Finite Group Databasebeta

← #30648639

Subgroups — #30648639

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861168C211yesmaximal, characteristic, complemented
2#22550214C212yesmaximal, complemented
3#22550209C211yesmaximal, characteristic, complemented
4#22550204C211yesmaximal, characteristic, complemented
5#22550197C212yesmaximal, complemented
6#22550192C211yesmaximal, characteristic, complemented
7#22550187C211yesmaximal, characteristic, complemented
8#22549912C212yesmaximal, complemented
9#22549872C212yesmaximal, complemented
10C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
11#525×C127C211yesmaximal, characteristic
12C22×C127⋊C8C2212yes—
13#16861168—21no—
14#16861168—21no—
15#16861168C2212yes—
16#16861168C2212yes—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C22⋊Dic127—21no—
20C2×C4×Dic127—21no—
21#16861142C2212yescomplemented
22#16861142C2212yescomplemented
23#16861140C2212yescomplemented
24#16861140C2212yescomplemented
25#16861133C2212yescomplemented
26#16861133C2212yescomplemented
27#16861130C2212yescomplemented
28#16861130C2212yescomplemented
29C22⋊(C127⋊C8)C2211yescharacteristic
30C22⋊(C127⋊C8)C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C2×#41×C127C2211yescharacteristic
33C23×C4×C127C2211yescharacteristic
34#123×C127—21no—
35C22⋊Q8×C127C2211yescharacteristic
36C22⋊Q8×C127C2211yescharacteristic
37#121×C127C2211yescharacteristic
38#121×C127C2211yescharacteristic
39#120×C127—21no—
40C4×D4×C127—21no—
41C4×D4×C127—21no—
42#117×C127C2211yescharacteristic
43C22×Dic127—21no—
44C22×Dic127—21no—
45C2×C127⋊C8—24no—
46C2×C127⋊C8C2312yes—
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—21no—
49C4⋊Dic127—21no—
50C4⋊Dic127C2312yes—
51C4⋊Dic127C2312yes—
52C22⋊Dic127—22no—
53C22⋊Dic127—22no—
54C4×Dic127—22no—
55C4×Dic127—22no—
56C24×C127D411yescharacteristic
57#41×C127—24no—
58C2×Q8×C127C2311yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127C2311yescharacteristic
63C22×C4×C127—24no—
64C22×C4×C127—21no—
65C22×C4×C127D411yescharacteristic
66C22×C4×C127D411yescharacteristic
67C22×C4×C127D411yescharacteristic
68C22×C4×C127C2311yescharacteristic
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127—21no—
72C4⋊C4×C127—21no—
73C4⋊C4×C127C2311yescharacteristic
74C4⋊C4×C127C2311yescharacteristic
75C22⋊C4×C127—21no—
76C22⋊C4×C127—21no—
77C22⋊C4×C127—21no—
78C22⋊C4×C127—21no—
79C22⋊C4×C127—21no—
80C42×C127—21no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—42no—
84C2×Dic127—42no—
85C127⋊C8—22no—
86C127⋊C8—22no—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127—21no—
90C23×C127—41no—
91C23×C127C2×D411yescharacteristic
92C23×C127C2×D411yescharacteristic
93C23×C127C2×D411yescharacteristic
94Q8×C127—22no—
95D4×C127—22no—
96D4×C127—42no—
97D4×C127—42no—
98D4×C127—42no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—42no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127#4114yes—
112C2×C4×C127C2411yescharacteristic
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—44no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127#12312yes—
125C22×C127#13512yes—
126C22×C127C22×D411yescharacteristic
127C4×C127—21no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—41no—
131C4×C127—41no—
132C4×C127#12312yes—
133C4×C127C2×#4112yes—
134Dic127—42no—
135Dic127—42no—
136C2×C127—22no—
137C2×C127—22no—
138C2×C127—41no—
139C2×C127—81no—
140C2×C127C2×#12311yescharacteristic
141C2×C127C2×#13511yescharacteristic
142C2×C127#57511yescharacteristic
143#3091—1271nomaximal, Sylow
144C127#309111yescharacteristic, complemented, Sylow
145#525D12711yescharacteristic
146C2×#117—1271no—
147#485—1272no—
148#484—1271no—
149#483—1271no—
150#482—1272no—
151#481—1271no—
152#480—1271no—
153#427—1272no—
154#416—1272no—
155C2×C22⋊C8—1271no—
156C2×#41C2×D12711yescharacteristic
157C23×C4C2×D12711yescharacteristic
158C22×C8—1272no—
159#123—21no—
160C22⋊Q8C2×D12711yescharacteristic
161C22⋊Q8C2×D12711yescharacteristic
162#121C2×D12711yescharacteristic
163#121C2×D12711yescharacteristic
164#120—21no—
165C4×D4—21no—
166C4×D4—21no—
167#117—1272no—
168#117—1272no—
169#117—2541no—
170#117—2541no—
171#117C2×D12711yescharacteristic
172C2×C4⋊C4—1271no—
173C2×C4⋊C4—1271no—
174C2×C22⋊C4—2541no—
175C2×C42—2541no—
176#107—1272no—
177#107—1272no—
178#106—1272no—
179#106—1272no—
180#103—1272no—
181#103—1272no—
182#102—1272no—
183#102—1272no—
184C22⋊C8—1271no—
185C22⋊C8—1271no—
186C22⋊C8—1272no—
187C24C22⋊D12711yescharacteristic
188#41—24no—
189C2×Q8C22×D12711yescharacteristic
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4C22×D12711yescharacteristic
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—2541no—
197C22×C4—2541no—
198C22×C4C22⋊D12711yescharacteristic
199C22×C4C22⋊D12711yescharacteristic
200C22×C4C22⋊D12711yescharacteristic
201C22×C4C22×D12711yescharacteristic
202C22×C4C22×D12711yescharacteristic
203C22×C4C22×D12711yescharacteristic
204C2×C8—1272no—
205C2×C8—1272no—
206C2×C8—2544no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—1272no—
210C4⋊C4—1272no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C4⋊C4C22×D12711yescharacteristic
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C22⋊C4—2542no—
221C22⋊C4—2542no—
222C42—21no—
223C42—2542no—
224C42—2542no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—41no—
229C23C2×C22⋊D12711yescharacteristic
230C23C2×C22⋊D12711yescharacteristic
231C23C2×C22⋊D12711yescharacteristic
232Q8—22no—
233D4—22no—
234D4—42no—
235D4—42no—
236D4—42no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—5082no—
247C2×C4—5082no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4C2×C22⋊D12711yescharacteristic
251C2×C4C2×C22⋊D12711yescharacteristic
252C2×C4C2×C22⋊D12711yescharacteristic
253C2×C4#1452324714yes—
254C2×C4C23×D12711yescharacteristic
255C8—2542no—
256C8—2542no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22#1686119012yes—
269C22#1686122912yes—
270C22C22×C22⋊D12711yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—5082no—
277C4—5082no—
278C4#1686119012yes—
279C4C2×#1452324712yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—81no—
284C2C2×#1686119011yescharacteristic
285C2C2×#1686122911yescharacteristic
286C2#2255067411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3091
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16861168C2semidirect
2. #22550214C2semidirect
3. #22550209C2semidirect
4. #22550204C2semidirect
5. #22550197C2semidirect
6. #22550192C2semidirect
7. #22550187C2semidirect
8. #22549912C2semidirect
9. #22549872C2semidirect
10. C2×C22⋊(C127⋊C8)C2semidirect
21. #16861142C22semidirect
22. #16861142C22semidirect
23. #16861140C22semidirect
24. #16861140C22semidirect
25. #16861133C22semidirect
26. #16861133C22semidirect
27. #16861130C22semidirect
28. #16861130C22semidirect
144. C127#3091semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225502142
#225502092
#225502042
#225501972
#225501922
#225501872
#225499122
#225498722
C2×C22⋊(C127⋊C8)2
#525×C1272
#3091127