Finite Group Databasebeta

← #30648871

Subgroups — #30648871

Order 16256.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22550412C211yesmaximal, characteristic, complemented
2C2×#16861171C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic, complemented
4#22550175C212yesmaximal, complemented
5#22550094C212yesmaximal, complemented
6#22550026C211yesmaximal, characteristic, complemented
7#22550012C211yesmaximal, characteristic
8#22549987C212yesmaximal, complemented
9#22549979C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#483×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523235C2212yes—
15C2×D1016C2212yes—
16#16861204—21no—
17#16861200C2211yescharacteristic, complemented
18#16861198C2211yescharacteristic, complemented
19#16861193—21no—
20#16861183C2211yescharacteristic
21Q8×Dic127C2211yescharacteristic, complemented
22#16861171—21no—
23#16861171—21no—
24#16861171C2212yes—
25#16861171C2212yes—
26#16861171C2211yescharacteristic, complemented
27(C22⋊D127)⋊C4—21no—
28#16861167—21no—
29C2×(C2×D127)⋊C4—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861142C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861131C2212yes—
34#16861129C2212yes—
35#16861129C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yescomplemented
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#106×C127C2212yescomplemented
41#103×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127—21no—
44C2×D508—24no—
45C2×D508—21no—
46C2×D508C2311yescharacteristic
47C2×D508C2312yes—
48C2×C22⋊D127—21no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—21no—
54C2×C4×D127—44no—
55#14523235—24no—
56D1016—24no—
57C2×C127⋊C8C2312yes—
58C4⋊Dic127—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—21no—
62(C2×D127)⋊C4—21no—
63(C2×D127)⋊C4—21no—
64(C2×D127)⋊C4—21no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—21no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×Q8×C127C2311yescharacteristic
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—24no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—42no—
86C22×D127—42no—
87D508—24no—
88D508—22no—
89D508—22no—
90D508#4114yes—
91C22⋊D127—42no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—41no—
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C4×D127—42no—
100C4×D127—44no—
101C4×D127—44no—
102C127⋊C8—22no—
103C23×C127C2×D411yescharacteristic
104Q8×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—42no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127AGL(1,ℤ/8)12yes—
116C22×C127C22×D411yescharacteristic
117C2×D127—24no—
118C2×D127—48no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C4×C127—21no—
125C4×C127—41no—
126C4×C127—42no—
127C4×C127—41no—
128C4×C127C2×#4112yes—
129Dic127—21no—
130Dic127—21no—
131Dic127—21no—
132Dic127—21no—
133Dic127—41no—
134C2×C127—22no—
135C2×C127#52511yescharacteristic
136C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C127#57511yescharacteristic
138D127—44no—
139D127—81no—
140#3114—1271nomaximal, Sylow
141C127#311411yescharacteristic, complemented, Sylow
142#529—1271no—
143C2×C4×D4—1271no—
144#483D12711yescharacteristic, complemented
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—2541no—
158#125—1271no—
159#124—1271no—
160#123—2541no—
161C22⋊Q8C2×D12711yescharacteristic
162#121—1271no—
163C4×Q8—1271no—
164C4×D4—1272no—
165C4×D4—1272no—
166C4×D4—1271no—
167C4×D4—2541no—
168C4×D4—2541no—
169C4×D4—2541no—
170#117—2541no—
171C2×C4⋊C4C2×D12711yescharacteristic
172C2×C22⋊C4—2541no—
173C2×C42—1271no—
174#107—1272no—
175#106C2×D12712yescomplemented
176C4⋊C8—1272no—
177#103C2×D12712yescomplemented
178#102—1272no—
179#102—1272no—
180#102—1272no—
181C22⋊C8—1271no—
182C22⋊C8C2×D12711yescharacteristic
183#97—1272no—
184C24—2541no—
185C2×Q8C22×D12711yescharacteristic
186C2×D4—1272no—
187C2×D4—1271no—
188C2×D4—2544no—
189C2×D4—2541no—
190C2×D4—2541no—
191C22×C4—21no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—5084no—
195C22×C4—2541no—
196C22×C4—2542no—
197C22×C4—2542no—
198C22×C4C22×D12711yescharacteristic
199SD16—2544no—
200D8—2544no—
201C2×C8—1272no—
202C2×C8C22×D12712yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—1271no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4C22×D12711yescharacteristic
210C4⋊C4C22×D12712yes—
211C22⋊C4—21no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C22⋊C4—2541no—
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C42—1271no—
219C42—1271no—
220C42—1271no—
221C42—1271no—
222C42—2541no—
223C23—5082no—
224C23—5082no—
225C23—2544no—
226C23—2541no—
227C23—2542no—
228C23D4×D12711yescharacteristic
229Q8—22no—
230D4—1274no—
231D4—5082no—
232D4—2544no—
233D4—2542no—
234D4—2542no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—1271no—
241C2×C4—1271no—
242C2×C4—1271no—
243C2×C4—1271no—
244C2×C4—5084no—
245C2×C4—5084no—
246C2×C4—5082no—
247C2×C4—5081no—
248C2×C4—2541no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—22no—
253C8—2542no—
254C22—21no—
255C22—21no—
256C22—5088no—
257C22—5081no—
258C22—5081no—
259C22—5081no—
260C22—5082no—
261C22—5082no—
262C22—2544no—
263C22#1686123612yes—
264C22C2×D4×D12711yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—5081no—
270C4—2541no—
271C4—2541no—
272C4—2541no—
273C4—2541no—
274C4#41×D12712yes—
275C2—22no—
276C2—10161no—
277C2—5084no—
278C2#2255038411yescharacteristic
279C2C2×#1686123611yescharacteristic
280C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#483×C127C2
socleC22×C127C22×D4
prime radical#22550012C2
hypercenter#483D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22550412C2semidirect
2. C2×#16861171C2semidirect
3. #22550189C2semidirect
4. #22550175C2semidirect
5. #22550094C2semidirect
6. #22550026C2semidirect
8. #22549987C2semidirect
9. #22549979C2semidirect
10. #22549930C2semidirect
11. #483×C127C2semidirect
17. #16861200C22semidirect
18. #16861198C22semidirect
21. Q8×Dic127C22semidirect
26. #16861171C22semidirect
31. #16861142C22semidirect
32. Dic127⋊C8C22semidirect
35. #16861129C22semidirect
37. #16861119C22semidirect
40. #106×C127C22semidirect
41. #103×C127C22semidirect
141. C127#3114semidirect
144. #483D127semidirect
175. #106C2×D127semidirect
177. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504122
C2×#168611712
#225501892
#225501752
#225500942
#225500262
#225500122
#225499872
#225499792
#225499302
#483×C1272
#3114127