Finite Group Databasebeta

← #30648824

Subgroups — #30648824

Order 16256.

444 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 123 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×D4×Dic127C211yesmaximal, characteristic
3#22550212C211yesmaximal, characteristic, complemented
4#22550209C211yesmaximal, characteristic, complemented
5#22550154C212yesmaximal, complemented
6#22550079C212yesmaximal, complemented
7#22550027C211yesmaximal, characteristic
8#22549977C212yesmaximal, complemented
9#22549973C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#450×C127C211yesmaximal, characteristic
12C23×Dic127—21no—
13C2×#14523236C2212yes—
14#16861208C2211yescharacteristic, complemented
15#16861207C2211yescharacteristic, complemented
16#16861199—21no—
17#16861188C2211yescharacteristic
18#16861186—21no—
19Q8×Dic127C2211yescharacteristic, complemented
20#16861174—21no—
21#16861174C2211yescharacteristic, complemented
22D4×Dic127—21no—
23D4×Dic127—21no—
24D4×Dic127C2212yes—
25D4×Dic127C2212yes—
26#16861165—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861134C2212yescomplemented
34#16861133C2212yescomplemented
35#16861130C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yescomplemented
38C22×D4×C127C2211yescharacteristic
39C2×SD16×C127C2212yes—
40C22⋊Q8×C127C2211yescharacteristic
41#102×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45C22×Dic127—22no—
46C22×Dic127—22no—
47C22×Dic127—21no—
48C22×Dic127—44no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51#14523236—24no—
52C2×C127⋊C8C2312yes—
53C4⋊Dic127—21no—
54C4⋊Dic127—21no—
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2312yes—
57Dic127⋊C4—21no—
58Dic127⋊C4—21no—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62C22⋊Dic127—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127—21no—
65C4×Dic127—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—
82Dic254—22no—
83Dic254—42no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—22no—
89C2×Dic127—44no—
90C2×Dic127—42no—
91C2×Dic127—44no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C127⋊C8—22no—
97C23×C127—24no—
98C23×C127—22no—
99C23×C127—42no—
100C23×C127—42no—
101C23×C127C2×D411yescharacteristic
102Q8×C127—22no—
103D4×C127—24no—
104D4×C127—22no—
105D4×C127#4114yes—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127—21no—
113C22×C127—21no—
114C22×C127—24no—
115C22×C127—48no—
116C22×C127—42no—
117C22×C127—42no—
118C22×C127#13712yes—
119C22×C127C22×D411yescharacteristic
120C4×C127—21no—
121C4×C127—41no—
122C4×C127—41no—
123C4×C127C2×#4112yes—
124Dic127—21no—
125Dic127—21no—
126Dic127—21no—
127Dic127—41no—
128Dic127—42no—
129Dic127—41no—
130Dic127AGL(1,ℤ/8)12yes—
131C2×C127—22no—
132C2×C127—44no—
133C2×C127#52511yescharacteristic
134C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C127C2×#13711yescharacteristic
136#3110—1271nomaximal, Sylow
137C127#311011yescharacteristic, complemented, Sylow
138#533—1271no—
139C2×C4×D4—1271no—
140#484—1271no—
141#484—1271no—
142#474—1272no—
143#461—1272no—
144#450—1271no—
145#450D12711yescharacteristic
146#440—1272no—
147#440—1272no—
148#432—1271no—
149C22×D4C2×D12711yescharacteristic
150C23×C4—2541no—
151C2×SD16—1272no—
152C2×SD16C2×D12712yes—
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⋊C4—1271no—
168C2×C22⋊C4—2541no—
169C2×C42—1271no—
170#107—1272no—
171#107—1272no—
172C4⋊C8—1272no—
173#103—1272no—
174#103—1272no—
175#102—1272no—
176#102C2×D12712yes—
177C22⋊C8—1271no—
178C22⋊C8C2×D12711yescharacteristic
179#97—1272no—
180C24—21no—
181C2×Q8—1271no—
182C2×Q8—2541no—
183C2×Q8C22×D12711yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D12712yes—
187C22×C4—1271no—
188C22×C4—1271no—
189C22×C4—5084no—
190C22×C4—2542no—
191C22×C4—2542no—
192C22×C4—2541no—
193C22×C4C22×D12711yescharacteristic
194SD16—24no—
195SD16—2544no—
196C2×C8—1272no—
197C2×C8C22×D12712yes—
198C4⋊C4—21no—
199C4⋊C4—1271no—
200C4⋊C4—1272no—
201C4⋊C4—2541no—
202C4⋊C4—2541no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4C22×D12711yescharacteristic
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—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D12711yescharacteristic
224Q8—22no—
225Q8—5082no—
226Q8—2542no—
227D4—24no—
228D4—22no—
229D4#1452324714yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—1271no—
234C2×C4—1271no—
235C2×C4—1271no—
236C2×C4—1271no—
237C2×C4—5084no—
238C2×C4—5082no—
239C2×C4—5084no—
240C2×C4—2542no—
241C2×C4—2541no—
242C2×C4—2541no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4D4×D12711yescharacteristic
246C2×C4C23×D12711yescharacteristic
247C8—22no—
248C8—2542no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22#1686123812yes—
256C22C2×D4×D12711yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—1272no—
261C4—5081no—
262C4—5082no—
263C4—5081no—
264C4—2541no—
265C4—2541no—
266C4—2541no—
267C4C2×#1452324712yes—
268C2—22no—
269C2—44no—
270C2#2255038211yescharacteristic
271C2AGL(1,ℤ/8)×D12711yescharacteristic
272C2C2×#1686123811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenter#450D127
hypocenterC127#3110
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. #22550428C2semidirect
3. #22550212C2semidirect
4. #22550209C2semidirect
5. #22550154C2semidirect
6. #22550079C2semidirect
8. #22549977C2semidirect
9. #22549973C2semidirect
10. #22549930C2semidirect
14. #16861208C22semidirect
15. #16861207C22semidirect
19. Q8×Dic127C22semidirect
21. #16861174C22semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. Dic127⋊C8C22semidirect
33. #16861134C22semidirect
34. #16861133C22semidirect
37. #16861119C22semidirect
137. C127#3110semidirect

Maximal subgroups

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

subgroupindex
#225504282
C2×D4×Dic1272
#225502122
#225502092
#225501542
#225500792
#225500272
#225499772
#225499732
#225499302
#450×C1272
#3110127