Finite Group Databasebeta

← #30650345

Subgroups — #30650345

Order 16256.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C212yesmaximal, complemented
2#22550548C211yesmaximal, characteristic, complemented
3#22550511C211yesmaximal, characteristic, complemented
4#22550252C211yesmaximal, characteristic, complemented
5#22550128C212yesmaximal, complemented
6#22550072C211yesmaximal, characteristic, complemented
7#22550020C212yesmaximal, complemented
8#22550011C212yesmaximal, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10#22549953C211yesmaximal, characteristic, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yescomplemented
13C2×D4×D127—21no—
14C2×D4×D127C2212yes—
15#16861236—28no—
16C2×#14523235C2211yescharacteristic
17C2×D1016C2212yes—
18C2×#14523233C2212yescomplemented
19C2×#14523233C2212yescomplemented
20C2×#14523231C2212yescomplemented
21#16861206—21no—
22#16861206C2211yescharacteristic, complemented
23#16861184—22no—
24#16861184C2212yescomplemented
25#16861180—21no—
26#16861180C2212yes—
27#16861177—22no—
28C4×D508—21no—
29C4×D508C2211yescharacteristic
30C4×D508C2211yescharacteristic
31#16861140C2211yescharacteristic, complemented
32#16861133C2211yescharacteristic, complemented
33#16861131C2212yescomplemented
34#16861129C2211yescharacteristic
35(C2×D127)⋊C8C2212yescomplemented
36C4×C127⋊C8C2211yescharacteristic, complemented
37C2×SD16×C127C2212yescomplemented
38#127×C127C2211yescharacteristic
39C4×Q8×C127C2211yescharacteristic, complemented
40C4⋊C8×C127C2211yescharacteristic, complemented
41#102×C127C2212yescomplemented
42C23×D127—22no—
43#14523249—24no—
44#14523249—42no—
45C2×D508—22no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—21no—
49C2×D508D411yescharacteristic
50C2×D508C2312yes—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54D4×D127—24no—
55D4×D127—42no—
56D4×D127—44no—
57C2×C4×D127—22no—
58C2×C4×D127—21no—
59C2×C4×D127C2312yes—
60#14523235D414yescomplemented
61D1016—24no—
62#14523233—24no—
63#14523233—42no—
64#14523231—24no—
65C2×C127⋊C8C2311yescharacteristic
66C2×C127⋊C8C2311yescharacteristic
67C4⋊Dic127D411yescharacteristic
68C4⋊Dic127C2311yescharacteristic
69C22⋊Dic127—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—21no—
72(C2×D127)⋊C4D412yes—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127C2312yes—
77SD16×C127—24no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127—21no—
80C4⋊C4×C127C2311yescharacteristic
81C42×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—44no—
86C22×D127—44no—
87C22×D127—82no—
88C22×D127C2×D412yes—
89D508—22no—
90D508—22no—
91D508—21no—
92D508—22no—
93D508—44no—
94D508—42no—
95D508—81no—
96D508—82no—
97D508C2×D412yes—
98C22⋊D127—44no—
99C22⋊D127—44no—
100C2×Dic127—21no—
101C2×Dic127C2×D412yes—
102C4×D127—22no—
103C4×D127—22no—
104C4×D127—44no—
105C4×D127—42no—
106C127⋊C8—21no—
107C127⋊C8C2×D412yescomplemented
108C23×C127—22no—
109Q8×C127—21no—
110Q8×C127C2×D412yescomplemented
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—42no—
114D4×C127—81no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C22×C127—42no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—41no—
130C2×D127—41no—
131C2×D127—42no—
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—88no—
135C4×C127—22no—
136C4×C127—21no—
137C4×C127—41no—
138C4×C127AGL(1,ℤ/8)12yescomplemented
139C4×C127C22×D411yescharacteristic
140C4×C127H(5,𝔽2)11yescharacteristic
141Dic127—22no—
142Dic127—41no—
143C2×C127—82no—
144C2×C127D4211yescharacteristic
145C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C127#57611yescharacteristic
147D127—42no—
148D127—42no—
149D127—82no—
150#3274—1271nomaximal, Sylow
151C127#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1272no—
153#550—1271no—
154D42—1271no—
155#493—1271no—
156#469—1272no—
157#460D12711yescharacteristic, complemented
158#459—1271no—
159#449—1272no—
160#447—1272no—
161C4×SD16—1271no—
162#436—1271no—
163C2×#41—1272no—
164C22×D4—1272no—
165C22×D4—2541no—
166AGL(1,ℤ/8)—2548no—
167C2×SD16—1271no—
168C2×SD16C2×D12712yescomplemented
169C2×D8—1272no—
170C2×D8—1272no—
171C2×D8—1272no—
172C2×M16—1272no—
173#127—1271no—
174#127—2541no—
175#127C2×D12711yescharacteristic
176#121—1272no—
177#121—1272no—
178#121—2542no—
179#121—2541no—
180#120—2542no—
181C4×Q8C2×D12711yescharacteristic, complemented
182C4×D4—1271no—
183C4×D4—1271no—
184C4×D4—2541no—
185#106—1271no—
186C4⋊C8C2×D12711yescharacteristic, complemented
187#103—1271no—
188#102—1272no—
189#102—1271no—
190#102C2×D12712yescomplemented
191C22⋊C8—1272no—
192C4×C8—1271no—
193C24—2542no—
194#41—5082no—
195#41—2544no—
196C2×Q8C22×D12711yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—1271no—
200C2×D4—1271no—
201C2×D4—1272no—
202C2×D4—5084no—
203C2×D4—5082no—
204C2×D4—2542no—
205C2×D4—2542no—
206C2×D4—2542no—
207C2×D4—2544no—
208C2×D4—2542no—
209C2×D4—2541no—
210C2×D4—2541no—
211C2×D4C22×D12712yes—
212C22×C4—1272no—
213C22×C4—2542no—
214C22×C4—2541no—
215SD16—24no—
216SD16—1274no—
217D8—5082no—
218D8—2544no—
219D8—2544no—
220M16—2544no—
221C2×C8—1271no—
222C2×C8—1271no—
223C2×C8C22×D12712yes—
224C4⋊C4—21no—
225C4⋊C4—1271no—
226C4⋊C4—1271no—
227C4⋊C4C22×D12711yescharacteristic
228C22⋊C4—1272no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4—2541no—
232C42—21no—
233C42C22×D12711yescharacteristic
234C23—22no—
235C23—1272no—
236C23—10162no—
237C23—5084no—
238C23—5084no—
239C23—2542no—
240C23—2541no—
241Q8—21no—
242Q8D4×D12712yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—1272no—
248D4—10161no—
249D4—10162no—
250D4—5084no—
251D4—5084no—
252D4—5084no—
253D4—5082no—
254D4—2542no—
255D4—2542no—
256D4—2542no—
257D4—2541no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—1272no—
261C2×C4—5084no—
262C2×C4—5082no—
263C2×C4—2541no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4D4×D12711yescharacteristic
267C2×C4D4×D12711yescharacteristic
268C2×C4C23×D12711yescharacteristic
269C8—22no—
270C8—1272no—
271C8—2541no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—10168no—
276C22—5082no—
277C22—5082no—
278C22—5082no—
279C22—5081no—
280C22—5081no—
281C22—2542no—
282C22—2542no—
283C22—2542no—
284C22—2542no—
285C22C2×D4×D12711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—5081no—
290C4—2542no—
291C4#1686123612yescomplemented
292C4C2×D4×D12711yescharacteristic
293C4#1686125511yescharacteristic
294C2—82no—
295C2—10162no—
296C2—5082no—
297C2—5082no—
298C2#2255051011yescharacteristic
299C2C2×#1686123611yescharacteristic
300C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
hypercenter#460D127
hypocenterC127#3274
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. #22550548C2semidirect
3. #22550511C2semidirect
4. #22550252C2semidirect
5. #22550128C2semidirect
6. #22550072C2semidirect
7. #22550020C2semidirect
8. #22550011C2semidirect
9. C4×#14523235C2semidirect
10. #22549953C2semidirect
11. #460×C127C2semidirect
12. C2×#14523249C22semidirect
18. C2×#14523233C22semidirect
19. C2×#14523233C22semidirect
20. C2×#14523231C22semidirect
22. #16861206C22semidirect
24. #16861184C22semidirect
31. #16861140C22semidirect
32. #16861133C22semidirect
33. #16861131C22semidirect
35. (C2×D127)⋊C8C22semidirect
36. C4×C127⋊C8C22semidirect
37. C2×SD16×C127C22semidirect
39. C4×Q8×C127C22semidirect
40. C4⋊C8×C127C22semidirect
41. #102×C127C22semidirect
60. #14523235D4semidirect
107. C127⋊C8C2×D4semidirect
110. Q8×C127C2×D4semidirect
138. C4×C127AGL(1,ℤ/8)semidirect
151. C127#3274semidirect
157. #460D127semidirect
168. C2×SD16C2×D127semidirect
181. C4×Q8C2×D127semidirect
186. C4⋊C8C2×D127semidirect
190. #102C2×D127semidirect
242. Q8D4×D127semidirect
291. C4#16861236semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#225505482
#225505112
#225502522
#225501282
#225500722
#225500202
#225500112
C4×#145232352
#225499532
#460×C1272
#3274127