Finite Group Databasebeta

← #30648864

Subgroups — #30648864

Order 16256.

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

normal subgroups only

This group has 450 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×C4×D508C211yesmaximal, characteristic, complemented
2#22550204C211yesmaximal, characteristic, complemented
3#22550187C211yesmaximal, characteristic, complemented
4#22550173C212yesmaximal, complemented
5#22550088C212yesmaximal, complemented
6#22550026C211yesmaximal, characteristic, complemented
7#22550009C211yesmaximal, characteristic
8#22549986C212yesmaximal, complemented
9#22549974C212yesmaximal, complemented
10#22549933C211yesmaximal, characteristic, complemented
11#529×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523235C2212yes—
15C2×#14523233C2212yes—
16C4×D508—21no—
17C4×D508—21no—
18C4×D508C2212yes—
19C4×D508C2212yes—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×(C2×D127)⋊C4—21no—
22#16861142C2212yescomplemented
23#16861140C2212yescomplemented
24C4⋊(C127⋊C8)C2212yescomplemented
25#16861133C2212yescomplemented
26#16861130C2212yescomplemented
27#16861129C2212yes—
28#16861129C2212yes—
29C22⋊(C127⋊C8)C2211yescharacteristic
30C22⋊(C127⋊C8)C2211yescharacteristic
31#16861120C2212yescomplemented
32#126×C127—21no—
33#125×C127C2211yescharacteristic, complemented
34#124×C127C2211yescharacteristic, complemented
35#123×C127—21no—
36C22⋊Q8×C127C2211yescharacteristic
37#121×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic, complemented
39C4×D4×C127—21no—
40C4×D4×C127C2211yescharacteristic, complemented
41#117×C127—21no—
42C2×C42×C127C2211yescharacteristic
43C23×D127—21no—
44C2×D508—24no—
45C2×D508—21no—
46C2×D508C2312yes—
47C22×Dic127—21no—
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127—44no—
51#14523235—24no—
52#14523233—24no—
53C2×C127⋊C8C2312yes—
54C2×C127⋊C8C2312yes—
55C4⋊Dic127—21no—
56C4⋊Dic127C2312yes—
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—22no—
59C2×Q8×C127C2311yescharacteristic
60C2×D4×C127—21no—
61C2×D4×C127C2311yescharacteristic
62C22×C4×C127—21no—
63C22×C4×C127D411yescharacteristic
64C22×C4×C127D411yescharacteristic
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—21no—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2311yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C22⋊C4×C127—21no—
74C22⋊C4×C127—21no—
75C22⋊C4×C127—21no—
76C22⋊C4×C127—21no—
77C42×C127—21no—
78C42×C127D411yescharacteristic
79C42×C127D411yescharacteristic
80C42×C127C2311yescharacteristic
81C42×C127C2311yescharacteristic
82C22×D127—24no—
83C22×D127—22no—
84C22×D127—42no—
85C22×D127—42no—
86D508—24no—
87D508—22no—
88D508#4114yes—
89C2×Dic127—22no—
90C2×Dic127—42no—
91C4×D127—44no—
92C4×D127—44no—
93C127⋊C8—22no—
94C127⋊C8—22no—
95C23×C127—21no—
96C23×C127C2×D411yescharacteristic
97Q8×C127—22no—
98D4×C127—22no—
99D4×C127—42no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—41no—
106C2×C4×C127—42no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127AGL(1,ℤ/8)12yes—
119C22×C127C22×D411yescharacteristic
120C2×D127—24no—
121C2×D127—48no—
122C2×D127—42no—
123C2×D127—42no—
124C4×C127—21no—
125C4×C127—21no—
126C4×C127—21no—
127C4×C127—21no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—41no—
131C4×C127—41no—
132C4×C127C2×#4112yes—
133Dic127—42no—
134C2×C127—22no—
135C2×C127—81no—
136C2×C127#52511yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127#57511yescharacteristic
139D127—44no—
140#3114—1271nomaximal, Sylow
141C127#311411yescharacteristic, complemented, Sylow
142#529D12711yescharacteristic, complemented
143C2×C4×D4—1271no—
144#483—1271no—
145#480—1271no—
146#478—1272no—
147#463—1272no—
148#450—1271no—
149#447—1271no—
150#442—1272no—
151#440—1272no—
152#432—1271no—
153C22×D4—1271no—
154C23×C4—2541no—
155C2×SD16—1272no—
156C2×D8—1272no—
157#126—21no—
158#125C2×D12711yescharacteristic, complemented
159#124C2×D12711yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D12711yescharacteristic
162#121C2×D12711yescharacteristic
163C4×Q8C2×D12711yescharacteristic, complemented
164C4×D4—21no—
165C4×D4—1272no—
166C4×D4—1272no—
167C4×D4—2541no—
168C4×D4—2541no—
169C4×D4C2×D12711yescharacteristic, complemented
170#117—21no—
171C2×C4⋊C4—1271no—
172C2×C22⋊C4—2541no—
173C2×C42C2×D12711yescharacteristic
174#107—1272no—
175#106—1272no—
176C4⋊C8—1272no—
177#103—1272no—
178#102—1272no—
179#102—1272no—
180#102—1272no—
181C22⋊C8—1271no—
182C22⋊C8—1271no—
183#97—1272no—
184C24—2541no—
185C2×Q8C22×D12711yescharacteristic
186C2×D4—21no—
187C2×D4—1272no—
188C2×D4—2544no—
189C2×D4—2541no—
190C2×D4C22×D12711yescharacteristic
191C22×C4—21no—
192C22×C4—5084no—
193C22×C4—2541no—
194C22×C4—2542no—
195C22×C4—2542no—
196C22×C4C22⋊D12711yescharacteristic
197C22×C4C22⋊D12711yescharacteristic
198C22×C4C22×D12711yescharacteristic
199SD16—2544no—
200D8—2544no—
201C2×C8—1272no—
202C2×C8—1272no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—1272no—
208C4⋊C4—2541no—
209C4⋊C4C22×D12711yescharacteristic
210C4⋊C4C22×D12711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—2542no—
217C22⋊C4—2542no—
218C42—21no—
219C42C22⋊D12711yescharacteristic
220C42C22⋊D12711yescharacteristic
221C42C22×D12711yescharacteristic
222C42C22×D12711yescharacteristic
223C23—21no—
224C23—5082no—
225C23—5082no—
226C23—2544no—
227C23—2542no—
228C23C2×C22⋊D12711yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—1274no—
233D4—2544no—
234D4—2542no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—5084no—
243C2×C4—5084no—
244C2×C4—5082no—
245C2×C4—2542no—
246C2×C4C2×C22⋊D12711yescharacteristic
247C2×C4C2×C22⋊D12711yescharacteristic
248C2×C4C2×C22⋊D12711yescharacteristic
249C2×C4C2×C22⋊D12711yescharacteristic
250C2×C4C2×C22⋊D12711yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—2542no—
253C8—2542no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—5088no—
260C22—5082no—
261C22—5082no—
262C22—2544no—
263C22#1686123412yes—
264C22C22×C22⋊D12711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—5082no—
274C4#41×D12712yes—
275C2—22no—
276C2—81no—
277C2—5084no—
278C2#2255038311yescharacteristic
279C2C2×#1686123411yescharacteristic
280C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#529×C127C2
socleC22×C127C22×D4
prime radical#22550009C2
hypercenter#529D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D508C2semidirect
2. #22550204C2semidirect
3. #22550187C2semidirect
4. #22550173C2semidirect
5. #22550088C2semidirect
6. #22550026C2semidirect
8. #22549986C2semidirect
9. #22549974C2semidirect
10. #22549933C2semidirect
11. #529×C127C2semidirect
22. #16861142C22semidirect
23. #16861140C22semidirect
24. C4⋊(C127⋊C8)C22semidirect
25. #16861133C22semidirect
26. #16861130C22semidirect
31. #16861120C22semidirect
33. #125×C127C22semidirect
34. #124×C127C22semidirect
38. C4×Q8×C127C22semidirect
40. C4×D4×C127C22semidirect
141. C127#3114semidirect
142. #529D127semidirect
158. #125C2×D127semidirect
159. #124C2×D127semidirect
163. C4×Q8C2×D127semidirect
169. C4×D4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5082
#225502042
#225501872
#225501732
#225500882
#225500262
#225500092
#225499862
#225499742
#225499332
#529×C1272
#3114127