Finite Group Databasebeta

← #30651867

Subgroups — #30651867

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#22550677C212yesmaximal, complemented
2#22550312C212yesmaximal, complemented
3#22550310C214yesmaximal, complemented
4#22550308C211yesmaximal, characteristic, complemented
5#22550305C212yesmaximal, complemented
6D16×D127C212yesmaximal, complemented
7#22550292C211yesmaximal, characteristic, complemented
8C2×D16×C127C211yesmaximal, characteristic, complemented
9#16861254—22no—
10#16861235—22no—
11#16861233—24no—
12#16861231C2212yescomplemented
13#16861228C2214yescomplemented
14D8×D127C2214yescomplemented
15#16861215C2211yescharacteristic
16Dic1016C2211yescharacteristic, complemented
17#16861154C2212yescomplemented
18#16861153C2214yescomplemented
19D2032C2211yescharacteristic, complemented
20#16861150C2214yescomplemented
21#16861149C2211yescharacteristic, complemented
22#16861148C2212yescomplemented
23C16×D127C2212yescomplemented
24C2×D8×C127C2212yescomplemented
25D16×C127C2212yescomplemented
26D16×C127C2212yescomplemented
27C2×C16×C127C2211yescharacteristic, complemented
28#14523248—22no—
29#14523248D411yescharacteristic
30#14523247—24no—
31#14523247—42no—
32C2×C22⋊D127—44no—
33D4×D127—24no—
34D4×D127—42no—
35Dic508C2312yes—
36#14523237—22no—
37#14523236—24no—
38D1016C2312yes—
39#14523233—24no—
40#14523232D411yescharacteristic
41#14523231D412yes—
42C8×D127C2312yes—
43C127⋊C16C2312yes—
44C2×D4×C127—22no—
45D8×C127—22no—
46D8×C127C2314yes—
47C2×C8×C127C2311yescharacteristic
48C16×C127C2311yescharacteristic
49C16×C127C2311yescharacteristic
50C22×D127—44no—
51Dic254—22no—
52Dic254D811yescharacteristic
53D508—22no—
54D508D811yescharacteristic
55C22⋊D127—42no—
56C22⋊D127—44no—
57C22⋊D127—44no—
58C22⋊D127D812yes—
59C2×Dic127—44no—
60C4×D127—42no—
61C4×D127C2×D412yes—
62C127⋊C8C2×D412yes—
63C23×C127—42no—
64D4×C127—24no—
65D4×C127—42no—
66C2×C4×C127C2×D411yescharacteristic
67C8×C127C2×D411yescharacteristic
68C8×C127C2411yescharacteristic
69C22×C127—44no—
70C22×C127—162no—
71C22×C127C2×D811yescharacteristic
72C2×D127—42no—
73C2×D127—164no—
74C2×D127C2×D812yes—
75C4×C127C2×D811yescharacteristic
76C4×C127C22×D411yescharacteristic
77Dic127—42no—
78Dic127C2×D812yes—
79C2×C127—21no—
80C2×C127—84no—
81C2×C127C22×D811yescharacteristic
82D127—22no—
83D127—82no—
84#3415—1271nomaximal, Sylow
85C127#341511yescharacteristic, complemented, Sylow
86#576—1272no—
87#509—1272no—
88#509—1274no—
89#508—1272no—
90#508—1271no—
91C2×D16—1272no—
92C2×D16D12711yescharacteristic, complemented
93#504—1271no—
94H(5,𝔽2)—2542no—
95AGL(1,ℤ/8)—2544no—
96AGL(1,ℤ/8)—2542no—
97#135—1274no—
98#135—1272no—
99C2×D8—1274no—
100C2×D8C2×D12712yescomplemented
101#131—1271no—
102Q32—1271no—
103SD32—1274no—
104SD32—1272no—
105D16—1274no—
106D16—1271no—
107D16C2×D12712yescomplemented
108D16C2×D12712yescomplemented
109M32—1271no—
110M32—1272no—
111C2×C16—1272no—
112C2×C16C2×D12711yescharacteristic, complemented
113#41—1271no—
114#41—5082no—
115#41—2544no—
116#41—2542no—
117C2×D4—22no—
118C2×D4—5082no—
119C2×D4—5084no—
120C2×D4—2544no—
121Q16—1272no—
122SD16—2544no—
123SD16—2542no—
124D8—22no—
125D8—1272no—
126D8—2544no—
127D8C22×D12714yes—
128M16—1271no—
129M16—1272no—
130C2×C8—1272no—
131C2×C8C22×D12711yescharacteristic
132C16—1272no—
133C16C22×D12711yescharacteristic
134C16C22×D12711yescharacteristic
135C23—42no—
136C23—5084no—
137Q8—1271no—
138Q8—2542no—
139D4—24no—
140D4—42no—
141D4—1272no—
142D4—1271no—
143D4—5084no—
144D4—5084no—
145D4—5082no—
146D4—2542no—
147C2×C4—1272no—
148C2×C4—5084no—
149C2×C4—5082no—
150C2×C4D4×D12711yescharacteristic
151C8—1272no—
152C8D4×D12711yescharacteristic
153C8C23×D12711yescharacteristic
154C22—44no—
155C22—162no—
156C22—1272no—
157C22—20324no—
158C22—5082no—
159C22D8×D12711yescharacteristic
160C4—1272no—
161C4—5082no—
162C4D8×D12711yescharacteristic
163C4C2×D4×D12711yescharacteristic
164C2—21no—
165C2—84no—
166C2—10162no—
167C2—2542no—
168C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
FittingC2×D16×C127C2
socleC2×C127C22×D8
hypercenterC2×D16D127
hypocenterC127#3415
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #22550677C2semidirect
2. #22550312C2semidirect
3. #22550310C2semidirect
4. #22550308C2semidirect
5. #22550305C2semidirect
6. D16×D127C2semidirect
7. #22550292C2semidirect
8. C2×D16×C127C2semidirect
12. #16861231C22semidirect
13. #16861228C22semidirect
14. D8×D127C22semidirect
16. Dic1016C22semidirect
17. #16861154C22semidirect
18. #16861153C22semidirect
19. D2032C22semidirect
20. #16861150C22semidirect
21. #16861149C22semidirect
22. #16861148C22semidirect
23. C16×D127C22semidirect
24. C2×D8×C127C22semidirect
25. D16×C127C22semidirect
26. D16×C127C22semidirect
27. C2×C16×C127C22semidirect
85. C127#3415semidirect
92. C2×D16D127semidirect
100. C2×D8C2×D127semidirect
107. D16C2×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
#225506772
#225503122
#225503102
#225503082
#225503052
D16×D1272
#225502922
C2×D16×C1272
#3415127