Finite Group Databasebeta

← #30650749

Subgroups — #30650749

Order 16256.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550561C211yesmaximal, characteristic, complemented
2#22550510C211yesmaximal, characteristic
3#22550239C211yesmaximal, characteristic, complemented
4#22550203C211yesmaximal, characteristic, complemented
5#22550179C211yesmaximal, characteristic, complemented
6#22550123C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550075C211yesmaximal, characteristic, complemented
9#22550029C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22549975C211yesmaximal, characteristic, complemented
12#22549944C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic, complemented
14C2×#16861129C211yesmaximal, characteristic, complemented
15#480×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×#14523237C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523231C2211yescharacteristic, complemented
23C22×C127⋊C8C2211yescharacteristic, complemented
24#16861205C2211yescharacteristic
25#16861204—21no—
26#16861200C2211yescharacteristic, complemented
27#16861199—21no—
28#16861192—21no—
29#16861189C2211yescharacteristic, complemented
30#16861188C2211yescharacteristic, complemented
31#16861184C2211yescharacteristic
32#16861182—21no—
33#16861179—21no—
34#16861177—21no—
35#16861174C2211yescharacteristic, complemented
36#16861171C2211yescharacteristic
37C4×C22⋊D127C2211yescharacteristic
38#16861168—21no—
39#16861167C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic, complemented
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52#121×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic, complemented
54#107×C127C2211yescharacteristic, complemented
55#107×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C23×D127—21no—
61C2×Dic254C2311yescharacteristic
62C2×D508—22no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—41no—
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127D411yescharacteristic
72D4×D127—22no—
73D4×D127—41no—
74D4×D127—44no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523237—22no—
78#14523237—41no—
79#14523236—41no—
80#14523231—41no—
81C2×C127⋊C8—41no—
82C2×C127⋊C8C2311yescharacteristic
83C2×C127⋊C8C2311yescharacteristic
84C4⋊Dic127—21no—
85C4⋊Dic127—21no—
86C4⋊Dic127C2311yescharacteristic
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4D411yescharacteristic
91C22⋊Dic127—21no—
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×D4×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—42no—
115C22×D127—42no—
116C22×D127—42no—
117C22×D127—42no—
118C22×D127—81no—
119C22×D127—81no—
120C22×D127C2×D411yescharacteristic
121Dic254—21no—
122Dic254—21no—
123D508—21no—
124D508—21no—
125D508—21no—
126D508—42no—
127D508—42no—
128D508#4112yes—
129C22⋊D127—22no—
130C22⋊D127—42no—
131C22⋊D127—42no—
132C22⋊D127—81no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127C2×D411yescharacteristic
136C2×Dic127C2×D411yescharacteristic
137C2×Dic127C2×D411yescharacteristic
138C4×D127—21no—
139C4×D127—21no—
140C4×D127—42no—
141C127⋊C8—21no—
142C127⋊C8—21no—
143C23×C127—21no—
144C23×C127C2×D411yescharacteristic
145D4×C127—21no—
146D4×C127—21no—
147D4×C127—42no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—42no—
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127AGL(1,ℤ/8)12yes—
164C22×C127C22×D411yescharacteristic
165C2×D127—21no—
166C2×D127—22no—
167C2×D127—21no—
168C2×D127—21no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—84no—
175C2×D127—84no—
176C4×C127—21no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—41no—
180C4×C127C2×#4111yescharacteristic
181C4×C127H(5,𝔽2)11yescharacteristic
182Dic127—21no—
183Dic127—21no—
184Dic127—21no—
185Dic127—41no—
186Dic127—41no—
187C2×C127—22no—
188C2×C127—81no—
189C2×C127#54611yescharacteristic
190C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C127#57711yescharacteristic
192D127—41no—
193D127—42no—
194D127—81no—
195#3310—1271nomaximal, Sylow
196C127#331011yescharacteristic, complemented, Sylow
197#552—1271no—
198D42—1271no—
199#490—1271no—
200#482—1271no—
201#480D12711yescharacteristic, complemented
202#478—1271no—
203#468—1271no—
204#465—1271no—
205#460—1271no—
206#450—1271no—
207#450—1271no—
208#440—1271no—
209#435—1271no—
210#418—1271no—
211C2×#102—1271no—
212C22×D4—1271no—
213C22×D4—1271no—
214C22×D4—2541no—
215C2×SD16—1271no—
216C2×SD16—1271no—
217C2×SD16—1271no—
218C2×M16—1271no—
219C22×C8—1271no—
220#127—1271no—
221#126—2541no—
222#125—1271no—
223#125—2541no—
224#123—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8—1271no—
227#121—1271no—
228#121—2541no—
229#121C2×D12711yescharacteristic
230#120—2541no—
231#120—2541no—
232C4×Q8—1271no—
233C4×D4—1271no—
234C4×D4—1271no—
235#117—1271no—
236#117—2541no—
237C2×C4⋊C4C2×D12711yescharacteristic, complemented
238#107C2×D12711yescharacteristic, complemented
239#107C2×D12711yescharacteristic, complemented
240#106—1271no—
241C4⋊C8—1271no—
242#103—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102C2×D12711yescharacteristic, complemented
251#102C2×D12711yescharacteristic, complemented
252C22⋊C8—1271no—
253C22⋊C8C2×D12711yescharacteristic, complemented
254#97—1271no—
255C24—2541no—
256C24—2541no—
257C2×Q8—1271no—
258C2×D4—21no—
259C2×D4—1271no—
260C2×D4—1271no—
261C2×D4—1271no—
262C2×D4—5081no—
263C2×D4—5084no—
264C2×D4—5081no—
265C2×D4—2542no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2542no—
271C2×D4—2541no—
272C2×D4C22×D12711yescharacteristic
273C22×C4—21no—
274C22×C4—1271no—
275C22×C4—2541no—
276C22×C4C22×D12711yescharacteristic
277SD16—5081no—
278SD16—5081no—
279SD16—2542no—
280M16—5081no—
281C2×C8—1271no—
282C2×C8—1271no—
283C2×C8—5081no—
284C2×C8C22×D12711yescharacteristic
285C2×C8C22×D12711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—1271no—
288C4⋊C4—1271no—
289C4⋊C4—2541no—
290C4⋊C4—2541no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4C22×D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C4⋊C4C22×D12711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—1271no—
299C22⋊C4—1271no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C42—1271no—
306C42—2541no—
307C42—2541no—
308C23—21no—
309C23—1271no—
310C23—10161no—
311C23—10161no—
312C23—5082no—
313C23—5082no—
314C23—5082no—
315C23—5082no—
316C23—2541no—
317C23—2541no—
318C23D4×D12711yescharacteristic
319Q8—2541no—
320Q8—2541no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—1272no—
325D4—10161no—
326D4—5082no—
327D4—5082no—
328D4—5082no—
329D4—5082no—
330D4—2542no—
331D4—2541no—
332D4—2541no—
333D4—2541no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—1271no—
341C2×C4—1271no—
342C2×C4—1271no—
343C2×C4—5082no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4D4×D12711yescharacteristic
349C2×C4C23×D12711yescharacteristic
350C8—21no—
351C8—21no—
352C8—2541no—
353C8—2541no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—10164no—
360C22—10164no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—5081no—
366C22—2541no—
367C22—2541no—
368C22—2541no—
369C22—2542no—
370C22#1686123312yes—
371C22C2×D4×D12711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—5081no—
377C4—5081no—
378C4—2541no—
379C4—2541no—
380C4—2541no—
381C4#41×D12711yescharacteristic
382C4#1686125511yescharacteristic
383C2—22no—
384C2—81no—
385C2—10161no—
386C2—5082no—
387C2—5081no—
388C2#2255051811yescharacteristic
389C2C2×#1686123311yescharacteristic
390C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#480×C127C2
socleC22×C127C22×D4
prime radical#22550510C2
hypercenter#480D127
hypocenterC127#3310
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #22550561C2semidirect
3. #22550239C2semidirect
4. #22550203C2semidirect
5. #22550179C2semidirect
6. #22550123C2semidirect
7. #22550103C2semidirect
8. #22550075C2semidirect
9. #22550029C2semidirect
10. #22550028C2semidirect
11. #22549975C2semidirect
12. #22549944C2semidirect
13. #22549879C2semidirect
14. C2×#16861129C2semidirect
15. #480×C127C2semidirect
20. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
22. C2×#14523231C22semidirect
23. C22×C127⋊C8C22semidirect
26. #16861200C22semidirect
29. #16861189C22semidirect
30. #16861188C22semidirect
35. #16861174C22semidirect
39. #16861167C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861132C22semidirect
44. #16861131C22semidirect
45. #16861130C22semidirect
48. #16861129C22semidirect
49. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861119C22semidirect
53. C2×C4⋊C4×C127C22semidirect
54. #107×C127C22semidirect
55. #107×C127C22semidirect
56. #102×C127C22semidirect
57. #102×C127C22semidirect
58. C22⋊C8×C127C22semidirect
196. C127#3310semidirect
201. #480D127semidirect
237. C2×C4⋊C4C2×D127semidirect
238. #107C2×D127semidirect
239. #107C2×D127semidirect
250. #102C2×D127semidirect
251. #102C2×D127semidirect
253. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505612
#225505102
#225502392
#225502032
#225501792
#225501232
#225501032
#225500752
#225500292
#225500282
#225499752
#225499442
#225498792
C2×#168611292
#480×C1272
#3310127