Finite Group Databasebeta

← #30648670

Subgroups — #30648670

Order 16256.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861183C211yesmaximal, characteristic
2#22550200C214yesmaximal, complemented
3#22550194C212yesmaximal, complemented
4#22550189C212yesmaximal, complemented
5#22549919C212yesmaximal, complemented
6#22549880C212yesmaximal, complemented
7#22549840C211yesmaximal, characteristic, complemented
8C2×#117×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C2×#14523232C2212yes—
12#16861183—21no—
13#16861183—21no—
14#16861183C2212yes—
15#16861183C2212yes—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4—21no—
18#16861142C2214yescomplemented
19#16861140C2214yescomplemented
20#16861129C2214yescomplemented
21#16861129C2214yescomplemented
22C22⋊(C127⋊C8)C2212yescomplemented
23C22⋊(C127⋊C8)C2212yescomplemented
24C23×C4×C127C2211yescharacteristic
25#117×C127—21no—
26#117×C127—21no—
27#117×C127C2212yes—
28#117×C127C2212yescomplemented
29C2×C4⋊C4×C127C2211yescharacteristic, complemented
30C2×C4⋊C4×C127C2211yescharacteristic, complemented
31C2×C22⋊C4×C127—21no—
32C2×C42×C127—21no—
33C23×D127—21no—
34C2×D508—21no—
35C2×D508—24no—
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C22⋊D127—22no—
39C2×C22⋊D127—44no—
40C22×Dic127—21no—
41#14523232—24no—
42C2×C127⋊C8C2314yes—
43C4⋊Dic127—21no—
44C4⋊Dic127C2312yes—
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—22no—
47C24×C127D411yescharacteristic
48C22×C4×C127—24no—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127D411yescharacteristic
54C22×C4×C127C2311yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59C4⋊C4×C127C2312yes—
60C4⋊C4×C127C2312yes—
61C22⋊C4×C127—22no—
62C22⋊C4×C127—22no—
63C42×C127—22no—
64C42×C127—22no—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—42no—
68C22×D127—42no—
69D508—24no—
70D508—42no—
71C22⋊D127—44no—
72C22⋊D127—44no—
73C2×Dic127—22no—
74C2×Dic127—42no—
75C127⋊C8—24no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—41no—
79C23×C127C2×D411yescharacteristic
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—42no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4114yes—
95C2×C4×C127C2411yescharacteristic
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—44no—
104C22×C127#12312yes—
105C22×C127AGL(1,ℤ/8)12yes—
106C22×C127C22×D411yescharacteristic
107C2×D127—42no—
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—84no—
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—42no—
114C4×C127—42no—
115C4×C127#12312yes—
116C4×C127C2×#4112yes—
117Dic127—42no—
118C2×C127—22no—
119C2×C127—22no—
120C2×C127—41no—
121C2×C127C2×#12311yescharacteristic
122C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C127#57511yescharacteristic
124D127—82no—
125#3094—1271nomaximal, Sylow
126C127#309411yescharacteristic, complemented, Sylow
127C2×#121—1271no—
128C2×#117D12711yescharacteristic, complemented
129#482—1274no—
130#481—1272no—
131#480—1272no—
132#428—1272no—
133#418—1272no—
134#407—1271no—
135C22×D4—1271no—
136C22×D4—2541no—
137C23×C4C2×D12711yescharacteristic
138C2×M16—1272no—
139#121—1272no—
140#121—1272no—
141#121—2541no—
142#121—2541no—
143#117—21no—
144#117—21no—
145#117C2×D12712yes—
146#117C2×D12712yescomplemented
147C2×C4⋊C4—1271no—
148C2×C4⋊C4C2×D12711yescharacteristic, complemented
149C2×C4⋊C4C2×D12711yescharacteristic, complemented
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2541no—
152C2×C42—21no—
153#107—1274no—
154#106—1274no—
155#102—1274no—
156#102—1274no—
157C22⋊C8—1272no—
158C22⋊C8—1272no—
159C24—2541no—
160C24C22⋊D12711yescharacteristic
161C2×D4—1272no—
162C2×D4—5084no—
163C2×D4—2542no—
164C2×D4—2542no—
165C2×D4—2541no—
166C2×D4—2544no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2541no—
171C22×C4C22⋊D12711yescharacteristic
172C22×C4C22⋊D12711yescharacteristic
173C22×C4C22⋊D12711yescharacteristic
174C22×C4C22×D12711yescharacteristic
175C22×C4C22×D12711yescharacteristic
176C22×C4C22×D12711yescharacteristic
177M16—2544no—
178C2×C8—1274no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1272no—
182C4⋊C4—2541no—
183C4⋊C4C22×D12712yes—
184C4⋊C4C22×D12712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—5082no—
195C23—5082no—
196C23—5082no—
197C23—2542no—
198C23C2×C22⋊D12711yescharacteristic
199C23C2×C22⋊D12711yescharacteristic
200C23C2×C22⋊D12711yescharacteristic
201D4—5084no—
202D4—5084no—
203D4—5082no—
204D4—2544no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—5082no—
215C2×C4—2542no—
216C2×C4C2×C22⋊D12711yescharacteristic
217C2×C4C2×C22⋊D12711yescharacteristic
218C2×C4C2×C22⋊D12711yescharacteristic
219C2×C4#1452324814yes—
220C2×C4C23×D12711yescharacteristic
221C8—2544no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—10164no—
231C22—5082no—
232C22—5082no—
233C22—5082no—
234C22#1686119212yes—
235C22#1686123412yes—
236C22C22×C22⋊D12711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—5082no—
242C4#1686119212yes—
243C4C2×#1452324812yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—10162no—
248C2C2×#1686119211yescharacteristic
249C2C2×#1686123411yescharacteristic
250C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#117×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861183C2
hypercenterC2×#117D127
hypocenterC127#3094
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22550200C2semidirect
3. #22550194C2semidirect
4. #22550189C2semidirect
5. #22549919C2semidirect
6. #22549880C2semidirect
7. #22549840C2semidirect
8. C2×#117×C127C2semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. #16861129C22semidirect
21. #16861129C22semidirect
22. C22⋊(C127⋊C8)C22semidirect
23. C22⋊(C127⋊C8)C22semidirect
28. #117×C127C22semidirect
29. C2×C4⋊C4×C127C22semidirect
30. C2×C4⋊C4×C127C22semidirect
126. C127#3094semidirect
128. C2×#117D127semidirect
146. #117C2×D127semidirect
148. C2×C4⋊C4C2×D127semidirect
149. C2×C4⋊C4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
#225502002
#225501942
#225501892
#225499192
#225498802
#225498402
C2×#117×C1272
#3094127