Finite Group Databasebeta

← #22550385

Subgroups — #22550385

Order 8128.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14523248C211yesmaximal, characteristic, complemented
2C22×C4×D127C211yesmaximal, characteristic, complemented
3#16861194C211yesmaximal, characteristic, complemented
4#16861191C211yesmaximal, characteristic, complemented
5#16861187C211yesmaximal, characteristic, complemented
6#16861186C211yesmaximal, characteristic, complemented
7#16861182C211yesmaximal, characteristic, complemented
8#16861181C211yesmaximal, characteristic, complemented
9#16861179C211yesmaximal, characteristic, complemented
10#16861177C211yesmaximal, characteristic, complemented
11C4×D508C211yesmaximal, characteristic, complemented
12C4×C22⋊D127C211yesmaximal, characteristic, complemented
13C4×C22⋊D127C211yesmaximal, characteristic, complemented
14#16861166C211yesmaximal, characteristic, complemented
15C4×D4×C127C211yesmaximal, characteristic, complemented
16C23×D127C2211yescharacteristic
17#14523248—24no—
18C2×Dic254C2211yescharacteristic, complemented
19C2×D508C2211yescharacteristic
20C2×D508C2211yescharacteristic
21C2×C22⋊D127C2211yescharacteristic
22C2×C22⋊D127C2211yescharacteristic
23C2×C22⋊D127C2211yescharacteristic
24C2×C22⋊D127C2211yescharacteristic, complemented
25C22×Dic127C2211yescharacteristic
26C2×C4×D127—24no—
27C2×C4×D127C2211yescharacteristic
28C2×C4×D127C2211yescharacteristic
29C2×C4×D127C2211yescharacteristic
30C2×C4×D127C2211yescharacteristic
31C2×C4×D127C2211yescharacteristic, complemented
32C4⋊Dic127C2211yescharacteristic, complemented
33Dic127⋊C4C2211yescharacteristic, complemented
34Dic127⋊C4C2211yescharacteristic, complemented
35Dic127⋊C4C2211yescharacteristic, complemented
36Dic127⋊C4C2211yescharacteristic, complemented
37C22⋊Dic127C2211yescharacteristic
38C22⋊Dic127C2211yescharacteristic, complemented
39(C2×D127)⋊C4C2211yescharacteristic
40(C2×D127)⋊C4C2211yescharacteristic
41(C2×D127)⋊C4C2211yescharacteristic
42(C2×D127)⋊C4C2211yescharacteristic
43(C2×D127)⋊C4C2211yescharacteristic, complemented
44(C2×D127)⋊C4C2211yescharacteristic, complemented
45C4×Dic127C2211yescharacteristic, complemented
46C2×D4×C127C2211yescharacteristic
47C22×C4×C127C2211yescharacteristic
48C22×C4×C127C2211yescharacteristic, complemented
49C4⋊C4×C127C2211yescharacteristic, complemented
50C22⋊C4×C127C2211yescharacteristic
51C22⋊C4×C127C2211yescharacteristic, complemented
52C42×C127C2211yescharacteristic, complemented
53C22×D127—22no—
54C22×D127—22no—
55C22×D127—22no—
56C22×D127C2311yescharacteristic
57C22×D127C2311yescharacteristic
58C22×D127C2311yescharacteristic
59Dic254—22no—
60D508—22no—
61D508—22no—
62C22⋊D127—22no—
63C22⋊D127—22no—
64C22⋊D127—22no—
65C22⋊D127—22no—
66C2×Dic127—22no—
67C2×Dic127C2311yescharacteristic
68C2×Dic127C2311yescharacteristic
69C2×Dic127C2311yescharacteristic
70C2×Dic127C2311yescharacteristic
71C2×Dic127C2311yescharacteristic
72C4×D127—22no—
73C4×D127—22no—
74C4×D127—22no—
75C4×D127—22no—
76C4×D127D414yescomplemented
77C23×C127C2311yescharacteristic
78C23×C127C2311yescharacteristic
79D4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127C2311yescharacteristic
83C2×C4×C127C2311yescharacteristic
84C2×C4×C127C2311yescharacteristic
85C2×C4×C127C2311yescharacteristic
86C2×C4×C127C2311yescharacteristic
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127#4112yescomplemented
93C22×C127C2411yescharacteristic
94C2×D127—28no—
95C2×D127—21no—
96C2×D127—21no—
97C2×D127—21no—
98C2×D127—21no—
99C2×D127—21no—
100C2×D127—21no—
101C2×D127—21no—
102C2×D127C2×D412yes—
103C2×D127#4112yescomplemented
104C4×C127—21no—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—21no—
108C4×C127C2×D412yes—
109Dic127—21no—
110Dic127—21no—
111Dic127—21no—
112Dic127—21no—
113Dic127C2×D412yes—
114C2×C127—22no—
115C2×C127—41no—
116C2×C127C22×D411yescharacteristic
117C2×C127C2×#4111yescharacteristic
118C2×C127C2×#4111yescharacteristic
119D127—22no—
120D127—22no—
121D127—41no—
122C127#52511yescharacteristic, complemented, Sylow
123#525—1271nomaximal, Sylow
124C2×#41—1271no—
125C23×C4—1271no—
126#123—1271no—
127#123—1271no—
128C22⋊Q8—1271no—
129C22⋊Q8—1271no—
130#121—1271no—
131#121—1271no—
132#120—1271no—
133#120—1271no—
134C4×D4—1271no—
135C4×D4—1271no—
136C4×D4—1271no—
137C4×D4D12711yescharacteristic, complemented
138#117—1271no—
139C24—1271no—
140#41—2544no—
141C2×Q8—1271no—
142C2×D4—1271no—
143C2×D4—1271no—
144C2×D4—1271no—
145C2×D4—1271no—
146C2×D4—1271no—
147C2×D4—1271no—
148C2×D4C2×D12711yescharacteristic
149C22×C4—1271no—
150C22×C4—1271no—
151C22×C4—1271no—
152C22×C4—1271no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—2544no—
156C22×C4C2×D12711yescharacteristic
157C22×C4C2×D12711yescharacteristic, complemented
158C4⋊C4—1271no—
159C4⋊C4—1271no—
160C4⋊C4—1271no—
161C4⋊C4—1271no—
162C4⋊C4—1271no—
163C4⋊C4C2×D12711yescharacteristic, complemented
164C22⋊C4—1271no—
165C22⋊C4—1271no—
166C22⋊C4—1271no—
167C22⋊C4—1271no—
168C22⋊C4—1271no—
169C22⋊C4—1271no—
170C22⋊C4—1271no—
171C22⋊C4—1271no—
172C22⋊C4C2×D12711yescharacteristic
173C22⋊C4C2×D12711yescharacteristic, complemented
174C42—1271no—
175C42C2×D12711yescharacteristic, complemented
176C23—1271no—
177C23—1271no—
178C23—1271no—
179C23—2542no—
180C23—2542no—
181C23—2542no—
182C23C22×D12711yescharacteristic
183C23C22×D12711yescharacteristic
184Q8—2542no—
185D4—22no—
186D4—2542no—
187D4—2542no—
188D4—2542no—
189D4—2542no—
190D4—2542no—
191D4—2542no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—1271no—
195C2×C4—1274no—
196C2×C4—1271no—
197C2×C4—1271no—
198C2×C4—1271no—
199C2×C4—1271no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4—2542no—
203C2×C4—2542no—
204C2×C4—2542no—
205C2×C4C22×D12711yescharacteristic
206C2×C4C22×D12711yescharacteristic
207C2×C4C22×D12711yescharacteristic
208C2×C4C22×D12711yescharacteristic
209C2×C4C22×D12711yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—1272no—
216C22—1272no—
217C22—2548no—
218C22—2541no—
219C22—2541no—
220C22—2541no—
221C22—2541no—
222C22—2541no—
223C22—2541no—
224C22—2541no—
225C22#1452324812yescomplemented
226C22C23×D12711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—1272no—
232C4—2541no—
233C4—2541no—
234C4—2541no—
235C4—2541no—
236C4D4×D12712yes—
237C2—22no—
238C2—41no—
239C2—5081no—
240C2—2542no—
241C2—2542no—
242C2C2×D4×D12711yescharacteristic
243C2#41×D12711yescharacteristic
244C2C2×#1452324811yescharacteristic
subgroupG/H
centerC2×C4C22×D127
commutator, socleC22×C127C24
FrattiniC22C23×D127
FittingC4×D4×C127C2
hypercenterC4×D4D127
hypocenterC127#525
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14523248C2semidirect
2. C22×C4×D127C2semidirect
3. #16861194C2semidirect
4. #16861191C2semidirect
5. #16861187C2semidirect
6. #16861186C2semidirect
7. #16861182C2semidirect
8. #16861181C2semidirect
9. #16861179C2semidirect
10. #16861177C2semidirect
11. C4×D508C2semidirect
12. C4×C22⋊D127C2semidirect
13. C4×C22⋊D127C2semidirect
14. #16861166C2semidirect
15. C4×D4×C127C2semidirect
18. C2×Dic254C22semidirect
24. C2×C22⋊D127C22semidirect
31. C2×C4×D127C22semidirect
32. C4⋊Dic127C22semidirect
33. Dic127⋊C4C22semidirect
34. Dic127⋊C4C22semidirect
35. Dic127⋊C4C22semidirect
36. Dic127⋊C4C22semidirect
38. C22⋊Dic127C22semidirect
43. (C2×D127)⋊C4C22semidirect
44. (C2×D127)⋊C4C22semidirect
45. C4×Dic127C22semidirect
48. C22×C4×C127C22semidirect
49. C4⋊C4×C127C22semidirect
51. C22⋊C4×C127C22semidirect
52. C42×C127C22semidirect
76. C4×D127D4semidirect
92. C22×C127#41semidirect
103. C2×D127#41semidirect
122. C127#525semidirect
137. C4×D4D127semidirect
157. C22×C4C2×D127semidirect
163. C4⋊C4C2×D127semidirect
173. C22⋊C4C2×D127semidirect
175. C42C2×D127semidirect
225. C22#14523248semidirect