Finite Group Databasebeta

← #30650760

Subgroups — #30650760

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#22550522C211yesmaximal, characteristic
2#22550238C211yesmaximal, characteristic, complemented
3#22550217C211yesmaximal, characteristic, complemented
4#22550200C211yesmaximal, characteristic, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550132C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550098C211yesmaximal, characteristic, complemented
9#22550036C211yesmaximal, characteristic, complemented
10#22550026C211yesmaximal, characteristic, complemented
11#22549974C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549880C211yesmaximal, characteristic, complemented
14#22549870C211yesmaximal, characteristic, complemented
15#556×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×#14523238C2211yescharacteristic, complemented
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21C2×#14523232C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic, complemented
23#16861197C2211yescharacteristic
24#16861194—21no—
25#16861188C2211yescharacteristic
26#16861183C2211yescharacteristic
27#16861182—21no—
28#16861177—21no—
29C4×D508C2211yescharacteristic
30C22⋊C4×D127—21no—
31#16861142C2211yescharacteristic, complemented
32#16861140C2211yescharacteristic, complemented
33#16861140C2211yescharacteristic, complemented
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35#16861133C2211yescharacteristic, complemented
36#16861132C2211yescharacteristic, complemented
37#16861132C2211yescharacteristic, complemented
38#16861132C2211yescharacteristic, complemented
39#16861129C2211yescharacteristic
40#16861129C2211yescharacteristic
41#16861129C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44#16861129C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic, complemented
47#16861120C2211yescharacteristic, complemented
48C4⋊Q8×C127C2211yescharacteristic, complemented
49#126×C127—21no—
50#126×C127—21no—
51#124×C127—21no—
52C22⋊Q8×C127C2211yescharacteristic
53C22⋊Q8×C127C2211yescharacteristic, complemented
54C4×Q8×C127—21no—
55C4×Q8×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic
57#117×C127C2211yescharacteristic
58#117×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—41no—
62C2×Dic254C2311yescharacteristic
63C2×D508—22no—
64C2×D508—21no—
65C2×D508—41no—
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72#14523238—41no—
73#14523235—22no—
74#14523235—41no—
75#14523232—41no—
76C2×C127⋊C8—41no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82Dic127⋊C4—21no—
83C22⋊Dic127—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88C2×Q8×C127—21no—
89C2×Q8×C127C2311yescharacteristic
90C2×D4×C127D411yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C4⋊C4×C127—21no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127D411yescharacteristic
98C4⋊C4×C127C2311yescharacteristic
99C4⋊C4×C127C2311yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127D411yescharacteristic
106C22⋊C4×C127D411yescharacteristic
107C42×C127—21no—
108C42×C127—21no—
109C42×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116Dic254—21no—
117Dic254—21no—
118D508—21no—
119D508—21no—
120D508—21no—
121D508—42no—
122D508#4112yes—
123C22⋊D127—42no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C4×D127—42no—
128C4×D127—81no—
129C127⋊C8—21no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127C2×D411yescharacteristic
134C23×C127C2×D411yescharacteristic
135Q8×C127—21no—
136Q8×C127—21no—
137Q8×C127—42no—
138D4×C127—41no—
139C2×C4×C127—21no—
140C2×C4×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×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127C22×D411yescharacteristic
159C2×D127—22no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—84no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127—41no—
174C4×C127C2×#4111yescharacteristic
175C4×C127H(5,𝔽2)11yescharacteristic
176Dic127—41no—
177Dic127—41no—
178C2×C127—41no—
179C2×C127—41no—
180C2×C127#54611yescharacteristic
181C2×C127#57511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—42no—
184D127—81no—
185#3311—1271nomaximal, Sylow
186C127#331111yescharacteristic, complemented, Sylow
187#556D12711yescharacteristic, complemented
188#546—1271no—
189#489—1271no—
190#485—1271no—
191#482—1271no—
192#476—1271no—
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—21no—
211#126—21no—
212#124—21no—
213#124—1271no—
214#123—2541no—
215C22⋊Q8—1271no—
216C22⋊Q8C2×D12711yescharacteristic
217C22⋊Q8C2×D12711yescharacteristic, complemented
218#121—1271no—
219#121—2541no—
220#120—2541no—
221C4×Q8—21no—
222C4×Q8C2×D12711yescharacteristic, complemented
223C4×D4—1271no—
224C4×D4C2×D12711yescharacteristic
225#117C2×D12711yescharacteristic
226#117C2×D12711yescharacteristic, complemented
227C2×C22⋊C4—2541no—
228#107—1271no—
229#106—1271no—
230#106—1271no—
231C4⋊C8—1271no—
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#102—1271no—
241#102—1271no—
242C22⋊C8—1271no—
243C22⋊C8—1271no—
244#97—1271no—
245C24—2541no—
246#41—5081no—
247#41—2542no—
248C2×Q8—21no—
249C2×Q8—1271no—
250C2×Q8C22×D12711yescharacteristic
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—5081no—
254C2×D4—2542no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4—2541no—
258C2×D4C22⋊D12711yescharacteristic
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4C22×D12711yescharacteristic
262C22×C4C22×D12711yescharacteristic
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266M16—5081no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—1271no—
271C2×C8—5081no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4C22⋊D12711yescharacteristic
279C4⋊C4C22×D12711yescharacteristic
280C4⋊C4C22×D12711yescharacteristic
281C4⋊C4C22×D12711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4C22⋊D12711yescharacteristic
292C22⋊C4C22⋊D12711yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D12711yescharacteristic
297C23—10161no—
298C23—5082no—
299C23—5082no—
300C23—2541no—
301C23—2541no—
302C23C2×C22⋊D12711yescharacteristic
303C23C2×C22⋊D12711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—2541no—
308Q8—2541no—
309D4—41no—
310D4—1272no—
311D4—10161no—
312D4—5082no—
313D4—5082no—
314D4—2541no—
315D4—2541no—
316D4—2541no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—10161no—
326C2×C4—5082no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4C2×C22⋊D12711yescharacteristic
330C2×C4C2×C22⋊D12711yescharacteristic
331C2×C4C2×C22⋊D12711yescharacteristic
332C2×C4C2×C22⋊D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—2541no—
335C8—2541no—
336C8—2541no—
337C8—2541no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—10164no—
345C22—5081no—
346C22—5081no—
347C22—5081no—
348C22—5081no—
349C22—5081no—
350C22—2542no—
351C22C22×C22⋊D12711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—5081no—
361C4—5081no—
362C4#41×D12711yescharacteristic
363C4#1686125511yescharacteristic
364C2—41no—
365C2—41no—
366C2—10161no—
367C2—5082no—
368C2#2255051411yescharacteristic
369C2#2255067411yescharacteristic
370C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#556×C127C2
socleC22×C127C22×D4
prime radical#22550522C2
hypercenter#556D127
hypocenterC127#3311
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #22550238C2semidirect
3. #22550217C2semidirect
4. #22550200C2semidirect
5. #22550163C2semidirect
6. #22550132C2semidirect
7. #22550107C2semidirect
8. #22550098C2semidirect
9. #22550036C2semidirect
10. #22550026C2semidirect
11. #22549974C2semidirect
12. #22549948C2semidirect
13. #22549880C2semidirect
14. #22549870C2semidirect
15. #556×C127C2semidirect
18. C2×#14523238C22semidirect
22. C22×C127⋊C8C22semidirect
31. #16861142C22semidirect
32. #16861140C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. #16861133C22semidirect
36. #16861132C22semidirect
37. #16861132C22semidirect
38. #16861132C22semidirect
46. C22⋊(C127⋊C8)C22semidirect
47. #16861120C22semidirect
48. C4⋊Q8×C127C22semidirect
53. C22⋊Q8×C127C22semidirect
55. C4×Q8×C127C22semidirect
58. #117×C127C22semidirect
186. C127#3311semidirect
187. #556D127semidirect
209. C4⋊Q8C2×D127semidirect
217. C22⋊Q8C2×D127semidirect
222. C4×Q8C2×D127semidirect
226. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505222
#225502382
#225502172
#225502002
#225501632
#225501322
#225501072
#225500982
#225500362
#225500262
#225499742
#225499482
#225498802
#225498702
#556×C1272
#3311127