Finite Group Databasebeta

← #30648667

Subgroups — #30648667

Order 16256.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D127C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3#22550202C212yesmaximal, complemented
4#22550198C212yesmaximal, complemented
5#22550194C211yesmaximal, characteristic, complemented
6#22550190C211yesmaximal, characteristic, complemented
7#22550187C211yesmaximal, characteristic, complemented
8#22549915C212yesmaximal, complemented
9#22549879C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#481×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127—21no—
13C2×D4×D127C2211yescharacteristic
14C22×C4×D127C2211yescharacteristic
15C2×#14523231C2212yes—
16#16861184C2211yescharacteristic
17#16861183C2211yescharacteristic
18#16861182—21no—
19#16861181—21no—
20#16861180C2211yescharacteristic
21#16861167—21no—
22#16861167—21no—
23#16861167C2212yes—
24#16861167C2212yescomplemented
25C2×C4⋊Dic127C2211yescharacteristic, complemented
26C4⋊C4×D127C2211yescharacteristic
27C2×(C2×D127)⋊C4—21no—
28C22⋊C4×D127—21no—
29C2×C4×Dic127—21no—
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32#16861140C2212yescomplemented
33#16861131C2212yescomplemented
34#16861130C2212yescomplemented
35#16861129C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic, complemented
37(C2×D127)⋊C8C2212yescomplemented
38#121×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic, complemented
40#106×C127C2212yescomplemented
41#102×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic, complemented
43C23×D127—21no—
44C23×D127D411yescharacteristic
45C2×D508—21no—
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C22⋊D127—21no—
49C2×C22⋊D127—21no—
50D4×D127—24no—
51D4×D127—44no—
52C22×Dic127—21no—
53C22×Dic127D411yescharacteristic
54C2×C4×D127—24no—
55C2×C4×D127—21no—
56C2×C4×D127D411yescharacteristic
57C2×C4×D127D411yescharacteristic
58C2×C4×D127C2311yescharacteristic
59C2×C4×D127C2311yescharacteristic
60#14523231—24no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64C4⋊Dic127C2311yescharacteristic
65Dic127⋊C4—21no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—21no—
71C4×Dic127—22no—
72C4×Dic127—22no—
73C2×D4×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—41no—
86C22×D127—42no—
87C22×D127—42no—
88C22×D127—42no—
89C22×D127C2×D411yescharacteristic
90C22×D127C2×D411yescharacteristic
91D508—22no—
92D508—42no—
93C22⋊D127—42no—
94C22⋊D127—42no—
95C22⋊D127—42no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127—21no—
99C2×Dic127—42no—
100C2×Dic127C2×D411yescharacteristic
101C2×Dic127C2×D411yescharacteristic
102C4×D127—22no—
103C4×D127—22no—
104C4×D127—22no—
105C4×D127—42no—
106C4×D127#4114yes—
107C127⋊C8—22no—
108C23×C127—21no—
109C23×C127C2×D411yescharacteristic
110D4×C127—22no—
111D4×C127—42no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—22no—
115C2×C4×C127—42no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127AGL(1,ℤ/8)12yes—
125C22×C127C22×D411yescharacteristic
126C2×D127—21no—
127C2×D127—21no—
128C2×D127—21no—
129C2×D127—21no—
130C2×D127—21no—
131C2×D127—44no—
132C2×D127—41no—
133C2×D127—41no—
134C2×D127—41no—
135C2×D127—84no—
136C2×D127#12312yes—
137C4×C127—21no—
138C4×C127—41no—
139C4×C127—42no—
140C4×C127C2×#4112yes—
141Dic127—21no—
142Dic127—41no—
143Dic127—42no—
144Dic127#12312yes—
145C2×C127—22no—
146C2×C127—81no—
147C2×C127C2×#12311yescharacteristic
148C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C127#57511yescharacteristic
150D127—22no—
151D127—41no—
152D127—81no—
153#3094—1271nomaximal, Sylow
154C127#309411yescharacteristic, complemented, Sylow
155C2×#121—1271no—
156C2×#117—1271no—
157#482—1272no—
158#482—1272no—
159#481—1271no—
160#481D12711yescharacteristic, complemented
161#480—1271no—
162#480—1271no—
163#428—1272no—
164#418—1272no—
165#407—1271no—
166C22×D4—1271no—
167C22×D4—2541no—
168C23×C4—1271no—
169C2×M16—1272no—
170#121—1271no—
171#121—1271no—
172#121—1271no—
173#121—2541no—
174#121—2541no—
175#121C2×D12711yescharacteristic
176#117—1272no—
177#117—1272no—
178#117—2541no—
179#117—2541no—
180C2×C4⋊C4—1271no—
181C2×C4⋊C4—1271no—
182C2×C4⋊C4C2×D12711yescharacteristic, complemented
183C2×C22⋊C4—2541no—
184C2×C22⋊C4—2541no—
185C2×C42—2541no—
186#107—1272no—
187#107—1272no—
188#106—1272no—
189#106C2×D12712yescomplemented
190#102—1272no—
191#102—1272no—
192#102—1272no—
193#102C2×D12712yescomplemented
194C22⋊C8—1271no—
195C22⋊C8—1272no—
196C22⋊C8C2×D12711yescharacteristic, complemented
197C24—1271no—
198C24—2541no—
199C2×D4—21no—
200C2×D4—1271no—
201C2×D4—5084no—
202C2×D4—2541no—
203C2×D4—2544no—
204C2×D4—2541no—
205C2×D4—2541no—
206C2×D4—2541no—
207C2×D4C22×D12711yescharacteristic
208C22×C4—21no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—1271no—
213C22×C4—1271no—
214C22×C4—2544no—
215C22×C4—2541no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218M16—2544no—
219C2×C8—1272no—
220C2×C8C22×D12712yes—
221C4⋊C4—21no—
222C4⋊C4—1272no—
223C4⋊C4—1271no—
224C4⋊C4—2541no—
225C4⋊C4—2541no—
226C4⋊C4C22×D12712yes—
227C4⋊C4C22×D12711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2541no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C22⋊C4—2541no—
233C22⋊C4—2541no—
234C42—2542no—
235C42—2542no—
236C23—21no—
237C23—1271no—
238C23—1271no—
239C23—5081no—
240C23—5082no—
241C23—5082no—
242C23—5082no—
243C23—2542no—
244C23—2541no—
245C23—2542no—
246C23D4×D12711yescharacteristic
247D4—22no—
248D4—42no—
249D4—5082no—
250D4—5082no—
251D4—5082no—
252D4—5082no—
253D4—2542no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—1271no—
259C2×C4—1274no—
260C2×C4—1271no—
261C2×C4—5082no—
262C2×C4—5082no—
263C2×C4—2542no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4—2542no—
267C2×C4—2541no—
268C2×C4—2542no—
269C2×C4D4×D12711yescharacteristic
270C2×C4C23×D12711yescharacteristic
271C8—22no—
272C8—2542no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—1272no—
279C22—10164no—
280C22—5084no—
281C22—5081no—
282C22—5081no—
283C22—5081no—
284C22—2541no—
285C22—2541no—
286C22—2541no—
287C22—2541no—
288C22—2541no—
289C22#1686123612yes—
290C22C2×D4×D12711yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—1272no—
295C4—5081no—
296C4—5082no—
297C4—2541no—
298C4#41×D12712yes—
299C2—22no—
300C2—81no—
301C2—10161no—
302C2—5081no—
303C2—2542no—
304C2#123×D12711yescharacteristic
305C2C2×#1686123611yescharacteristic
306C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#481×C127C2
socleC22×C127C22×D4
prime radical#121×D127C2
hypercenter#481D127
hypocenterC127#3094
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#16861167C2semidirect
3. #22550202C2semidirect
4. #22550198C2semidirect
5. #22550194C2semidirect
6. #22550190C2semidirect
7. #22550187C2semidirect
8. #22549915C2semidirect
9. #22549879C2semidirect
10. #22549838C2semidirect
11. #481×C127C2semidirect
24. #16861167C22semidirect
25. C2×C4⋊Dic127C22semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. #16861140C22semidirect
33. #16861131C22semidirect
34. #16861130C22semidirect
35. #16861129C22semidirect
36. C22⋊(C127⋊C8)C22semidirect
37. (C2×D127)⋊C8C22semidirect
39. C2×C4⋊C4×C127C22semidirect
40. #106×C127C22semidirect
41. #102×C127C22semidirect
42. C22⋊C8×C127C22semidirect
154. C127#3094semidirect
160. #481D127semidirect
182. C2×C4⋊C4C2×D127semidirect
189. #106C2×D127semidirect
193. #102C2×D127semidirect
196. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#121×D1272
C2×#168611672
#225502022
#225501982
#225501942
#225501902
#225501872
#225499152
#225498792
#225498382
#481×C1272
#3094127