Finite Group Databasebeta

← #30650549

Subgroups — #30650549

Order 16256.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550584C211yesmaximal, characteristic, complemented
2#22550510C211yesmaximal, characteristic
3#22550221C211yesmaximal, characteristic, complemented
4#22550196C211yesmaximal, characteristic, complemented
5#22550192C211yesmaximal, characteristic, complemented
6#22550165C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550026C211yesmaximal, characteristic, complemented
10#22550025C211yesmaximal, characteristic, complemented
11#22549961C211yesmaximal, characteristic, complemented
12C8×C22⋊D127C211yesmaximal, characteristic, complemented
13C2×#16861131C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic, complemented
21C2×C8×D127C2211yescharacteristic, complemented
22#16861208C2211yescharacteristic, complemented
23#16861205C2211yescharacteristic
24#16861199—21no—
25#16861189—21no—
26#16861186—21no—
27#16861185C2211yescharacteristic, complemented
28#16861184C2211yescharacteristic
29#16861182—21no—
30#16861179—21no—
31#16861177—21no—
32Q8×Dic127C2211yescharacteristic, complemented
33#16861171C2211yescharacteristic
34C4×C22⋊D127C2211yescharacteristic
35(C22⋊D127)⋊C4—21no—
36C2×C4⋊Dic127C2211yescharacteristic, complemented
37C4⋊C4×D127C2211yescharacteristic, complemented
38#16861140C2211yescharacteristic, 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
51C8×Dic127C2211yescharacteristic, complemented
52C2×SD16×C127C2211yescharacteristic, complemented
53C22×C8×C127C2211yescharacteristic, complemented
54C22⋊Q8×C127C2211yescharacteristic, complemented
55#121×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C23×D127—21no—
61C2×Dic254—21no—
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—
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79#14523235—22no—
80#14523235—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—41no—
84C4⋊Dic127—21no—
85C4⋊Dic127—21no—
86C4⋊Dic127C2311yescharacteristic
87C4⋊Dic127C2311yescharacteristic
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4D411yescharacteristic
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C4×Dic127C2311yescharacteristic
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103SD16×C127—41no—
104C2×C8×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
121Dic254—81no—
122D508—21no—
123D508—21no—
124D508—21no—
125D508—42no—
126D508—42no—
127D508#4112yes—
128C22⋊D127—42no—
129C22⋊D127—42no—
130C22⋊D127—81no—
131C22⋊D127SD1614yescomplemented
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—81no—
135C2×Dic127C2×D411yescharacteristic
136C2×Dic127C2×D411yescharacteristic
137C2×Dic127C2×D411yescharacteristic
138C4×D127—21no—
139C4×D127—21no—
140C4×D127—42no—
141C4×D127—81no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C23×C127—21no—
145C23×C127C2×D411yescharacteristic
146Q8×C127—21no—
147Q8×C127—21no—
148D4×C127—21no—
149D4×C127—21no—
150D4×C127—42no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C8×C127—21no—
159C8×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127C2×SD1612yes—
166C22×C127C22×D411yescharacteristic
167C2×D127—21no—
168C2×D127—21no—
169C2×D127—22no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—84no—
176C2×D127—84no—
177C2×D127C2×SD1612yes—
178C4×C127—21no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C2×#4111yescharacteristic
183C4×C127H(5,𝔽2)11yescharacteristic
184Dic127—21no—
185Dic127—21no—
186Dic127—41no—
187Dic127—41no—
188Dic127C2×SD1612yes—
189C2×C127—22no—
190C2×C127—81no—
191C2×C127#54611yescharacteristic
192C2×C127C22×SD1611yescharacteristic
193C2×C127#57611yescharacteristic
194D127—22no—
195D127—42no—
196D127—81no—
197#3295—1271nomaximal, Sylow
198C127#329511yescharacteristic, complemented, Sylow
199#554—1271no—
200D42—1271no—
201#486—1271no—
202#481—1271no—
203#481—1271no—
204#476—1271no—
205#465—1271no—
206#465D12711yescharacteristic, complemented
207#460—1271no—
208#450—1271no—
209#450—1271no—
210C4×SD16—1271no—
211C8×D4—1271no—
212C2×#102—1271no—
213C2×#102—1271no—
214C22×D4—1271no—
215C22×D4—1271no—
216C22×D4—2541no—
217C2×SD16—1271no—
218C2×SD16—1271no—
219C2×SD16C2×D12711yescharacteristic, complemented
220C22×C8—1271no—
221C22×C8C2×D12711yescharacteristic, complemented
222C4⋊Q8—1271no—
223#127—1271no—
224#125—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8—2541no—
227C22⋊Q8—2541no—
228C22⋊Q8C2×D12711yescharacteristic, complemented
229#121—1271no—
230#121—2541no—
231#121C2×D12711yescharacteristic
232#120—2541no—
233#120—2541no—
234C4×Q8—1271no—
235C4×D4—1271no—
236C4×D4—1271no—
237C4×D4—2541no—
238C2×C4⋊C4—1271no—
239C2×C4⋊C4—1271no—
240#106—1271no—
241#106—1271no—
242#106C2×D12711yescharacteristic, complemented
243C4⋊C8—1271no—
244#103C2×D12711yescharacteristic, complemented
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251#102—1271no—
252#102—1271no—
253#102C2×D12711yescharacteristic, complemented
254C22⋊C8—1271no—
255C22⋊C8—1271no—
256C4×C8—1271no—
257C24—2541no—
258C24—2541no—
259C2×Q8—2541no—
260C2×Q8C22×D12711yescharacteristic
261C2×D4—21no—
262C2×D4—1271no—
263C2×D4—1271no—
264C2×D4—1271no—
265C2×D4—5081no—
266C2×D4—5084no—
267C2×D4—5081no—
268C2×D4—2542no—
269C2×D4—2541no—
270C2×D4—2541no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2542no—
274C2×D4—2541no—
275C2×D4C22×D12711yescharacteristic
276C22×C4—1271no—
277C22×C4—2541no—
278C22×C4—2541no—
279C22×C4—2541no—
280C22×C4C22×D12711yescharacteristic
281SD16—41no—
282SD16—5081no—
283SD16—2542no—
284C2×C8—41no—
285C2×C8—1271no—
286C2×C8—1271no—
287C2×C8—5081no—
288C2×C8C22×D12711yescharacteristic
289C2×C8C22×D12711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1271no—
292C4⋊C4—1271no—
293C4⋊C4—1271no—
294C4⋊C4—2541no—
295C4⋊C4—2541no—
296C4⋊C4—2541no—
297C4⋊C4—2541no—
298C4⋊C4—2541no—
299C4⋊C4C22×D12711yescharacteristic
300C4⋊C4C22×D12711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—1271no—
304C22⋊C4—1271no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C42—1271no—
309C42—2541no—
310C23—21no—
311C23—1271no—
312C23—10161no—
313C23—10161no—
314C23—5082no—
315C23—5082no—
316C23—5082no—
317C23—5082no—
318C23—2541no—
319C23—2541no—
320C23D4×D12711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—10161no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—1274no—
328D4—1272no—
329D4—10161no—
330D4—5082no—
331D4—5082no—
332D4—5082no—
333D4—5082no—
334D4—2541no—
335D4—2541no—
336D4—2541no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—1271no—
343C2×C4—1271no—
344C2×C4—1271no—
345C2×C4—10161no—
346C2×C4—10161no—
347C2×C4—5082no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4D4×D12711yescharacteristic
353C2×C4C23×D12711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2541no—
357C8—2541no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—1272no—
364C22—10164no—
365C22—10164no—
366C22—5081no—
367C22—5081no—
368C22—5081no—
369C22—5081no—
370C22—5081no—
371C22—2542no—
372C22—2541no—
373C22—2541no—
374C22SD16×D12712yes—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1272no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4#41×D12711yescharacteristic
386C4#1686125411yescharacteristic
387C2—22no—
388C2—81no—
389C2—10161no—
390C2—5082no—
391C2—2542no—
392C2#2255051511yescharacteristic
393C2C2×SD16×D12711yescharacteristic
394C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
prime radical#22550510C2
hypercenter#465D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22550584C2semidirect
3. #22550221C2semidirect
4. #22550196C2semidirect
5. #22550192C2semidirect
6. #22550165C2semidirect
7. #22550108C2semidirect
8. #22550074C2semidirect
9. #22550026C2semidirect
10. #22550025C2semidirect
11. #22549961C2semidirect
12. C8×C22⋊D127C2semidirect
13. C2×#16861131C2semidirect
14. #102×D127C2semidirect
15. #465×C127C2semidirect
20. C2×#14523235C22semidirect
21. C2×C8×D127C22semidirect
22. #16861208C22semidirect
27. #16861185C22semidirect
32. Q8×Dic127C22semidirect
36. C2×C4⋊Dic127C22semidirect
37. C4⋊C4×D127C22semidirect
38. #16861140C22semidirect
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. C8×Dic127C22semidirect
52. C2×SD16×C127C22semidirect
53. C22×C8×C127C22semidirect
54. C22⋊Q8×C127C22semidirect
56. #106×C127C22semidirect
57. #103×C127C22semidirect
58. #102×C127C22semidirect
131. C22⋊D127SD16semidirect
198. C127#3295semidirect
206. #465D127semidirect
219. C2×SD16C2×D127semidirect
221. C22×C8C2×D127semidirect
228. C22⋊Q8C2×D127semidirect
242. #106C2×D127semidirect
244. #103C2×D127semidirect
253. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505842
#225505102
#225502212
#225501962
#225501922
#225501652
#225501082
#225500742
#225500262
#225500252
#225499612
C8×C22⋊D1272
C2×#168611312
#102×D1272
#465×C1272
#3295127