Finite Group Databasebeta

← #29834365

Subgroups — #29834365

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840196C211yesmaximal, characteristic, complemented
2C2×#16840193C211yesmaximal, characteristic, complemented
3#21456301C211yesmaximal, characteristic, complemented
4#21454689C211yesmaximal, characteristic, complemented
5#21447884C211yesmaximal, characteristic, complemented
6#21447878C211yesmaximal, characteristic, complemented
7#21447876C211yesmaximal, characteristic, complemented
8#21447860C211yesmaximal, characteristic, complemented
9#21447858C211yesmaximal, characteristic, complemented
10#21447854C211yesmaximal, characteristic, complemented
11#21447852C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447530C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C2×#14519800C2211yescharacteristic
18C2×#14519796C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840330—21no—
21#16840241—21no—
22#16840196—24no—
23#16840193—24no—
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519721C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519716C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840152—21no—
33#16840148—21no—
34#16840147C2211yescharacteristic
35#16840139C2211yescharacteristic
36#16840137C2211yescharacteristic
37#16839862—21no—
38#16839859—21no—
39C4×Dic244C2211yescharacteristic, complemented
40#16839856C2211yescharacteristic, complemented
41D8×Dic61C2211yescharacteristic, complemented
42#107×D61C2211yescharacteristic
43#16839760—21no—
44#16839716C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839580C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic
52#16839562C2211yescharacteristic
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic
57C4×D8×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758224—21no—
63C2×D4×D61—21no—
64#14519804—44no—
65#14519800—22no—
66#14519800—22no—
67#14519796—22no—
68#14519796D414yescomplemented
69C2×Dic244C2311yescharacteristic
70C2×#11758200—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×D488—21no—
74C2×D488C2311yescharacteristic
75C2×D488C2311yescharacteristic
76C2×#11758199—21no—
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80C2×#11758189D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519774—21no—
83#14519773—21no—
84#14519766—21no—
85#14519750—41no—
86#14519750—41no—
87#14519747—41no—
88#14519747—41no—
89C4×Dic122—21no—
90#14519742—21no—
91D4×Dic61—21no—
92C4×C22⋊D61—41no—
93#14519737—41no—
94#14519734D411yescharacteristic
95#14519733—41no—
96C2×#11758196—41no—
97C4⋊C4×D61—21no—
98#14519723—22no—
99#14519722—22no—
100#14519722—22no—
101#14519721—22no—
102#14519718—22no—
103#14519718—22no—
104#14519716—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108#14519711C2311yescharacteristic
109#14519710—21no—
110#14519702—21no—
111#14519700—21no—
112#14519698—21no—
113#14519688D411yescharacteristic
114#14519687D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×SD16×C61—21no—
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120#124×C61—21no—
121C4×D4×C61—21no—
122SD32×C61—22no—
123D16×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#102×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C23×D61—41no—
130C2×Q8×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C22×C4×C61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140#11758226—42no—
141#11758226—42no—
142#11758224—24no—
143#11758224—42no—
144D4×D61—42no—
145D4×D61—42no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—41no—
149C2×D244—41no—
150C2×D244—41no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×C4×D61C2×D411yescharacteristic
154C2×Dic122—21no—
155Q16×C61—21no—
156Q16×C61—21no—
157SD16×C61—42no—
158D8×C61—21no—
159D8×C61—21no—
160D8×C61—21no—
161D8×C61C2×D412yes—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2411yescharacteristic
164C4⋊C4×C61—21no—
165C22⋊C4×C61—41no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C42×C61C2×D411yescharacteristic
169C22⋊Dic61—41no—
170#11758201—42no—
171#11758201—42no—
172#11758200—24no—
173#11758199—42no—
174#11758199—42no—
175#11758198—41no—
176#11758198—41no—
177#11758198D811yescharacteristic
178#11758198D811yescharacteristic
179C4⋊Dic61—21no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196D811yescharacteristic
184#11758196D811yescharacteristic
185C4×Dic61—41no—
186C4×Dic61—41no—
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244C2×D412yes—
192D488—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—42no—
197#11758189—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C23×C61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—82no—
209C22×D61—161no—
210C22×D61—161no—
211C22×D61C2×D811yescharacteristic
212Q8×C61—41no—
213Q8×C61—41no—
214D4×C61—22no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—161no—
221C2×C4×C61C2×D811yescharacteristic
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C22⋊D61—44no—
225C22⋊D61—82no—
226C22⋊D61—161no—
227C22⋊D61—161no—
228C2×Dic61—41no—
229C2×Dic61—44no—
230C2×Dic61—41no—
231C2×Dic61—161no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234C2×Dic61C2×D811yescharacteristic
235D244—41no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—82no—
240D244—161no—
241D244—161no—
242C4×D61—82no—
243C4×D61—161no—
244C4×D61#12112yes—
245C4×D61AGL(1,ℤ/8)12yes—
246Dic122—22no—
247Dic122—41no—
248C8×C61—21no—
249C8×C61C22×D411yescharacteristic
250C8×C61C2×#4111yescharacteristic
251#92418—21no—
252#92418#12112yes—
253#33589—611nomaximal, Sylow
254C22×C61—42no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61C22×D811yescharacteristic
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—164no—
270C2×D61#45912yes—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C2×#12111yescharacteristic
276C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—42no—
280Dic61—81no—
281Dic61#45912yes—
282C2×#509—611no—
283C2×#508—611no—
284#3155—611no—
285#2951—611no—
286#2211—611no—
287#2211—611no—
288#2210—611no—
289#2210D6111yescharacteristic, complemented
290#2207—611no—
291#2207—611no—
292#2206—611no—
293#2206—611no—
294#2151—611no—
295#2139—611no—
296C2×#357—611no—
297C2×C61—82no—
298C2×C61—161no—
299C2×C61C2×#45911yescharacteristic
300C2×C61#341511yescharacteristic
301C2×C61#341611yescharacteristic
302D61—22no—
303D61—161no—
304D61—161no—
305C2×AGL(1,ℤ/8)—1221no—
306C2×#135—611no—
307C2×#135—611no—
308C22×D8—611no—
309#535—1221no—
310#520—1221no—
311#509—1224no—
312#508—1224no—
313C2×Q32—611no—
314C2×SD32—611no—
315C2×SD32—611no—
316C2×SD32—611no—
317C2×SD32C2×D6111yescharacteristic, complemented
318C2×D16—611no—
319C2×D16—611no—
320C2×D16C2×D6111yescharacteristic
321C2×M32—611no—
322C22×C16—611no—
323#496—1221no—
324#496—1221no—
325#495—611no—
326#495C2×D6111yescharacteristic
327#493—611no—
328#493—611no—
329#440—1221no—
330#440—1221no—
331C4×Q16—611no—
332C4×Q16—611no—
333C4×D8—611no—
334C4×D8C2×D6111yescharacteristic, complemented
335C2×#107—611no—
336#417—1221no—
337#405—611no—
338C4⋊C16—611no—
339C4⋊C16—611no—
340C4⋊C16—611no—
341C4⋊C16C2×D6111yescharacteristic, complemented
342#358—611no—
343#358C2×D6111yescharacteristic, complemented
344#357—611no—
345#357—611no—
346#357—611no—
347#357—611no—
348#357—611no—
349#357C2×D6111yescharacteristic
350C61#3358911yescharacteristic, complemented, Sylow
351C2×#41—1221no—
352C2×#41—1221no—
353C22×D4—1221no—
354AGL(1,ℤ/8)—2444no—
355#135—614no—
356#135—1222no—
357#135—1222no—
358#135—1222no—
359C2×Q16—611no—
360C2×Q16C22×D6111yescharacteristic
361C2×SD16—21no—
362C2×SD16—1221no—
363C2×SD16—1221no—
364C2×SD16—1221no—
365C2×D8—611no—
366C2×D8—611no—
367C2×D8—1222no—
368C2×D8—1222no—
369C2×D8—1221no—
370C2×D8—1221no—
371C2×D8—1221no—
372C2×D8C22×D6111yescharacteristic
373C2×D8C22×D6111yescharacteristic
374C2×M16—611no—
375C22×C8—611no—
376#127—1221no—
377#127—1221no—
378#124—21no—
379#124—1221no—
380#121—2441no—
381#121—2441no—
382#120—2441no—
383#120—2441no—
384C4×Q8—1221no—
385C4×Q8—1221no—
386C4×D4—21no—
387C4×D4—1221no—
388C4×D4—2441no—
389C4×D4—2441no—
390#117—611no—
391#117—2441no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—2441no—
394Q32—1222no—
395SD32—22no—
396SD32—1222no—
397SD32—1222no—
398SD32—1222no—
399D16—22no—
400D16—1222no—
401D16—1222no—
402M32—1222no—
403C2×C16—611no—
404C2×C16—611no—
405C2×C16—1222no—
406C2×C16C22×D6111yescharacteristic
407C2×C16C22×D6111yescharacteristic
408#107—611no—
409#107—1221no—
410#107C22×D6111yescharacteristic
411#103—1221no—
412#103—1221no—
413#102—21no—
414#102—1221no—
415#97—611no—
416#97—611no—
417C4×C8—611no—
418C4×C8C22×D6111yescharacteristic
419C24—2441no—
420#41—1224no—
421#41—2442no—
422#41—2442no—
423#41—2442no—
424C2×Q8—21no—
425C2×Q8—1221no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—2441no—
431C2×D4—2442no—
432C2×D4—2442no—
433C2×D4—2441no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2441no—
439C22×C4—41no—
440C22×C4—611no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445Q16—21no—
446Q16—21no—
447Q16—612no—
448Q16—1221no—
449SD16—42no—
450SD16—1224no—
451SD16—2442no—
452SD16—2442no—
453D8—21no—
454D8—21no—
455D8—21no—
456D8—1221no—
457D8—1221no—
458D8—1221no—
459D8—1221no—
460D8—2442no—
461D8—2442no—
462D8—2442no—
463D8D4×D6112yes—
464M16—1222no—
465C2×C8—611no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—611no—
469C2×C8D4×D6111yescharacteristic
470C2×C8C23×D6111yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—611no—
482C22⋊C4—611no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—611no—
487C42—2441no—
488C42—2441no—
489C42D4×D6111yescharacteristic
490C16—21no—
491C16—21no—
492C16—1221no—
493C16—1221no—
494C23—41no—
495C23—41no—
496C23—611no—
497C23—9761no—
498C23—9761no—
499C23—4882no—
500C23—2441no—
501C23—2441no—
502Q8—41no—
503Q8—41no—
504Q8—1222no—
505Q8—2441no—
506D4—22no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—9761no—
511D4—9761no—
512D4—9761no—
513D4—9761no—
514D4—4882no—
515D4—4882no—
516D4—2444no—
517D4—2441no—
518D4—2441no—
519D4—2441no—
520D4—2441no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—161no—
524C2×C4—611no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—9761no—
530C2×C4—9761no—
531C2×C4—4882no—
532C2×C4—2444no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4D8×D6111yescharacteristic
536C2×C4D8×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C8—21no—
539C8—612no—
540C8—1221no—
541C8C2×D4×D6111yescharacteristic
542C8#41×D6111yescharacteristic
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—612no—
550C22—1221no—
551C22—1221no—
552C22—9764no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22C2×D8×D6111yescharacteristic
560C4—21no—
561C4—21no—
562C4—81no—
563C4—81no—
564C4—612no—
565C4—1221no—
566C4—1221no—
567C4—4881no—
568C4—2442no—
569C4#121×D6111yescharacteristic
570C4AGL(1,ℤ/8)×D6111yescharacteristic
571C2—82no—
572C2—161no—
573C2—1222no—
574C2—9761no—
575C2—9761no—
576C2#459×D6111yescharacteristic
577C2#2145895911yescharacteristic
578C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33589
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. C2×#16840193C2semidirect
3. #21456301C2semidirect
4. #21454689C2semidirect
5. #21447884C2semidirect
6. #21447878C2semidirect
7. #21447876C2semidirect
8. #21447860C2semidirect
9. #21447858C2semidirect
10. #21447854C2semidirect
11. #21447852C2semidirect
12. #21447576C2semidirect
13. #21447530C2semidirect
14. #357×D61C2semidirect
15. C61×#2210C2semidirect
24. C2×#14519723C22semidirect
26. C2×#14519722C22semidirect
29. C2×#14519718C22semidirect
39. C4×Dic244C22semidirect
40. #16839856C22semidirect
41. D8×Dic61C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839580C22semidirect
47. #16839580C22semidirect
48. #16839566C22semidirect
50. #16839564C22semidirect
53. #16839562C22semidirect
54. C2×SD32×C61C22semidirect
57. C4×D8×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
68. #14519796D4semidirect
289. #2210D61semidirect
317. C2×SD32C2×D61semidirect
334. C4×D8C2×D61semidirect
341. C4⋊C16C2×D61semidirect
343. #358C2×D61semidirect
350. C61#33589semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
C2×#168401932
#214563012
#214546892
#214478842
#214478782
#214478762
#214478602
#214478582
#214478542
#214478522
#214475762
#214475302
#357×D612
C61×#22102
#3358961