Finite Group Databasebeta

← #30651868

Subgroups — #30651868

Order 16256.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#22550678C212yesmaximal, complemented
2#22550313C214yesmaximal, complemented
3#22550311C212yesmaximal, complemented
4#22550309C212yesmaximal, complemented
5C2×D2032C211yesmaximal, characteristic, complemented
6D16×D127C212yesmaximal, complemented
7#22550292C211yesmaximal, characteristic, complemented
8#508×C127C211yesmaximal, characteristic, complemented
9#16861255—22no—
10#16861236—24no—
11#16861234—22no—
12#16861232C2214yescomplemented
13C2×D1016C2212yescomplemented
14D8×D127C2214yescomplemented
15#16861215C2211yescharacteristic
16#16861152C2214yescomplemented
17D2032C2212yescomplemented
18D2032C2212yescomplemented
19#16861150C2214yescomplemented
20#16861149C2211yescharacteristic, complemented
21#16861148C2212yescomplemented
22C16×D127C2212yescomplemented
23#135×C127C2212yescomplemented
24Q32×C127C2211yescharacteristic, complemented
25SD32×C127C2212yescomplemented
26D16×C127C2211yescharacteristic, complemented
27C2×C16×C127C2211yescharacteristic, complemented
28#14523249—24no—
29#14523248—42no—
30#14523248D411yescharacteristic
31C2×D508—22no—
32C2×D508—42no—
33D4×D127—24no—
34D4×D127—44no—
35#14523235—24no—
36D1016—22no—
37D1016C2314yes—
38#14523233—24no—
39#14523232D411yescharacteristic
40#14523231D412yes—
41C8×D127C2312yes—
42C127⋊C16C2312yes—
43#41×C127—22no—
44Q16×C127C2312yes—
45SD16×C127—22no—
46D8×C127C2312yes—
47C2×C8×C127C2311yescharacteristic
48C16×C127C2311yescharacteristic
49C16×C127C2311yescharacteristic
50C22×D127—42no—
51C22×D127—44no—
52Dic254D811yescharacteristic
53D508—24no—
54D508—44no—
55D508—42no—
56D508D811yescharacteristic
57C22⋊D127—44no—
58C22⋊D127D812yes—
59C4×D127—44no—
60C4×D127C2×D412yes—
61C127⋊C8C2×D412yes—
62Q8×C127—22no—
63D4×C127—22no—
64D4×C127—42no—
65C2×C4×C127—42no—
66C2×C4×C127C2×D411yescharacteristic
67C8×C127C2×D411yescharacteristic
68C8×C127C2411yescharacteristic
69C22×C127—42no—
70C22×C127C2×D811yescharacteristic
71C2×D127—44no—
72C2×D127—164no—
73C2×D127—162no—
74C2×D127C2×D812yes—
75C4×C127—42no—
76C4×C127C2×D811yescharacteristic
77C4×C127C22×D411yescharacteristic
78Dic127C2×D812yes—
79C2×C127—21no—
80C2×C127—82no—
81C2×C127C22×D811yescharacteristic
82D127—22no—
83D127—84no—
84#3415—1271nomaximal, Sylow
85C127#341511yescharacteristic, complemented, Sylow
86#576—1272no—
87#509—1272no—
88#509—1274no—
89#508—1272no—
90#508D12711yescharacteristic, complemented
91C2×D16—1272no—
92C2×D16—1271no—
93#504—1271no—
94H(5,𝔽2)—2542no—
95AGL(1,ℤ/8)—2544no—
96AGL(1,ℤ/8)—2542no—
97#135—1274no—
98#135C2×D12712yescomplemented
99C2×D8—1274no—
100C2×D8—1272no—
101#131—1271no—
102Q32C2×D12711yescharacteristic, complemented
103SD32—1274no—
104SD32C2×D12712yescomplemented
105D16—1274no—
106D16—1272no—
107D16—1272no—
108D16C2×D12711yescharacteristic, complemented
109M32—1271no—
110M32—1272no—
111C2×C16—1272no—
112C2×C16C2×D12711yescharacteristic, complemented
113#41—22no—
114#41—1271no—
115#41—5082no—
116#41—2544no—
117C2×D4—5082no—
118C2×D4—5084no—
119C2×D4—2544no—
120C2×D4—2542no—
121Q16C22×D12712yes—
122SD16—22no—
123SD16—2544no—
124D8—1274no—
125D8—2544no—
126D8—2542no—
127D8C22×D12712yes—
128M16—1271no—
129M16—1272no—
130C2×C8—1272no—
131C2×C8C22×D12711yescharacteristic
132C16—1272no—
133C16C22×D12711yescharacteristic
134C16C22×D12711yescharacteristic
135C23—5084no—
136C23—5082no—
137Q8—22no—
138Q8—1271no—
139D4—22no—
140D4—42no—
141D4—1272no—
142D4—1271no—
143D4—5084no—
144D4—5084no—
145D4—5082no—
146D4—2544no—
147C2×C4—42no—
148C2×C4—1272no—
149C2×C4—5084no—
150C2×C4D4×D12711yescharacteristic
151C8—1272no—
152C8D4×D12711yescharacteristic
153C8C23×D12711yescharacteristic
154C22—42no—
155C22—1272no—
156C22—20322no—
157C22—20324no—
158C22—5084no—
159C22D8×D12711yescharacteristic
160C4—42no—
161C4—1272no—
162C4D8×D12711yescharacteristic
163C4C2×D4×D12711yescharacteristic
164C2—21no—
165C2—82no—
166C2—10164no—
167C2—2542no—
168C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#508×C127C2
socleC2×C127C22×D8
hypercenter#508D127
hypocenterC127#3415
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22550678C2semidirect
2. #22550313C2semidirect
3. #22550311C2semidirect
4. #22550309C2semidirect
5. C2×D2032C2semidirect
6. D16×D127C2semidirect
7. #22550292C2semidirect
8. #508×C127C2semidirect
12. #16861232C22semidirect
13. C2×D1016C22semidirect
14. D8×D127C22semidirect
16. #16861152C22semidirect
17. D2032C22semidirect
18. D2032C22semidirect
19. #16861150C22semidirect
20. #16861149C22semidirect
21. #16861148C22semidirect
22. C16×D127C22semidirect
23. #135×C127C22semidirect
24. Q32×C127C22semidirect
25. SD32×C127C22semidirect
26. D16×C127C22semidirect
27. C2×C16×C127C22semidirect
85. C127#3415semidirect
90. #508D127semidirect
98. #135C2×D127semidirect
102. Q32C2×D127semidirect
104. SD32C2×D127semidirect
108. D16C2×D127semidirect
112. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506782
#225503132
#225503112
#225503092
C2×D20322
D16×D1272
#225502922
#508×C1272
#3415127