
369#102—38no—
370#102—34no—
371#102—34no—
372#102—38no—
373#102C22×D318yes—
374#102C22×D318yes—
375#97—34no—
376#97—32no—
377#97C22×D311yescharacteristic
378#97C22×D311yescharacteristic
379#97C22×D312yes—
380C4×C8—34no—
381C4×C8—32no—
382C23×C3—24no—
383C23×C3—44no—
384C23×C3—84no—
385C23×C3C22×D411yescharacteristic
386C22×D3—22no—
387C22×D3—48no—
388C22×D3—88no—
389C22×D3C22×D412yes—
390C3×D4—28no—
391C3×D4—48no—
392C3×D4—44no—
393C3×D4—84no—
394C2×C3×C4—24no—
395C2×C3×C4—42no—
396C2×C3×C4—82no—
397C2×C3×C4C22×D412yes—
398C2×C3×C4C22×D412yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4114yes—
401C2×C3×C4C2511yescharacteristic
402#67—44no—
403#67—48no—
404#67—48no—
405#67—88no—
406#67#12418yes—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—48no—
411C2×Dic3C22×D414yes—
412C2×Dic3C22×D412yes—
413D12—24no—
414D12—44no—
415D12#12414yes—
416C4×D3—44no—
417C4×D3—44no—
418C4×D3—84no—
419C4×D3C2×#4118yes—
420Dic6—24no—
421Dic6—84no—
422Dic6—82no—
423Dic6#12414yes—
424C3×C8—22no—
425C3×C8—22no—
426C3⋊C8—24no—
427C24—22no—
428C24—64no—
429#41—38no—
430#41—616no—
431#41—68no—
432#41—128no—
433C2×Q8—31no—
434C2×Q8—32no—
435C2×Q8—64no—
436C2×Q8—62no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—32no—
442C2×D4—31no—
443C2×D4—32no—
444C2×D4—616no—
445C2×D4—64no—
446C2×D4—68no—
447C2×D4—64no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—84no—
451C2×D4—128no—
452C2×D4—1216no—
453C2×D4—248no—
454C2×D4C23×D314yes—
455C22×C4—22no—
456C22×C4—32no—
457C22×C4—34no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—62no—
461C22×C4D3×D412yes—
462C22×C4C23×D311yescharacteristic
463M16—68no—
464M16—64no—
465C2×C8—24no—
466C2×C8—34no—
467C2×C8—68no—
468C2×C8C23×D312yes—
469C2×C8C23×D312yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—34no—
474C4⋊C4—32no—
475C4⋊C4—32no—
476C4⋊C4—32no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4C23×D314yes—
486C22⋊C4—32no—
487C22⋊C4—34no—
488C22⋊C4—64no—
489C22⋊C4—68no—
490C22⋊C4—62no—
491C22⋊C4—64no—
492C42—34no—
493C42—32no—
494C42—64no—
495C42—62no—
496C42D3×D412yes—
497C42C23×D311yescharacteristic
498C42C23×D311yescharacteristic
499C22×C3—22no—
500C22×C3—48no—
501C22×C3—44no—
502C22×C3—164no—
503C22×C3C2×#12412yes—
504C22×C3C23×D411yescharacteristic
505C2×D3—24no—
506C2×D3—44no—
507C2×D3—42no—
508C2×D3—168no—
509C2×D3C2×#12414yes—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—44no—
513C3×C4C2×#12412yes—
514C3×C4C22×#4112yes—
515Dic3—24no—
516Dic3—42no—
517Dic3—42no—
518Dic3—42no—
519Dic3C2×#12414yes—
520C23—24no—
521C23—32no—
522C23—44no—
523C23—62no—
524C23—84no—
525C23—128no—
526C23—248no—
527C23C2×D3×D411yescharacteristic
528Q8—34no—
529Q8—64no—
530Q8—242no—
531Q8—244no—
532D4—28no—
533D4—38no—
534D4—34no—
535D4—48no—
536D4—44no—
537D4—64no—
538D4—84no—
539D4—124no—
540D4—128no—
541D4—128no—
542D4—124no—
543D4—248no—
544C2×C4—24no—
545C2×C4—32no—
546C2×C4—38no—
547C2×C4—34no—
548C2×C4—42no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—82no—
553C2×C4—124no—
554C2×C4—128no—
555C2×C4—124no—
556C2×C4—244no—
557C2×C4C2×D3×D412yes—
558C2×C4C2×D3×D412yes—
559C2×C4C2×D3×D411yescharacteristic
560C2×C4D3×#4114yes—
561C2×C4C24×D311yescharacteristic
562C8—22no—
563C8—22no—
564C8—64no—
565C2×C3—22no—
566C2×C3—84no—
567C2×C3C22×#12411yescharacteristic
568C2×C3#358512yes—
569D3—24no—
570D3—82no—
571C22—22no—
572C22—34no—
573C22—44no—
574C22—48no—
575C22—64no—
576C22—122no—
577C22—124no—
578C22—164no—
579C22—488no—
580C22D3×#12412yes—
581C22C22×D3×D411yescharacteristic
582C4—22no—
583C4—22no—
584C4—34no—
585C4—44no—
586C4—64no—
587C4—122no—
588C4—122no—
589C4—122no—
590C4D3×#12412yes—
591C4C2×D3×#4112yes—
592C3#6062611yescharacteristic, complemented, Sylow
593C2—22no—
594C2—64no—
595C2—84no—
596C2—242no—
597C2C2×D3×#12411yescharacteristic
598C2#8719212yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#490C2
socleC22×C3C23×D4
prime radical#86195C2
hypercenterC2×#490D3
hypocenterC3#60626
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
2. #86098C2semidirect
3. C2×C3×#490C2semidirect
4. #83159C2semidirect
5. #83012C2semidirect
6. #81716C2semidirect
7. #81715C2semidirect
8. #81714C2semidirect
9. #81713C2semidirect
10. D3×#490C2semidirect
11. #78742C2semidirect
12. #78737C2semidirect
13. #78497C2semidirect
14. #78486C2semidirect
18. C2×C3×#125C22semidirect
28. #5689C22semidirect
29. #5688C22semidirect
30. D3×#125C22semidirect
34. #5500C22semidirect
37. #5470C22semidirect
38. C3×#490C22semidirect
39. C3×#490C22semidirect
40. C3×#490C22semidirect
41. C2×C3×#102C22semidirect
42. C2×C3×#97C22semidirect
43. #5215C22semidirect
44. #5113C22semidirect
45. #5101C22semidirect
46. #5071C22semidirect
47. #5071C22semidirect
48. #5070C22semidirect
49. #5069C22semidirect
50. #5068C22semidirect
51. #5067C22semidirect
52. #4966C22semidirect
53. C4×#208C22semidirect
54. #4772C22semidirect
55. #4771C22semidirect
56. #4770C22semidirect
57. D3×#102C22semidirect
58. #4768C22semidirect
59. #4765C22semidirect
60. #4764C22semidirect
61. #4760C22semidirect
62. #4714C22semidirect
63. #4713C22semidirect
64. #4713C22semidirect
65. #4708C22semidirect
66. #4707C22semidirect
67. D3×#97C22semidirect
68. #4703C22semidirect
69. #4703C22semidirect
81. C2×#490D3semidirect
189. C2×#125C2×D3semidirect
201. #490C2×D3semidirect
202. #490C2×D3semidirect
203. #490C2×D3semidirect
217. C2×#102C2×D3semidirect
223. C2×#97C2×D3semidirect
592. C3#60626semidirect

Maximal subgroups

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

subgroupindex
#861952
#860982
C2×C3×#4902
#831592
#830122
#817162
#817152
#817142
#817132
D3×#4902
#787422
#787372
#784972
#784862
#606263