Finite Group Databasebeta

← #30650746

Subgroups — #30650746

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#22550560C211yesmaximal, characteristic, complemented
2#22550510C211yesmaximal, characteristic
3#22550239C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550191C211yesmaximal, characteristic, complemented
6#22550179C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550026C211yesmaximal, characteristic, complemented
10#22550025C211yesmaximal, characteristic, complemented
11#22549979C211yesmaximal, characteristic, complemented
12#22549945C211yesmaximal, characteristic, complemented
13#22549882C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic, complemented
21C2×C8×D127C2211yescharacteristic, complemented
22#16861205C2211yescharacteristic
23#16861202—21no—
24#16861200C2211yescharacteristic, complemented
25#16861199—21no—
26#16861191—21no—
27#16861185C2211yescharacteristic, complemented
28#16861184C2211yescharacteristic
29#16861182—21no—
30#16861179—21no—
31#16861177—21no—
32Q8×Dic127C2211yescharacteristic, complemented
33#16861171C2211yescharacteristic
34C4×C22⋊D127C2211yescharacteristic
35#16861168C2211yescharacteristic, complemented
36#16861167—21no—
37C4⋊C4×D127C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861140C2211yescharacteristic, complemented
40Dic127⋊C8C2211yescharacteristic, complemented
41#16861131C2211yescharacteristic
42#16861131C2211yescharacteristic
43#16861131C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic, complemented
46#16861130C2211yescharacteristic, complemented
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52C2×SD16×C127C2211yescharacteristic, complemented
53C2×M16×C127C2211yescharacteristic, complemented
54C22⋊Q8×C127C2211yescharacteristic, complemented
55#121×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C23×D127—21no—
61C2×D508—22no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508—41no—
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127D411yescharacteristic
71D4×D127—22no—
72D4×D127—41no—
73D4×D127—44no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523235—22no—
78#14523235—41no—
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C8×D127—41no—
82C4⋊Dic127—21no—
83C4⋊Dic127—21no—
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127D411yescharacteristic
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127—21no—
98C4×Dic127C2311yescharacteristic
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103SD16×C127—41no—
104M16×C127—41no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
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
121D508—21no—
122D508—21no—
123D508—21no—
124D508—42no—
125D508—42no—
126D508#4112yes—
127C22⋊D127—22no—
128C22⋊D127—42no—
129C22⋊D127—42no—
130C22⋊D127—81no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127C2×D411yescharacteristic
134C2×Dic127C2×D411yescharacteristic
135C2×Dic127C2×D411yescharacteristic
136C4×D127—21no—
137C4×D127—21no—
138C4×D127—42no—
139C4×D127—42no—
140C127⋊C8—21no—
141C127⋊C8—21no—
142C23×C127—21no—
143C23×C127C2×D411yescharacteristic
144Q8×C127—21no—
145Q8×C127—21no—
146D4×C127—21no—
147D4×C127—21no—
148D4×C127—42no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127C22×D411yescharacteristic
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—22no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—84no—
174C2×D127—84no—
175C2×D127AGL(1,ℤ/8)12yes—
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—41no—
188C2×C127—81no—
189C2×C127#54611yescharacteristic
190C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C127#57711yescharacteristic
192D127—22no—
193D127—42no—
194D127—81no—
195#3310—1271nomaximal, Sylow
196C127#331011yescharacteristic, complemented, Sylow
197#552—1271no—
198D42—1271no—
199#490—1271no—
200#482—1271no—
201#480—1271no—
202#478—1271no—
203#468D12711yescharacteristic, complemented
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×SD16C2×D12711yescharacteristic, complemented
218C2×M16C2×D12711yescharacteristic, complemented
219C22×C8—1271no—
220#127—1271no—
221#126—2541no—
222#125—1271no—
223#125—2541no—
224#123—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8C2×D12711yescharacteristic, complemented
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⋊C4—1271no—
238#107—1271no—
239#107C2×D12711yescharacteristic, complemented
240#106—1271no—
241C4⋊C8—1271no—
242#103C2×D12711yescharacteristic, complemented
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251#102C2×D12711yescharacteristic, complemented
252C22⋊C8—1271no—
253C22⋊C8—1271no—
254#97—1271no—
255C24—2541no—
256C24—2541no—
257C2×Q8C22×D12711yescharacteristic
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—1271no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4C22×D12711yescharacteristic
277SD16—41no—
278SD16—5081no—
279SD16—2542no—
280M16—41no—
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—1271no—
290C4⋊C4—2541no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4—2541no—
295C4⋊C4C22×D12711yescharacteristic
296C4⋊C4C22×D12711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1271no—
300C22⋊C4—1271no—
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—21no—
320Q8—21no—
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—1271no—
340C2×C4—1271no—
341C2×C4—1271no—
342C2×C4—5082no—
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—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—1272no—
361C22—10164no—
362C22—10164no—
363C22—5081no—
364C22—5081no—
365C22—5081no—
366C22—5081no—
367C22—5081no—
368C22—2542no—
369C22—2541no—
370C22—2541no—
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#1686125411yescharacteristic
383C2—41no—
384C2—81no—
385C2—10161no—
386C2—5082no—
387C2—2542no—
388C2#2255051511yescharacteristic
389C2AGL(1,ℤ/8)×D12711yescharacteristic
390C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
prime radical#22550510C2
hypercenter#468D127
hypocenterC127#3310
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #22550560C2semidirect
3. #22550239C2semidirect
4. #22550197C2semidirect
5. #22550191C2semidirect
6. #22550179C2semidirect
7. #22550108C2semidirect
8. #22550074C2semidirect
9. #22550026C2semidirect
10. #22550025C2semidirect
11. #22549979C2semidirect
12. #22549945C2semidirect
13. #22549882C2semidirect
14. #102×D127C2semidirect
15. #468×C127C2semidirect
20. C2×#14523235C22semidirect
21. C2×C8×D127C22semidirect
24. #16861200C22semidirect
27. #16861185C22semidirect
32. Q8×Dic127C22semidirect
35. #16861168C22semidirect
37. C4⋊C4×D127C22semidirect
38. #16861142C22semidirect
39. #16861140C22semidirect
40. Dic127⋊C8C22semidirect
43. #16861131C22semidirect
44. #16861131C22semidirect
45. #16861130C22semidirect
46. #16861130C22semidirect
48. #16861129C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861119C22semidirect
52. C2×SD16×C127C22semidirect
53. C2×M16×C127C22semidirect
54. C22⋊Q8×C127C22semidirect
56. #107×C127C22semidirect
57. #103×C127C22semidirect
58. #102×C127C22semidirect
196. C127#3310semidirect
203. #468D127semidirect
217. C2×SD16C2×D127semidirect
218. C2×M16C2×D127semidirect
226. C22⋊Q8C2×D127semidirect
239. #107C2×D127semidirect
242. #103C2×D127semidirect
251. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505602
#225505102
#225502392
#225501972
#225501912
#225501792
#225501082
#225500742
#225500262
#225500252
#225499792
#225499452
#225498822
#102×D1272
#468×C1272
#3310127