Finite Group Databasebeta

← #30650563

Subgroups — #30650563

Order 16256.

408 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#22550561C211yesmaximal, characteristic, complemented
2#22550517C211yesmaximal, characteristic
3#22550232C211yesmaximal, characteristic, complemented
4#22550202C211yesmaximal, characteristic, complemented
5#22550175C211yesmaximal, characteristic, complemented
6#22550115C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550101C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550029C211yesmaximal, characteristic, complemented
11#22549963C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14C2×#16861129C211yesmaximal, characteristic, complemented
15#483×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×#14523238C2211yescharacteristic, complemented
19C2×#14523237C2211yescharacteristic
20C2×#14523237C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic, complemented
23#16861204—21no—
24#16861200C2211yescharacteristic, complemented
25#16861199—21no—
26#16861198C2211yescharacteristic
27#16861194—21no—
28#16861192—21no—
29#16861189C2211yescharacteristic, complemented
30#16861188C2211yescharacteristic
31#16861184—21no—
32#16861184C2211yescharacteristic
33#16861179—21no—
34#16861174C2211yescharacteristic, complemented
35#16861171C2211yescharacteristic
36C4×C22⋊D127C2211yescharacteristic
37#16861168—21no—
38#16861167C2211yescharacteristic, complemented
39C22⋊C4×D127—21no—
40#16861142C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22⋊Q8×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic
54#106×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C23×D127—21no—
60#14523249—22no—
61#14523249—41no—
62C2×Dic254C2311yescharacteristic
63C2×D508—22no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127D411yescharacteristic
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74#14523238—41no—
75#14523237—22no—
76#14523237—41no—
77C2×C127⋊C8—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127—21no—
82C4⋊Dic127—21no—
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4D411yescharacteristic
88C22⋊Dic127—21no—
89C22⋊Dic127D411yescharacteristic
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)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C4×Dic127—21no—
99C4×Dic127C2311yescharacteristic
100C2×Q8×C127C2311yescharacteristic
101C22×C4×C127—21no—
102C22×C4×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127—21no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117Dic254—21no—
118Dic254—21no—
119D508—21no—
120D508—21no—
121D508—21no—
122D508—42no—
123D508—42no—
124D508—81no—
125D508#4112yes—
126C22⋊D127—22no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—42no—
135C4×D127—81no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127C2×D411yescharacteristic
139Q8×C127—21no—
140Q8×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—42no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127#13512yes—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—22no—
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—84no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127C2×#4111yescharacteristic
172C4×C127H(5,𝔽2)11yescharacteristic
173Dic127—21no—
174Dic127—21no—
175Dic127—21no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—22no—
179C2×C127#54611yescharacteristic
180C2×C127C2×#13511yescharacteristic
181C2×C127#57711yescharacteristic
182D127—41no—
183D127—42no—
184D127—81no—
185#3296—1271nomaximal, Sylow
186C127#329611yescharacteristic, complemented, Sylow
187#552—1271no—
188#546—1271no—
189#488—1271no—
190#483D12711yescharacteristic, complemented
191#482—1271no—
192#478—1271no—
193#467—1271no—
194#465—1271no—
195#464—1271no—
196#452—1271no—
197#450—1271no—
198C4×SD16—1271no—
199C8×D4—1271no—
200#416—1271no—
201C2×#102—1271no—
202C2×#41—1271no—
203C22×D4—1271no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C22×C8—1271no—
208C22×C8—1271no—
209#126—2541no—
210#125—1271no—
211#125—2541no—
212#124—1271no—
213#123—2541no—
214#123—2541no—
215C22⋊Q8—1271no—
216C22⋊Q8—1271no—
217C22⋊Q8C2×D12711yescharacteristic
218#121—1271no—
219#121—2541no—
220#120—2541no—
221C4×Q8—1271no—
222C4×D4—1271no—
223C4×D4—1271no—
224#117—1271no—
225#117—2541no—
226C2×C4⋊C4C2×D12711yescharacteristic
227C2×C22⋊C4—2541no—
228#107—1271no—
229#106C2×D12711yescharacteristic, complemented
230#106C2×D12711yescharacteristic, complemented
231C4⋊C8—1271no—
232#103—1271no—
233#103—1271no—
234#103C2×D12711yescharacteristic, complemented
235#103C2×D12711yescharacteristic, complemented
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242C22⋊C8—1271no—
243C22⋊C8C2×D12711yescharacteristic
244C4×C8—1271no—
245C24—2541no—
246#41—5081no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8C22×D12711yescharacteristic
250C2×D4—1271no—
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—5081no—
254C2×D4—2542no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4—2541no—
258C22×C4—21no—
259C22×C4—1271no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266C2×C8—1271no—
267C2×C8—1271no—
268C2×C8—5081no—
269C2×C8—5081no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22×D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—1271no—
286C22⋊C4—1271no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—1271no—
295C42—2541no—
296C42—2541no—
297C23—1271no—
298C23—10161no—
299C23—5082no—
300C23—5082no—
301C23—2541no—
302C23—2541no—
303C23D4×D12711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2541no—
307Q8—2541no—
308D4—1272no—
309D4—10161no—
310D4—5082no—
311D4—5082no—
312D4—2542no—
313D4—2541no—
314D4—2541no—
315D4—2541no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—1271no—
324C2×C4—1271no—
325C2×C4—1271no—
326C2×C4—10161no—
327C2×C4—5082no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4D4×D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2541no—
337C8—2541no—
338C22—21no—
339C22—21no—
340C22—10164no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—2541no—
347C22—2541no—
348C22—2542no—
349C22—2541no—
350C22#1686123012yes—
351C22C2×D4×D12711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5081no—
358C4—5081no—
359C4—2541no—
360C4—2541no—
361C4—2541no—
362C4#41×D12711yescharacteristic
363C4#1686125511yescharacteristic
364C2—22no—
365C2—10161no—
366C2—5082no—
367C2—5081no—
368C2#2255051811yescharacteristic
369C2C2×#1686123011yescharacteristic
370C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#483×C127C2
socleC22×C127C22×D4
prime radical#22550517C2
hypercenter#483D127
hypocenterC127#3296
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22550561C2semidirect
3. #22550232C2semidirect
4. #22550202C2semidirect
5. #22550175C2semidirect
6. #22550115C2semidirect
7. #22550106C2semidirect
8. #22550101C2semidirect
9. #22550041C2semidirect
10. #22550029C2semidirect
11. #22549963C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. #22549871C2semidirect
14. C2×#16861129C2semidirect
15. #483×C127C2semidirect
18. C2×#14523238C22semidirect
22. C2×C8×D127C22semidirect
24. #16861200C22semidirect
29. #16861189C22semidirect
34. #16861174C22semidirect
38. #16861167C22semidirect
40. #16861142C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861133C22semidirect
43. #16861132C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
54. #106×C127C22semidirect
55. #106×C127C22semidirect
56. #103×C127C22semidirect
57. #103×C127C22semidirect
186. C127#3296semidirect
190. #483D127semidirect
229. #106C2×D127semidirect
230. #106C2×D127semidirect
234. #103C2×D127semidirect
235. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505612
#225505172
#225502322
#225502022
#225501752
#225501152
#225501062
#225501012
#225500412
#225500292
#225499632
(C22⋊D127)⋊C82
#225498712
C2×#168611292
#483×C1272
#3296127