Finite Group Databasebeta

← #30647579

Subgroups — #30647579

Order 16256.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D8×D127C211yesmaximal, characteristic
2#22550348C212yesmaximal, complemented
3#22549989C211yesmaximal, characteristic, complemented
4#22549988C211yesmaximal, characteristic, complemented
5#22549987C211yesmaximal, characteristic, complemented
6C4×#14523233C212yesmaximal, complemented
7#22549956C212yesmaximal, complemented
8#22549911C211yesmaximal, characteristic, complemented
9#22549879C212yesmaximal, complemented
10#22549833C211yesmaximal, characteristic, complemented
11#442×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127C2212yes—
13C2×D1016C2211yescharacteristic
14C2×#14523233C2212yes—
15D8×D127C418yescomplemented
16C2×#14523231C2211yescharacteristic
17C2×C8×D127C2211yescharacteristic
18C4×D508C2212yescomplemented
19D4×Dic127C2212yescomplemented
20#16861171C2212yescomplemented
21C4×C22⋊D127—22no—
22(C22⋊D127)⋊C4—22no—
23#16861167C2212yes—
24#16861166C2211yescharacteristic
25C2×(C2×D127)⋊C4—22no—
26C22⋊C4×D127—22no—
27#16861142C2212yescomplemented
28#16861141C2211yescharacteristic, complemented
29#16861131C2212yescomplemented
30#16861130C2212yescomplemented
31#16861129C2212yescomplemented
32#16861119C2211yescharacteristic, complemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34C8×Dic127C2211yescharacteristic, complemented
35C2×D8×C127C2211yescharacteristic
36C4×D4×C127C2212yescomplemented
37#106×C127C2211yescharacteristic, complemented
38#102×C127C2212yescomplemented
39#97×C127C2211yescharacteristic, complemented
40C23×D127—22no—
41C2×D508C2312yes—
42C2×C22⋊D127—22no—
43D4×D127—24no—
44D4×D127C2×C418yescomplemented
45C22×Dic127—22no—
46C2×C4×D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127C2311yescharacteristic
49D1016C2×C414yescomplemented
50#14523233C2×C418yescomplemented
51#14523231—22no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127C2×C414yescomplemented
55C4⋊Dic127C2312yes—
56Dic127⋊C4D412yes—
57C22⋊Dic127—22no—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4D412yes—
62C4×Dic127—22no—
63C4×Dic127C2311yescharacteristic
64C2×D4×C127C2312yes—
65C22×C4×C127—22no—
66D8×C127C2×C414yescomplemented
67C2×C8×C127C2311yescharacteristic
68C2×C8×C127C2311yescharacteristic
69C4⋊C4×C127C2312yes—
70C22⋊C4×C127—22no—
71C42×C127C2311yescharacteristic
72C22×D127—22no—
73C22×D127—28no—
74C22×D127—82no—
75C22×D127—82no—
76C22×D127C2×D411yescharacteristic
77D508—22no—
78D508C22×C414yes—
79C22⋊D127—28no—
80C2×Dic127—22no—
81C2×Dic127—82no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D411yescharacteristic
84C4×D127—82no—
85C4×D127—82no—
86C4×D127C22×C412yes—
87C4×D127#4112yes—
88C127⋊C8—21no—
89C127⋊C8C22×C412yes—
90C23×C127—22no—
91D4×C127—22no—
92D4×C127C22×C414yes—
93C2×C4×C127—22no—
94C2×C4×C127—82no—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—21no—
98C8×C127C22×C412yes—
99C22×C127—24no—
100C22×C127—42no—
101C22×C127—42no—
102C22×C127C22×D411yescharacteristic
103C2×D127—22no—
104C2×D127—24no—
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—88no—
108C2×D127C4×D412yes—
109C4×C127—22no—
110C4×C127—42no—
111C4×C127C23×C411yescharacteristic
112C4×C127C2×#4111yescharacteristic
113Dic127—22no—
114Dic127—42no—
115Dic127C4×D412yes—
116C2×C127—44no—
117C2×C127C2×C4×D411yescharacteristic
118C2×C127#57612yes—
119D127—22no—
120D127—44no—
121#2948—1271nomaximal, Sylow
122C127#294811yescharacteristic, complemented, Sylow
123C22×D8—1271no—
124#518—1272no—
125#442—1271no—
126#442—1271no—
127#442—1271no—
128#442D12711yescharacteristic, complemented
129C4×D8—1272no—
130C4×D8—1272no—
131#427—1271no—
132#418—1272no—
133#405—1271no—
134C22×D4—1272no—
135C2×D8—1272no—
136C2×D8—1278no—
137C2×D8—1271no—
138C2×D8C2×D12711yescharacteristic
139C2×M16—1271no—
140C22×C8—1271no—
141C4×D4—1272no—
142C4×D4—1272no—
143C4×D4—1272no—
144C4×D4—2542no—
145C4×D4—2542no—
146C4×D4C2×D12712yescomplemented
147#117—1271no—
148#117—1272no—
149C2×C22⋊C4—2542no—
150C2×C22⋊C4—2542no—
151#107—1272no—
152#106—1271no—
153#106C2×D12711yescharacteristic, complemented
154#102—1272no—
155#102—1272no—
156#102—1272no—
157#102C2×D12712yescomplemented
158#97—1271no—
159#97C2×D12711yescharacteristic, complemented
160C4×C8—1271no—
161C4×C8—1271no—
162C24—2542no—
163C2×D4—1278no—
164C2×D4—1272no—
165C2×D4—2544no—
166C2×D4—2542no—
167C2×D4C22×D12712yes—
168C22×C4—22no—
169C22×C4—1271no—
170C22×C4—2542no—
171C22×C4—2542no—
172C22×C4—2542no—
173D8—1278no—
174D8—1274no—
175D8C4×D12714yescomplemented
176M16—2542no—
177C2×C8—1271no—
178C2×C8—1274no—
179C2×C8—1271no—
180C2×C8C22×D12711yescharacteristic
181C2×C8C22×D12711yescharacteristic
182C4⋊C4—1272no—
183C4⋊C4—1272no—
184C4⋊C4C22×D12712yes—
185C22⋊C4—22no—
186C22⋊C4—1272no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C42—1271no—
192C42—2542no—
193C42C22×D12711yescharacteristic
194C23—22no—
195C23—1271no—
196C23—10162no—
197C23—10162no—
198C23—2548no—
199C23—2542no—
200D4—22no—
201D4—1274no—
202D4—2548no—
203D4—2542no—
204D4C2×C4×D12714yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—1271no—
208C2×C4—1272no—
209C2×C4—1272no—
210C2×C4—1272no—
211C2×C4—10162no—
212C2×C4—10162no—
213C2×C4—10162no—
214C2×C4—2542no—
215C2×C4D4×D12712yes—
216C2×C4C23×D12711yescharacteristic
217C8—21no—
218C8—1272no—
219C8—2541no—
220C8C2×C4×D12712yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—1272no—
225C22—10168no—
226C22—5082no—
227C22—5082no—
228C22—2544no—
229C22—2542no—
230C22C2×D4×D12711yescharacteristic
231C4—22no—
232C4—42no—
233C4—1272no—
234C4—5082no—
235C4—2542no—
236C4C22×C4×D12711yescharacteristic
237C4#41×D12711yescharacteristic
238C2—44no—
239C2—5084no—
240C2—2542no—
241C2C4×D4×D12711yescharacteristic
242C2#2255068012yes—
subgroupG/H
centerC22C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
prime radicalC2×D8×D127C2
cosocleC2×C4×C127C24
hypercenter#442D127
hypocenterC127#2948
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #22550348C2semidirect
3. #22549989C2semidirect
4. #22549988C2semidirect
5. #22549987C2semidirect
6. C4×#14523233C2semidirect
7. #22549956C2semidirect
8. #22549911C2semidirect
9. #22549879C2semidirect
10. #22549833C2semidirect
11. #442×C127C2semidirect
15. D8×D127C4semidirect
18. C4×D508C22semidirect
19. D4×Dic127C22semidirect
20. #16861171C22semidirect
27. #16861142C22semidirect
28. #16861141C22semidirect
29. #16861131C22semidirect
30. #16861130C22semidirect
31. #16861129C22semidirect
32. #16861119C22semidirect
33. C4×C127⋊C8C22semidirect
34. C8×Dic127C22semidirect
36. C4×D4×C127C22semidirect
37. #106×C127C22semidirect
38. #102×C127C22semidirect
39. #97×C127C22semidirect
44. D4×D127C2×C4semidirect
49. D1016C2×C4semidirect
50. #14523233C2×C4semidirect
54. C8×D127C2×C4semidirect
66. D8×C127C2×C4semidirect
122. C127#2948semidirect
128. #442D127semidirect
146. C4×D4C2×D127semidirect
153. #106C2×D127semidirect
157. #102C2×D127semidirect
159. #97C2×D127semidirect
175. D8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1272
#225503482
#225499892
#225499882
#225499872
C4×#145232332
#225499562
#225499112
#225498792
#225498332
#442×C1272
#2948127