Finite Group Databasebeta

← #22550674

Subgroups — #22550674

Order 8128.

262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 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
2#16861240C212yesmaximal, complemented
3#16861239C212yesmaximal, complemented
4#16861235C212yesmaximal, complemented
5#16861234C212yesmaximal, complemented
6#16861229C214yesmaximal, complemented
7C2×#14523232C211yesmaximal, characteristic, complemented
8C2×#41×C127C211yesmaximal, characteristic, complemented
9#14523248—21no—
10#14523248—21no—
11#14523248C2212yes—
12#14523248C2212yescomplemented
13C2×Dic254C2211yescharacteristic, complemented
14C2×D508C2211yescharacteristic, complemented
15C2×C22⋊D127—21no—
16C2×C4×D127—21no—
17#14523238C2214yescomplemented
18#14523236C2214yescomplemented
19#14523235C2214yescomplemented
20#14523233C2214yescomplemented
21#14523232C2212yescomplemented
22#14523232C2212yescomplemented
23C2×C127⋊C8C2212yescomplemented
24#41×C127—21no—
25#41×C127—21no—
26#41×C127C2212yes—
27#41×C127C2212yescomplemented
28C2×Q8×C127C2211yescharacteristic, complemented
29C2×D4×C127—21no—
30C2×D4×C127C2211yescharacteristic, complemented
31C22×C4×C127—21no—
32C22×C4×C127C2211yescharacteristic
33C22×D127—21no—
34Dic254—21no—
35Dic254C2312yes—
36D508—21no—
37D508C2312yes—
38C22⋊D127—22no—
39C22⋊D127—22no—
40C2×Dic127—21no—
41C4×D127—22no—
42C4×D127—22no—
43C127⋊C8C2314yes—
44C23×C127—21no—
45C23×C127D411yescharacteristic
46Q8×C127—21no—
47Q8×C127C2312yes—
48D4×C127—22no—
49D4×C127—22no—
50D4×C127—21no—
51D4×C127C2312yes—
52C2×C4×C127—21no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127D411yescharacteristic
56C2×C4×C127D411yescharacteristic
57C2×C4×C127D411yescharacteristic
58C2×C4×C127C2311yescharacteristic
59C2×C4×C127C2311yescharacteristic
60C2×C4×C127C2311yescharacteristic
61C22×C127—22no—
62C22×C127—41no—
63C22×C127—42no—
64C22×C127C2×D411yescharacteristic
65C22×C127C2×D411yescharacteristic
66C22×C127C2×D411yescharacteristic
67C2×D127—22no—
68C2×D127—42no—
69C4×C127—22no—
70C4×C127C2×D411yescharacteristic
71C4×C127C2×D411yescharacteristic
72C4×C127C2×D411yescharacteristic
73C4×C127C2411yescharacteristic
74Dic127—22no—
75C2×C127—21no—
76C2×C127—21no—
77C2×C127—21no—
78C2×C127—42no—
79C2×C127C22×D411yescharacteristic
80D127—42no—
81C127#57511yescharacteristic, complemented, Sylow
82#575—1271nomaximal, Sylow
83C2×#41—1271no—
84C2×#41D12711yescharacteristic, complemented
85#137—1272no—
86#137—1272no—
87AGL(1,ℤ/8)—1272no—
88AGL(1,ℤ/8)—1272no—
89#135—1274no—
90C2×M16—1271no—
91#41—21no—
92#41—21no—
93#41—1272no—
94#41—1272no—
95#41—2541no—
96#41—2541no—
97#41C2×D12712yes—
98#41C2×D12712yescomplemented
99C2×Q8—1271no—
100C2×Q8C2×D12711yescharacteristic, complemented
101C2×D4—21no—
102C2×D4—1271no—
103C2×D4—2541no—
104C2×D4C2×D12711yescharacteristic, complemented
105C22×C4—21no—
106C22×C4—2541no—
107C22×C4C2×D12711yescharacteristic
108Q16—1274no—
109SD16—1274no—
110SD16—1274no—
111D8—1274no—
112M16—1272no—
113M16—1272no—
114C2×C8—1272no—
115C23—21no—
116C23—2541no—
117C23C22⋊D12711yescharacteristic
118Q8—21no—
119Q8—1272no—
120Q8—2541no—
121Q8C22×D12712yes—
122D4—22no—
123D4—22no—
124D4—21no—
125D4—1272no—
126D4—2542no—
127D4—2542no—
128D4—2541no—
129D4C22×D12712yes—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—2541no—
134C2×C4—2542no—
135C2×C4—2542no—
136C2×C4C22⋊D12711yescharacteristic
137C2×C4C22⋊D12711yescharacteristic
138C2×C4C22⋊D12711yescharacteristic
139C2×C4C22×D12711yescharacteristic
140C2×C4C22×D12711yescharacteristic
141C2×C4C22×D12711yescharacteristic
142C8—1274no—
143C22—22no—
144C22—41no—
145C22—42no—
146C22—5082no—
147C22—2542no—
148C22C2×C22⋊D12711yescharacteristic
149C22C2×C22⋊D12711yescharacteristic
150C22C2×C22⋊D12711yescharacteristic
151C4—22no—
152C4—2542no—
153C4C2×C22⋊D12711yescharacteristic
154C4C2×C22⋊D12711yescharacteristic
155C4C2×C22⋊D12711yescharacteristic
156C4C23×D12711yescharacteristic
157C2—21no—
158C2—21no—
159C2—21no—
160C2—42no—
161C2—5082no—
162C2C22×C22⋊D12711yescharacteristic
subgroupG/H
centerC4C2×C22⋊D127
commutatorC4×C127C24
FrattiniC4C23×D127
FittingC2×#41×C127C2
socleC2×C127C22×D4
hypercenterC2×#41D127
hypocenterC127#575
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#14523248C2semidirect
2. #16861240C2semidirect
3. #16861239C2semidirect
4. #16861235C2semidirect
5. #16861234C2semidirect
6. #16861229C2semidirect
7. C2×#14523232C2semidirect
8. C2×#41×C127C2semidirect
12. #14523248C22semidirect
13. C2×Dic254C22semidirect
14. C2×D508C22semidirect
17. #14523238C22semidirect
18. #14523236C22semidirect
19. #14523235C22semidirect
20. #14523233C22semidirect
21. #14523232C22semidirect
22. #14523232C22semidirect
23. C2×C127⋊C8C22semidirect
27. #41×C127C22semidirect
28. C2×Q8×C127C22semidirect
30. C2×D4×C127C22semidirect
81. C127#575semidirect
84. C2×#41D127semidirect
98. #41C2×D127semidirect
100. C2×Q8C2×D127semidirect
104. C2×D4C2×D127semidirect