Finite Group Databasebeta

← #29835182

Subgroups — #29835182

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456470C211yesmaximal, characteristic, complemented
2#21454666C211yesmaximal, characteristic, complemented
3#21448013C211yesmaximal, characteristic, complemented
4#21448012C211yesmaximal, characteristic, complemented
5#21447973C211yesmaximal, characteristic, complemented
6#21447855C211yesmaximal, characteristic, complemented
7#21447854C211yesmaximal, characteristic, complemented
8#21447732C211yesmaximal, characteristic, complemented
9#21447732C211yesmaximal, characteristic, complemented
10#21447694C211yesmaximal, characteristic, complemented
11#21447692C211yesmaximal, characteristic, complemented
12#21447665C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14#21447625C211yesmaximal, characteristic, complemented
15C61×#2228C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840368—21no—
18#16840234—21no—
19C2×D976C2211yescharacteristic
20C2×D976C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519718C2211yescharacteristic
23#16840139C2211yescharacteristic, complemented
24#16840128—21no—
25#16840016—21no—
26#16839999C2211yescharacteristic
27#16839999C2211yescharacteristic
28#16839873—21no—
29#16839872—21no—
30#16839871—21no—
31C4×D488C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33#16839844C2211yescharacteristic
34#16839844C2211yescharacteristic, complemented
35#16839844C2211yescharacteristic, complemented
36#16839799C2211yescharacteristic
37#16839765—21no—
38#16839716C2211yescharacteristic
39#16839589C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41C4⋊#11758185C2211yescharacteristic, complemented
42#16839564C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic, complemented
44#16839562C2211yescharacteristic
45#16839562C2211yescharacteristic
46#16839562C2211yescharacteristic, complemented
47#16839562C2211yescharacteristic, complemented
48#16839541C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C61×#499C2211yescharacteristic, complemented
53C4×D8×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#366C2211yescharacteristic, complemented
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×D4×D61—21no—
60C2×D488—21no—
61C2×D488C2311yescharacteristic
62C2×D488C2311yescharacteristic
63C2×D488C2311yescharacteristic
64C2×#11758199—21no—
65D8×D61—22no—
66D8×D61—22no—
67C2×#11758189D411yescharacteristic
68C2×C8×D61C2311yescharacteristic
69#14519774—21no—
70#14519762—41no—
71#14519762—41no—
72#14519752—21no—
73#14519752—21no—
74#14519752—41no—
75#14519752—41no—
76C4×D244—21no—
77D4×Dic61—21no—
78#14519739—21no—
79#14519739—21no—
80#14519739—21no—
81C4×C22⋊D61—41no—
82#14519737—41no—
83#14519735—21no—
84#14519734D411yescharacteristic
85C2×#11758198—41no—
86C22⋊C4×D61—41no—
87D976—22no—
88D976—22no—
89#14519718—22no—
90#14519718—22no—
91C2×#11758185C2311yescharacteristic
92C2×#11758185C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519708—21no—
95#14519699—21no—
96#14519698—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519688D411yescharacteristic
103#14519687D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106#125×C61—21no—
107C4×D4×C61—21no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#106×C61—21no—
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C23×D61—41no—
117C2×D4×C61—21no—
118C22×C4×C61—41no—
119C2×C22⋊D61—41no—
120C22×Dic61—41no—
121D4×D61—24no—
122D4×D61—42no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—41no—
127C2×D244—41no—
128C2×D244—41no—
129C2×D244—41no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61C2×D411yescharacteristic
135D8×C61—21no—
136D8×C61#4112yes—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61C2×D411yescharacteristic
146C22⋊Dic61—41no—
147#11758199—24no—
148#11758198—41no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198D811yescharacteristic
156#11758198D811yescharacteristic
157C4⋊Dic61—21no—
158#11758196D811yescharacteristic
159#11758196D811yescharacteristic
160C4×Dic61—41no—
161C4×Dic61C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164D488—21no—
165D488—21no—
166D488—21no—
167D488—21no—
168D488—21no—
169D488—42no—
170D488#4112yes—
171#11758189—41no—
172C8×D61—21no—
173C8×D61—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C22×D61—44no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—161no—
184C22×D61—161no—
185C22×D61C2×D811yescharacteristic
186D4×C61—22no—
187D4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—161no—
192C2×C4×C61C2×D811yescharacteristic
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C22⋊D61—44no—
196C2×Dic61—41no—
197C2×Dic61—161no—
198C2×Dic61C2×D811yescharacteristic
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201D244—22no—
202D244—41no—
203D244—41no—
204D244—41no—
205D244—41no—
206D244—41no—
207D244—161no—
208D244—161no—
209D244—161no—
210D244—161no—
211C4×D61—21no—
212C4×D61—21no—
213C4×D61—161no—
214C4×D61—161no—
215C4×D61—161no—
216C4×D61—161no—
217C8×C61—21no—
218C8×C61C2×#4111yescharacteristic
219C8×C61C2×#4111yescharacteristic
220#92418—21no—
221#92418—21no—
222#33682—611nomaximal, Sylow
223C22×C61—42no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61C22×D811yescharacteristic
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—42no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—164no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61#52511yescharacteristic
246C4×C61#57511yescharacteristic
247Dic61—21no—
248Dic61—21no—
249Dic61—21no—
250Dic61—81no—
251#3176—611no—
252#2948—611no—
253#2234—611no—
254#2234—611no—
255#2228—611no—
256#2228D6111yescharacteristic, complemented
257#2206—611no—
258#2206—611no—
259#2184—611no—
260#2184—611no—
261#2177—611no—
262#2177—611no—
263C4×D16—611no—
264C4×D16—611no—
265#2164—611no—
266C2×C61—82no—
267C2×C61#310011yescharacteristic
268C2×C61#341511yescharacteristic
269C2×C61#341511yescharacteristic
270D61—41no—
271D61—82no—
272D61—161no—
273D61—161no—
274C22×D8—611no—
275#540—1221no—
276#518—1221no—
277C2×D16—611no—
278C2×D16—611no—
279C2×D16—611no—
280C2×D16—611no—
281#499C2×D6111yescharacteristic, complemented
282#493—611no—
283#490—1221no—
284#469—1221no—
285#466—611no—
286#466—611no—
287#442—1221no—
288#442—1221no—
289#442—1221no—
290C4×D8—611no—
291C4×D8—611no—
292C4×D8—611no—
293C4×D8—611no—
294C4×D8—611no—
295C4×D8C2×D6111yescharacteristic
296#427—611no—
297#418—1221no—
298#405—611no—
299#367—611no—
300#367C2×D6111yescharacteristic, complemented
301#366—611no—
302#366C2×D6111yescharacteristic, complemented
303C4⋊C16—611no—
304C4⋊C16C2×D6111yescharacteristic, complemented
305#357—611no—
306#357—611no—
307#357—611no—
308#357—611no—
309#357—611no—
310#357—611no—
311#357C2×D6111yescharacteristic
312#357C2×D6111yescharacteristic
313C22⋊C16—611no—
314C22⋊C16—611no—
315#346—611no—
316C4×C16—611no—
317C61#3368211yescharacteristic, complemented, Sylow
318C22×D4—1221no—
319C2×D8—611no—
320C2×D8—611no—
321C2×D8—611no—
322C2×D8—1221no—
323C2×D8—1222no—
324C2×D8—1222no—
325C2×D8—1221no—
326C2×D8C22×D6111yescharacteristic
327C2×M16—611no—
328C22×C8—611no—
329#127—1221no—
330#125—21no—
331#123—2441no—
332#123—2441no—
333#121—1221no—
334#121—1221no—
335#121—2441no—
336#121—2441no—
337C4×D4—21no—
338C4×D4—1221no—
339C4×D4—1221no—
340C4×D4—1221no—
341C4×D4—1221no—
342C4×D4—1221no—
343C4×D4—2441no—
344C4×D4—2441no—
345#117—611no—
346#117—1221no—
347C2×C22⋊C4—2441no—
348C2×C22⋊C4—2441no—
349D16—1222no—
350D16—1222no—
351D16—1222no—
352D16—1222no—
353C2×C16—611no—
354C2×C16—611no—
355C2×C16C22×D6111yescharacteristic
356C2×C16C22×D6111yescharacteristic
357#107—611no—
358#107C22×D6111yescharacteristic
359#107C22×D6111yescharacteristic
360#107C22×D6111yescharacteristic
361#106—21no—
362#106—1221no—
363#102—21no—
364#102—1221no—
365#102—1221no—
366#102—1221no—
367#102—1221no—
368#102—1221no—
369#102—1221no—
370#102—1221no—
371#97—611no—
372#97—611no—
373C4×C8—611no—
374C4×C8C22×D6111yescharacteristic
375C24—2441no—
376C2×D4—21no—
377C2×D4—1224no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—2441no—
382C2×D4—2441no—
383C2×D4—2441no—
384C2×D4—2442no—
385C2×D4—2441no—
386C2×D4—2441no—
387C22×C4—41no—
388C22×C4—611no—
389C22×C4—2441no—
390C22×C4—2441no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394D8—21no—
395D8—612no—
396D8—1224no—
397D8—1221no—
398D8—1221no—
399D8—1221no—
400D8—1221no—
401D8—1221no—
402D8—2442no—
403D8#1175822612yes—
404M16—2441no—
405C2×C8—611no—
406C2×C8—611no—
407C2×C8—1221no—
408C2×C8—1221no—
409C2×C8D4×D6111yescharacteristic
410C2×C8C23×D6111yescharacteristic
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—611no—
417C4⋊C4—611no—
418C4⋊C4—1221no—
419C22⋊C4—41no—
420C22⋊C4—611no—
421C22⋊C4—611no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4—2441no—
430C42—611no—
431C42—2441no—
432C42D4×D6111yescharacteristic
433C16—21no—
434C16—21no—
435C16—1221no—
436C16—1221no—
437C23—41no—
438C23—611no—
439C23—9761no—
440C23—9761no—
441C23—2444no—
442C23—2441no—
443C23—2441no—
444C23—2441no—
445D4—22no—
446D4—41no—
447D4—1222no—
448D4—9761no—
449D4—9761no—
450D4—9761no—
451D4—9761no—
452D4—2444no—
453D4—2441no—
454D4—2441no—
455D4—2441no—
456D4—2441no—
457D4—2441no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—161no—
462C2×C4—611no—
463C2×C4—611no—
464C2×C4—611no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—9761no—
468C2×C4—9761no—
469C2×C4—9761no—
470C2×C4—9761no—
471C2×C4—9761no—
472C2×C4—2441no—
473C2×C4D8×D6111yescharacteristic
474C2×C4D8×D6111yescharacteristic
475C2×C4C2×D4×D6111yescharacteristic
476C8—21no—
477C8—1221no—
478C8—1221no—
479C8#41×D6111yescharacteristic
480C8C2×#1175822611yescharacteristic
481C22—42no—
482C22—81no—
483C22—81no—
484C22—1221no—
485C22—1221no—
486C22—1221no—
487C22—9764no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22—4881no—
493C22—4881no—
494C22—4881no—
495C22—4881no—
496C22—2442no—
497C22C2×D8×D6111yescharacteristic
498C4—21no—
499C4—21no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—1221no—
504C4—1221no—
505C4—1221no—
506C4—4881no—
507C4#1684027511yescharacteristic
508C4#1684056111yescharacteristic
509C2—82no—
510C2—9761no—
511C2—9761no—
512C2—4882no—
513C2—2441no—
514C2#2145582111yescharacteristic
515C2#2145895711yescharacteristic
516C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2228C2
socleC22×C61C22×D8
hypercenter#2228D61
hypocenterC61#33682
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #21456470C2semidirect
2. #21454666C2semidirect
3. #21448013C2semidirect
4. #21448012C2semidirect
5. #21447973C2semidirect
6. #21447855C2semidirect
7. #21447854C2semidirect
8. #21447732C2semidirect
9. #21447732C2semidirect
10. #21447694C2semidirect
11. #21447692C2semidirect
12. #21447665C2semidirect
13. #21447663C2semidirect
14. #21447625C2semidirect
15. C61×#2228C2semidirect
23. #16840139C22semidirect
34. #16839844C22semidirect
35. #16839844C22semidirect
39. #16839589C22semidirect
40. #16839587C22semidirect
41. C4⋊#11758185C22semidirect
42. #16839564C22semidirect
43. #16839564C22semidirect
46. #16839562C22semidirect
47. #16839562C22semidirect
50. #16839537C22semidirect
51. C16×Dic61C22semidirect
52. C61×#499C22semidirect
54. C61×#367C22semidirect
55. C61×#366C22semidirect
56. C61×C4⋊C16C22semidirect
256. #2228D61semidirect
281. #499C2×D61semidirect
300. #367C2×D61semidirect
302. #366C2×D61semidirect
304. C4⋊C16C2×D61semidirect
317. C61#33682semidirect

Maximal subgroups

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

subgroupindex
#214564702
#214546662
#214480132
#214480122
#214479732
#214478552
#214478542
#214477322
#214476942
#214476922
#214476652
#214476632
#214476252
C61×#22282
#3368261