Finite Group Databasebeta

← #30650763

Subgroups — #30650763

Order 16256.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550600C211yesmaximal, characteristic, complemented
2#22550521C211yesmaximal, characteristic
3#22550238C211yesmaximal, characteristic, complemented
4#22550220C211yesmaximal, characteristic, complemented
5#22550202C211yesmaximal, characteristic, complemented
6#22550135C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550099C211yesmaximal, characteristic, complemented
9#22550039C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12#22549946C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#476×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×Dic508C2211yescharacteristic, complemented
19C2×#14523237C2211yescharacteristic
20C2×#14523236C2211yescharacteristic
21C2×#14523231C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic, complemented
23#16861204—21no—
24#16861204—21no—
25#16861198—21no—
26#16861197C2211yescharacteristic
27#16861193—21no—
28#16861189C2211yescharacteristic
29#16861189C2211yescharacteristic
30#16861189C2211yescharacteristic, complemented
31#16861184C2211yescharacteristic
32#16861183—21no—
33#16861179—21no—
34Q8×Dic127—21no—
35C4×Dic254C2211yescharacteristic, complemented
36C4×D508C2211yescharacteristic
37#16861167C2211yescharacteristic
38#16861167C2211yescharacteristic, complemented
39C2×(C2×D127)⋊C4—21no—
40#16861142C2211yescharacteristic, complemented
41#16861134C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861120C2211yescharacteristic, complemented
52C4⋊Q8×C127C2211yescharacteristic, complemented
53C4×D4×C127C2211yescharacteristic
54#106×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56C4⋊C8×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60#14523247—22no—
61#14523247—41no—
62C2×Dic254C2311yescharacteristic
63C2×Dic254C2311yescharacteristic
64C2×D508—21no—
65C2×D508D411yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69D4×D127—22no—
70D4×D127—41no—
71C22×Dic127—21no—
72C2×C4×D127C2311yescharacteristic
73C2×C4×D127C2311yescharacteristic
74Dic508—41no—
75#14523237—41no—
76#14523236—22no—
77#14523231—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127—21no—
82C4⋊Dic127—21no—
83C4⋊Dic127—21no—
84C4⋊Dic127D411yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C4×Dic127—21no—
99C4×Dic127—21no—
100C2×Q8×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127—21no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127—21no—
106C4⋊C4×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—42no—
113C22×D127—42no—
114C22×D127—81no—
115C22×D127C2×D411yescharacteristic
116C22×D127C2×D411yescharacteristic
117Dic254—21no—
118Dic254—21no—
119Dic254—21no—
120Dic254—21no—
121D508—21no—
122D508—21no—
123D508—41no—
124D508—81no—
125C22⋊D127—42no—
126C22⋊D127—42no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—42no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—21no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127—21no—
140Q8×C127—42no—
141D4×C127—21no—
142D4×C127#4112yes—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—22no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—84no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127C2×#4111yescharacteristic
172C4×C127H(5,𝔽2)11yescharacteristic
173Dic127—21no—
174Dic127—21no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—42no—
179C2×C127#54611yescharacteristic
180C2×C127#57511yescharacteristic
181C2×C127#57711yescharacteristic
182D127—41no—
183D127—41no—
184D127—81no—
185#3311—1271nomaximal, Sylow
186C127#331111yescharacteristic, complemented, Sylow
187#556—1271no—
188#546—1271no—
189#489—1271no—
190#485—1271no—
191#482—1271no—
192#476D12711yescharacteristic, complemented
193#470—1271no—
194#465—1271no—
195#464—1271no—
196#452—1271no—
197#450—1271no—
198#440—1271no—
199#435—1271no—
200#418—1271no—
201#416—1271no—
202C2×#41—1271no—
203C22×D4—1271no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C2×M16—1271no—
208C22×C8—1271no—
209C4⋊Q8C2×D12711yescharacteristic, complemented
210#126—2541no—
211#126—2541no—
212#124—1271no—
213#124—2541no—
214#123—2541no—
215C22⋊Q8—1271no—
216C22⋊Q8—1271no—
217C22⋊Q8—1271no—
218#121—1271no—
219#121—2541no—
220#120—2541no—
221C4×Q8—1271no—
222C4×Q8—2541no—
223C4×D4—1271no—
224C4×D4C2×D12711yescharacteristic
225#117—1271no—
226#117—1271no—
227C2×C22⋊C4—2541no—
228#107—1271no—
229#106C2×D12711yescharacteristic, complemented
230#106C2×D12711yescharacteristic, complemented
231C4⋊C8C2×D12711yescharacteristic, complemented
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102C2×D12711yescharacteristic
241#102C2×D12711yescharacteristic
242C22⋊C8—1271no—
243C22⋊C8—1271no—
244#97—1271no—
245C24—2541no—
246#41—5081no—
247#41—2542no—
248C2×Q8—21no—
249C2×Q8—1271no—
250C2×Q8—1271no—
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—5081no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4—2542no—
258C2×D4C22×D12711yescharacteristic
259C22×C4—21no—
260C22×C4—1271no—
261C22×C4—1271no—
262C22×C4—2541no—
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266M16—5081no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—5081no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4C22×D12711yescharacteristic
280C4⋊C4C22×D12711yescharacteristic
281C4⋊C4C22×D12711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—1271no—
284C22⋊C4—1271no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—2541no—
294C42—2541no—
295C42—2541no—
296C42C22×D12711yescharacteristic
297C23—21no—
298C23—1271no—
299C23—1271no—
300C23—10161no—
301C23—5082no—
302C23—5082no—
303C23—2541no—
304Q8—42no—
305Q8—2541no—
306Q8—2541no—
307Q8—2541no—
308Q8—2541no—
309D4—21no—
310D4—10161no—
311D4—5081no—
312D4—5082no—
313D4—5082no—
314D4—2541no—
315D4—2541no—
316D4#1452324912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—5082no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4D4×D12711yescharacteristic
332C2×C4D4×D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2541no—
337C8—2541no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—10164no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—2541no—
346C22—2541no—
347C22—2541no—
348C22—2541no—
349C22—2541no—
350C22—2541no—
351C22C2×D4×D12711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5081no—
358C4—5081no—
359C4—5081no—
360C4—2541no—
361C4—2541no—
362C4C2×#1452324911yescharacteristic
363C4#1686125511yescharacteristic
364C2—42no—
365C2—10161no—
366C2—5081no—
367C2—5081no—
368C2#2255051711yescharacteristic
369C2#2255067211yescharacteristic
370C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#476×C127C2
socleC22×C127C22×D4
prime radical#22550521C2
hypercenter#476D127
hypocenterC127#3311
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #22550600C2semidirect
3. #22550238C2semidirect
4. #22550220C2semidirect
5. #22550202C2semidirect
6. #22550135C2semidirect
7. #22550106C2semidirect
8. #22550099C2semidirect
9. #22550039C2semidirect
10. #22550028C2semidirect
11. #22549976C2semidirect
12. #22549946C2semidirect
13. #22549879C2semidirect
14. #22549869C2semidirect
15. #476×C127C2semidirect
18. C2×Dic508C22semidirect
22. C2×C8×D127C22semidirect
30. #16861189C22semidirect
35. C4×Dic254C22semidirect
38. #16861167C22semidirect
40. #16861142C22semidirect
41. #16861134C22semidirect
42. #16861132C22semidirect
43. #16861132C22semidirect
44. #16861132C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861120C22semidirect
52. C4⋊Q8×C127C22semidirect
54. #106×C127C22semidirect
55. #106×C127C22semidirect
56. C4⋊C8×C127C22semidirect
186. C127#3311semidirect
192. #476D127semidirect
209. C4⋊Q8C2×D127semidirect
229. #106C2×D127semidirect
230. #106C2×D127semidirect
231. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506002
#225505212
#225502382
#225502202
#225502022
#225501352
#225501062
#225500992
#225500392
#225500282
#225499762
#225499462
#225498792
#225498692
#476×C1272
#3311127