Finite Group Databasebeta

← #30648735

Subgroups — #30648735

Order 16256.

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

normal subgroups only

This group has 448 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
1#22550428C211yesmaximal, characteristic, complemented
2C2×#16861171C211yesmaximal, characteristic
3#22550206C211yesmaximal, characteristic, complemented
4#22550165C212yesmaximal, complemented
5#22550080C212yesmaximal, complemented
6#22550029C211yesmaximal, characteristic
7#22550026C211yesmaximal, characteristic
8#22549963C212yesmaximal, complemented
9#22549961C212yesmaximal, complemented
10#22549926C211yesmaximal, characteristic, complemented
11#483×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523237C2212yes—
15C2×#14523235C2212yes—
16#16861208C2211yescharacteristic, complemented
17#16861207C2211yescharacteristic, complemented
18#16861199—21no—
19#16861188C2211yescharacteristic
20#16861186—21no—
21Q8×Dic127C2211yescharacteristic, complemented
22#16861174—21no—
23#16861174C2211yescharacteristic, complemented
24#16861171—21no—
25#16861171—21no—
26#16861171C2212yes—
27#16861171C2212yes—
28#16861165—21no—
29C2×(C2×D127)⋊C4—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861140C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861132C2212yescomplemented
34#16861131C2212yes—
35#16861129C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37C8×Dic127C2212yescomplemented
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#106×C127C2212yescomplemented
41#103×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127—21no—
44C2×Dic254—21no—
45C2×Dic254C2311yescharacteristic
46C2×D508—24no—
47C2×D508—21no—
48C2×D508C2312yes—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—44no—
54#14523237—24no—
55#14523235—24no—
56C2×C127⋊C8C2312yes—
57C4⋊Dic127—21no—
58C4⋊Dic127C2311yescharacteristic
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67C4×Dic127—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—42no—
85C22×D127—42no—
86Dic254—22no—
87Dic254—42no—
88D508—24no—
89D508—22no—
90D508#4114yes—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—41no—
94C2×Dic127SD1614yescomplemented
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C4×D127—44no—
100C4×D127—44no—
101C127⋊C8—22no—
102C23×C127C2×D411yescharacteristic
103Q8×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127—21no—
113C22×C127—21no—
114C22×C127C2×SD1612yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—24no—
117C2×D127—48no—
118C2×D127—42no—
119C2×D127—42no—
120C4×C127—21no—
121C4×C127—41no—
122C4×C127—42no—
123C4×C127—41no—
124C4×C127C2×#4112yes—
125Dic127—21no—
126Dic127—21no—
127Dic127—41no—
128Dic127—41no—
129Dic127C2×SD1612yes—
130Dic127C2×SD1612yes—
131C2×C127—22no—
132C2×C127#52511yescharacteristic
133C2×C127C22×SD1611yescharacteristic
134C2×C127#57511yescharacteristic
135D127—44no—
136#3101—1271nomaximal, Sylow
137C127#310111yescharacteristic, complemented, Sylow
138#533—1271no—
139C2×C4×D4—1271no—
140#483—1271no—
141#483D12711yescharacteristic, complemented
142#476—1272no—
143#461—1272no—
144#450—1271no—
145#450—1271no—
146C4×SD16—1272no—
147C4×SD16—1272no—
148#431—1271no—
149C22×D4—1271no—
150C23×C4—2541no—
151C2×SD16—1272no—
152C2×SD16—1272no—
153C4⋊Q8—1271no—
154C4⋊Q8—1271no—
155#125—2541no—
156C22⋊Q8—1271no—
157C22⋊Q8—2541no—
158C22⋊Q8C2×D12711yescharacteristic
159C4×Q8—1271no—
160C4×Q8—1271no—
161C4×Q8—2541no—
162C4×D4—1272no—
163C4×D4—1272no—
164C4×D4—2541no—
165C4×D4—2541no—
166#117—2541no—
167C2×C4⋊C4C2×D12711yescharacteristic
168C2×C22⋊C4—2541no—
169C2×C42—1271no—
170#106—1272no—
171#106C2×D12712yescomplemented
172C4⋊C8—1272no—
173#103—1272no—
174#103C2×D12712yescomplemented
175#102—1272no—
176#102—1272no—
177C22⋊C8—1271no—
178C22⋊C8C2×D12711yescharacteristic
179C4×C8—1272no—
180C24—2541no—
181C2×Q8—1271no—
182C2×Q8—2541no—
183C2×Q8C22×D12711yescharacteristic
184C2×D4—1272no—
185C2×D4—2544no—
186C2×D4—2541no—
187C22×C4—21no—
188C22×C4—1271no—
189C22×C4—1271no—
190C22×C4—5084no—
191C22×C4—2542no—
192C22×C4—2542no—
193C22×C4C22×D12711yescharacteristic
194SD16—2544no—
195SD16—2544no—
196C2×C8—1272no—
197C2×C8C22×D12712yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—1271no—
201C4⋊C4—2541no—
202C4⋊C4—2541no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C4⋊C4C22×D12711yescharacteristic
208C4⋊C4C22×D12712yes—
209C22⋊C4—21no—
210C22⋊C4—2542no—
211C22⋊C4—2542no—
212C22⋊C4—2541no—
213C42—1271no—
214C42—1271no—
215C42—1271no—
216C42—1271no—
217C42—2541no—
218C42—2541no—
219C23—5082no—
220C23—5082no—
221C23—2544no—
222C23—2542no—
223C23D4×D12711yescharacteristic
224Q8—22no—
225Q8—5082no—
226Q8—2542no—
227D4—1274no—
228D4—2544no—
229D4—2542no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—1274no—
236C2×C4—1271no—
237C2×C4—1271no—
238C2×C4—1271no—
239C2×C4—1271no—
240C2×C4—5084no—
241C2×C4—5084no—
242C2×C4—5081no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4D4×D12711yescharacteristic
246C2×C4C23×D12711yescharacteristic
247C8—22no—
248C8—2542no—
249C22—21no—
250C22—21no—
251C22—5088no—
252C22—5082no—
253C22—5082no—
254C22—2544no—
255C22SD16×D12712yes—
256C22C2×D4×D12711yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—1272no—
262C4—1272no—
263C4—5081no—
264C4—5081no—
265C4—2541no—
266C4—2541no—
267C4#41×D12712yes—
268C2—22no—
269C2—5084no—
270C2#2255038411yescharacteristic
271C2C2×SD16×D12711yescharacteristic
272C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#483×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#483D127
hypocenterC127#3101
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. #22550428C2semidirect
3. #22550206C2semidirect
4. #22550165C2semidirect
5. #22550080C2semidirect
8. #22549963C2semidirect
9. #22549961C2semidirect
10. #22549926C2semidirect
11. #483×C127C2semidirect
16. #16861208C22semidirect
17. #16861207C22semidirect
21. Q8×Dic127C22semidirect
23. #16861174C22semidirect
31. #16861140C22semidirect
32. Dic127⋊C8C22semidirect
33. #16861132C22semidirect
37. C8×Dic127C22semidirect
40. #106×C127C22semidirect
41. #103×C127C22semidirect
94. C2×Dic127SD16semidirect
137. C127#3101semidirect
141. #483D127semidirect
171. #106C2×D127semidirect
174. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504282
C2×#168611712
#225502062
#225501652
#225500802
#225500292
#225500262
#225499632
#225499612
#225499262
#483×C1272
#3101127