Finite Group Databasebeta

← #22550386

Subgroups — #22550386

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×#14523247C211yesmaximal, characteristic, complemented
2C22×C4×D127C211yesmaximal, characteristic, complemented
3#16861191C212yesmaximal, complemented
4#16861189C211yesmaximal, characteristic, complemented
5#16861188C211yesmaximal, characteristic, complemented
6#16861180C211yesmaximal, characteristic, complemented
7#16861177C212yesmaximal, complemented
8D4×Dic127C212yesmaximal, complemented
9(C22⋊D127)⋊C4C212yesmaximal, complemented
10#16861167C211yesmaximal, characteristic, complemented
11#121×C127C211yesmaximal, characteristic, complemented
12C23×D127C2211yescharacteristic
13#14523247—24no—
14C2×Dic254C2211yescharacteristic, complemented
15C2×C22⋊D127C2212yes—
16C2×C22⋊D127C2212yes—
17C22×Dic127C2211yescharacteristic
18C22×Dic127C2212yescomplemented
19C2×C4×D127—24no—
20C2×C4×D127C2211yescharacteristic
21C2×C4×D127C2211yescharacteristic
22C2×C4×D127C2211yescharacteristic
23C2×C4×D127C2211yescharacteristic
24C4⋊Dic127C2211yescharacteristic, complemented
25C4⋊Dic127C2212yescomplemented
26Dic127⋊C4C2212yescomplemented
27C22⋊Dic127C2212yes—
28C22⋊Dic127C2212yescomplemented
29(C2×D127)⋊C4C2212yes—
30(C2×D127)⋊C4C2212yes—
31C4×Dic127C2212yescomplemented
32C2×D4×C127C2212yes—
33C2×D4×C127C2211yescharacteristic, complemented
34C22×C4×C127C2211yescharacteristic
35C4⋊C4×C127C2211yescharacteristic, complemented
36C22⋊C4×C127C2212yescomplemented
37C22×D127—22no—
38C22×D127—22no—
39C22×D127—22no—
40C22×D127C2311yescharacteristic
41C22×D127C2311yescharacteristic
42Dic254—22no—
43C22⋊D127—24no—
44C22⋊D127—24no—
45C2×Dic127—22no—
46C2×Dic127—24no—
47C2×Dic127C2312yes—
48C2×Dic127C2312yes—
49C2×Dic127C2311yescharacteristic
50C2×Dic127C2311yescharacteristic
51C4×D127—22no—
52C4×D127—22no—
53C4×D127—22no—
54C4×D127D414yescomplemented
55C23×C127C2312yes—
56C23×C127C2311yescharacteristic
57D4×C127—24no—
58D4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127C2311yescharacteristic
61C2×C4×C127C2312yes—
62C2×C4×C127C2311yescharacteristic
63C22×C127—21no—
64C22×C127—21no—
65C22×C127—22no—
66C22×C127—22no—
67C22×C127—22no—
68C22×C127#4112yescomplemented
69C22×C127C2411yescharacteristic
70C2×D127—28no—
71C2×D127—21no—
72C2×D127—21no—
73C2×D127—21no—
74C2×D127—21no—
75C2×D127C2×D412yes—
76C2×D127#4112yescomplemented
77C4×C127—21no—
78C4×C127—22no—
79C4×C127C2×D412yes—
80Dic127—21no—
81Dic127—22no—
82Dic127—22no—
83Dic127C2×D412yes—
84C2×C127—22no—
85C2×C127—42no—
86C2×C127C22×D411yescharacteristic
87C2×C127C2×#4111yescharacteristic
88C2×C127C2×#4111yescharacteristic
89D127—22no—
90D127—22no—
91C127#52511yescharacteristic, complemented, Sylow
92#525—1271nomaximal, Sylow
93C2×#41—1271no—
94C23×C4—1271no—
95#123—1272no—
96C22⋊Q8—1271no—
97C22⋊Q8—1271no—
98#121—1271no—
99#121D12711yescharacteristic, complemented
100#120—1272no—
101C4×D4—1272no—
102C4×D4—1272no—
103#117—1271no—
104C24—1271no—
105#41—2544no—
106C2×Q8—1271no—
107C2×D4—1272no—
108C2×D4—1272no—
109C2×D4C2×D12712yes—
110C2×D4C2×D12711yescharacteristic, complemented
111C22×C4—1271no—
112C22×C4—1271no—
113C22×C4—1272no—
114C22×C4—1271no—
115C22×C4—1271no—
116C22×C4—1271no—
117C22×C4—2544no—
118C22×C4C2×D12711yescharacteristic
119C4⋊C4—1272no—
120C4⋊C4—1271no—
121C4⋊C4—1272no—
122C4⋊C4C2×D12711yescharacteristic, complemented
123C22⋊C4—1272no—
124C22⋊C4—1272no—
125C22⋊C4—1272no—
126C22⋊C4—1272no—
127C22⋊C4C2×D12712yescomplemented
128C42—1272no—
129C23—1271no—
130C23—1271no—
131C23—2542no—
132C23—2542no—
133C23—2542no—
134C23C22×D12712yes—
135C23C22×D12711yescharacteristic
136Q8—2542no—
137D4—24no—
138D4—22no—
139D4—2544no—
140D4—2544no—
141C2×C4—22no—
142C2×C4—1271no—
143C2×C4—1274no—
144C2×C4—1272no—
145C2×C4—1271no—
146C2×C4—1272no—
147C2×C4—2542no—
148C2×C4—2542no—
149C2×C4—2542no—
150C2×C4—2544no—
151C2×C4—2542no—
152C2×C4C22×D12712yes—
153C2×C4C22×D12711yescharacteristic
154C2×C4C22×D12711yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—1272no—
161C22—1272no—
162C22—2548no—
163C22—2541no—
164C22—2541no—
165C22—2541no—
166C22—2541no—
167C22#1452324712yescomplemented
168C22C23×D12711yescharacteristic
169C4—21no—
170C4—22no—
171C4—1272no—
172C4—2541no—
173C4—2542no—
174C4—2542no—
175C4D4×D12712yes—
176C2—22no—
177C2—42no—
178C2—2542no—
179C2—2542no—
180C2C2×D4×D12711yescharacteristic
181C2#41×D12711yescharacteristic
182C2C2×#1452324711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#121×C127C2
hypercenter#121D127
hypocenterC127#525
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#14523247C2semidirect
2. C22×C4×D127C2semidirect
3. #16861191C2semidirect
4. #16861189C2semidirect
5. #16861188C2semidirect
6. #16861180C2semidirect
7. #16861177C2semidirect
8. D4×Dic127C2semidirect
9. (C22⋊D127)⋊C4C2semidirect
10. #16861167C2semidirect
11. #121×C127C2semidirect
14. C2×Dic254C22semidirect
18. C22×Dic127C22semidirect
24. C4⋊Dic127C22semidirect
25. C4⋊Dic127C22semidirect
26. Dic127⋊C4C22semidirect
28. C22⋊Dic127C22semidirect
31. C4×Dic127C22semidirect
33. C2×D4×C127C22semidirect
35. C4⋊C4×C127C22semidirect
36. C22⋊C4×C127C22semidirect
54. C4×D127D4semidirect
68. C22×C127#41semidirect
76. C2×D127#41semidirect
91. C127#525semidirect
99. #121D127semidirect
110. C2×D4C2×D127semidirect
122. C4⋊C4C2×D127semidirect
127. C22⋊C4C2×D127semidirect
167. C22#14523247semidirect