Finite Group Databasebeta

← #30648671

Subgroups — #30648671

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
2C2×#16861168C211yesmaximal, characteristic, complemented
3#22550201C214yesmaximal, complemented
4#22550195C212yesmaximal, complemented
5#22550190C212yesmaximal, complemented
6#22549918C212yesmaximal, complemented
7#22549882C212yesmaximal, complemented
8#407×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11#16861183—21no—
12#16861183—21no—
13#16861183C2212yes—
14#16861183C2212yes—
15#16861168—21no—
16#16861168—21no—
17#16861168C2212yes—
18#16861168C2212yescomplemented
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic, complemented
21C2×C4⋊Dic127C2211yescharacteristic, complemented
22C2×C22⋊Dic127—21no—
23C2×(C2×D127)⋊C4—21no—
24C2×C4×Dic127—21no—
25#16861143C2214yescomplemented
26#16861141C2214yescomplemented
27#16861131C2214yescomplemented
28#16861131C2214yescomplemented
29C23×C4×C127C2211yescharacteristic
30C2×M16×C127C2212yes—
31C22⋊C8×C127C2212yescomplemented
32C22⋊C8×C127C2212yescomplemented
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—
41C22×Dic127—21no—
42C22×Dic127—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127—21no—
46C4⋊Dic127C2312yes—
47C4⋊Dic127C2312yes—
48C4⋊Dic127C2312yes—
49C22⋊Dic127—22no—
50C22⋊Dic127—22no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—22no—
54C4×Dic127—22no—
55C24×C127D411yescharacteristic
56C22×C4×C127—24no—
57C22×C4×C127D411yescharacteristic
58C22×C4×C127D411yescharacteristic
59C22×C4×C127D411yescharacteristic
60C22×C4×C127C2311yescharacteristic
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63M16×C127—24no—
64C2×C8×C127C2314yes—
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—22no—
75C2×Dic127—22no—
76C2×Dic127—42no—
77C2×Dic127—42no—
78C2×Dic127—42no—
79C23×C127—22no—
80C23×C127—22no—
81C23×C127—41no—
82C23×C127C2×D411yescharacteristic
83C23×C127C2×D411yescharacteristic
84C23×C127C2×D411yescharacteristic
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4114yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—24no—
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#12312yes—
114C4×C127C2×#4112yes—
115Dic127—42no—
116Dic127—42no—
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×#117—1271no—
129#482—1274no—
130#481—1272no—
131#480—1272no—
132#428—1272no—
133#418—1272no—
134#407D12711yescharacteristic, complemented
135C22×D4—1271no—
136C22×D4—2541no—
137C23×C4C2×D12711yescharacteristic
138C2×M16C2×D12712yes—
139#121—1272no—
140#121—1272no—
141#121—2541no—
142#121—2541no—
143#117—1272no—
144#117—1272no—
145#117—2541no—
146#117—2541no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4—1271no—
150C2×C22⋊C4—2541no—
151C2×C22⋊C4—2541no—
152C2×C42—2541no—
153#107—1274no—
154#106—1274no—
155#102—1274no—
156#102—1274no—
157C22⋊C8C2×D12712yescomplemented
158C22⋊C8C2×D12712yescomplemented
159C24—2541no—
160C24D50811yescharacteristic
161C2×D4—1272no—
162C2×D4—5084no—
163C2×D4—2542no—
164C2×D4—2542no—
165C2×D4—2541no—
166C2×D4—2544no—
167C22×C4—24no—
168C22×C4—2541no—
169C22×C4—2541no—
170C22×C4—2541no—
171C22×C4D50811yescharacteristic
172C22×C4D50811yescharacteristic
173C22×C4D50811yescharacteristic
174C22×C4C22×D12711yescharacteristic
175C22×C4C22×D12711yescharacteristic
176C22×C4C22×D12711yescharacteristic
177M16—24no—
178C2×C8C22×D12714yes—
179C4⋊C4—1272no—
180C4⋊C4—1272no—
181C4⋊C4—1272no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C22⋊C4—2542no—
186C22⋊C4—2542no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C42—2542no—
190C42—2542no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—5082no—
195C23—5082no—
196C23—5082no—
197C23—2542no—
198C23C2×D50811yescharacteristic
199C23C2×D50811yescharacteristic
200C23C2×D50811yescharacteristic
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—5082no—
211C2×C4—5082no—
212C2×C4—5082no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4C2×D50811yescharacteristic
217C2×C4C2×D50811yescharacteristic
218C2×C4C2×D50811yescharacteristic
219C2×C4#1452324714yes—
220C2×C4C23×D12711yescharacteristic
221C8—24no—
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#1686119312yes—
235C22#1686123712yes—
236C22C22×D50811yescharacteristic
237C4—21no—
238C4—21no—
239C4—5082no—
240C4—5082no—
241C4—5082no—
242C4#1686119312yes—
243C4C2×#1452324712yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—10162no—
248C2C2×#1686119311yescharacteristic
249C2C2×#1686123711yescharacteristic
250C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861183C2
hypercenter#407D127
hypocenterC127#3094
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16861168C2semidirect
3. #22550201C2semidirect
4. #22550195C2semidirect
5. #22550190C2semidirect
6. #22549918C2semidirect
7. #22549882C2semidirect
8. #407×C127C2semidirect
18. #16861168C22semidirect
20. C2×C4⋊Dic127C22semidirect
21. C2×C4⋊Dic127C22semidirect
25. #16861143C22semidirect
26. #16861141C22semidirect
27. #16861131C22semidirect
28. #16861131C22semidirect
31. C22⋊C8×C127C22semidirect
32. C22⋊C8×C127C22semidirect
126. C127#3094semidirect
134. #407D127semidirect
157. C22⋊C8C2×D127semidirect
158. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×#168611682
#225502012
#225501952
#225501902
#225499182
#225498822
#407×C1272
#3094127