Finite Group Databasebeta

← #30648869

Subgroups — #30648869

Order 16256.

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

normal subgroups only

This group has 450 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#22550412C211yesmaximal, characteristic, complemented
2C2×D4×Dic127C211yesmaximal, characteristic, complemented
3#22550204C211yesmaximal, characteristic, complemented
4#22550190C211yesmaximal, characteristic, complemented
5#22550174C212yesmaximal, complemented
6#22550093C212yesmaximal, complemented
7#22550010C211yesmaximal, characteristic
8#22549985C212yesmaximal, complemented
9#22549977C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#450×C127C211yesmaximal, characteristic, complemented
12C23×Dic127—21no—
13C2×#14523233C2212yes—
14#16861204—21no—
15#16861200C2211yescharacteristic, complemented
16#16861198C2211yescharacteristic, complemented
17#16861193—21no—
18#16861183C2211yescharacteristic
19Q8×Dic127C2211yescharacteristic, complemented
20D4×Dic127—21no—
21D4×Dic127—21no—
22D4×Dic127C2212yes—
23D4×Dic127C2212yes—
24#16861171C2211yescharacteristic, complemented
25(C22⋊D127)⋊C4—21no—
26#16861167—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861143C2212yescomplemented
31#16861140C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861133C2212yescomplemented
34#16861131C2212yescomplemented
35#16861130C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yescomplemented
38C22×D4×C127C2211yescharacteristic
39C2×SD16×C127C2212yes—
40C22⋊Q8×C127C2211yescharacteristic
41#102×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—21no—
45C22×Dic127—22no—
46C22×Dic127—22no—
47C22×Dic127—21no—
48C22×Dic127—44no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51C2×C4×D127—21no—
52#14523233—24no—
53C2×C127⋊C8C2312yes—
54C4⋊Dic127—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2312yes—
59Dic127⋊C4—21no—
60C22⋊Dic127—22no—
61C22⋊Dic127—22no—
62(C2×D127)⋊C4—21no—
63(C2×D127)⋊C4—21no—
64(C2×D127)⋊C4—21no—
65(C2×D127)⋊C4—21no—
66C4×Dic127—21no—
67C4×Dic127D411yescharacteristic
68C4×Dic127D411yescharacteristic
69C4×Dic127C2311yescharacteristic
70C4×Dic127C2311yescharacteristic
71C24×C127—21no—
72C2×Q8×C127C2311yescharacteristic
73C2×D4×C127—24no—
74C2×D4×C127—21no—
75C2×D4×C127C2312yes—
76C22×C4×C127C2311yescharacteristic
77SD16×C127—24no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127—21no—
80C4⋊C4×C127C2311yescharacteristic
81C22⋊C4×C127—21no—
82C22×D127—21no—
83D508—22no—
84C22⋊D127—42no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—22no—
88C2×Dic127—41no—
89C2×Dic127—44no—
90C2×Dic127—44no—
91C2×Dic127—42no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—42no—
97C127⋊C8—22no—
98C23×C127—24no—
99C23×C127—22no—
100C23×C127—42no—
101C23×C127—42no—
102C23×C127C2×D411yescharacteristic
103Q8×C127—22no—
104D4×C127—24no—
105D4×C127—22no—
106D4×C127#4114yes—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—24no—
116C22×C127—48no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127AGL(1,ℤ/8)12yes—
120C22×C127C22×D411yescharacteristic
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C4×C127—21no—
125C4×C127—41no—
126C4×C127—41no—
127C4×C127C2×#4112yes—
128Dic127—21no—
129Dic127—21no—
130Dic127—21no—
131Dic127—21no—
132Dic127—42no—
133Dic127—41no—
134C2×C127—22no—
135C2×C127—44no—
136C2×C127#52511yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127#57511yescharacteristic
139D127—81no—
140#3114—1271nomaximal, Sylow
141C127#311411yescharacteristic, complemented, Sylow
142#529—1271no—
143C2×C4×D4—1271no—
144#483—1271no—
145#480—1271no—
146#478—1272no—
147#463—1272no—
148#450D12711yescharacteristic, complemented
149#447—1271no—
150#442—1272no—
151#440—1272no—
152#432—1271no—
153C22×D4C2×D12711yescharacteristic
154C23×C4—2541no—
155C2×SD16C2×D12712yes—
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⋊C4—1271no—
172C2×C22⋊C4—2541no—
173C2×C42—1271no—
174#107—1272no—
175#106—1272no—
176C4⋊C8—1272no—
177#103—1272no—
178#102—1272no—
179#102—1272no—
180#102C2×D12712yes—
181C22⋊C8—1271no—
182C22⋊C8C2×D12711yescharacteristic
183#97—1272no—
184C24—21no—
185C2×Q8C22×D12711yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—1271no—
189C2×D4—2541no—
190C2×D4C22×D12712yes—
191C22×C4—1271no—
192C22×C4—1271no—
193C22×C4—5084no—
194C22×C4—2541no—
195C22×C4—2542no—
196C22×C4—2542no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199SD16—24no—
200D8—2544no—
201C2×C8—1272no—
202C2×C8C22×D12712yes—
203C4⋊C4—21no—
204C4⋊C4—1271no—
205C4⋊C4—1272no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4C22×D12711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—2541no—
213C22⋊C4—2541no—
214C22⋊C4—2541no—
215C22⋊C4—2542no—
216C22⋊C4—2542no—
217C22⋊C4—2541no—
218C42—1271no—
219C42—1271no—
220C42—1271no—
221C42—1271no—
222C42—2541no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—2541no—
228C23D4×D12711yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—5082no—
233D4—2542no—
234D4#1452324714yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—1271no—
239C2×C4—1271no—
240C2×C4—1271no—
241C2×C4—1271no—
242C2×C4—5082no—
243C2×C4—5082no—
244C2×C4—5084no—
245C2×C4—5084no—
246C2×C4—5081no—
247C2×C4—2541no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—22no—
253C8—2542no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—5081no—
261C22—5081no—
262C22—5081no—
263C22#1686123612yes—
264C22C2×D4×D12711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—5082no—
269C4—5081no—
270C4—2541no—
271C4—2541no—
272C4—2541no—
273C4—2541no—
274C4C2×#1452324712yes—
275C2—22no—
276C2—44no—
277C2—10161no—
278C2#2255038211yescharacteristic
279C2C2×#1686123611yescharacteristic
280C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radical#22550010C2
hypercenter#450D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550412C2semidirect
2. C2×D4×Dic127C2semidirect
3. #22550204C2semidirect
4. #22550190C2semidirect
5. #22550174C2semidirect
6. #22550093C2semidirect
8. #22549985C2semidirect
9. #22549977C2semidirect
10. #22549930C2semidirect
11. #450×C127C2semidirect
15. #16861200C22semidirect
16. #16861198C22semidirect
19. Q8×Dic127C22semidirect
24. #16861171C22semidirect
30. #16861143C22semidirect
31. #16861140C22semidirect
32. Dic127⋊C8C22semidirect
33. #16861133C22semidirect
34. #16861131C22semidirect
37. #16861119C22semidirect
141. C127#3114semidirect
148. #450D127semidirect

Maximal subgroups

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

subgroupindex
#225504122
C2×D4×Dic1272
#225502042
#225501902
#225501742
#225500932
#225500102
#225499852
#225499772
#225499302
#450×C1272
#3114127