Finite Group Databasebeta

← #27184270

Subgroups — #27184270

Order 12928.

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#20265979C212yesmaximal, complemented
2#20265614C214yesmaximal, complemented
3#20265612C212yesmaximal, complemented
4#20265610C212yesmaximal, complemented
5C2×D1616C211yesmaximal, characteristic, complemented
6D16×D101C212yesmaximal, complemented
7#20265593C211yesmaximal, characteristic, complemented
8#508×C101C211yesmaximal, characteristic, complemented
9#15692301—22no—
10#15692282—24no—
11#15692280—22no—
12#15692278C2214yescomplemented
13C2×D808C2212yescomplemented
14D8×D101C2214yescomplemented
15#15692261C2211yescharacteristic
16#15692198C2214yescomplemented
17D1616C2212yescomplemented
18D1616C2212yescomplemented
19#15692196C2214yescomplemented
20#15692195C2211yescharacteristic, complemented
21#15692194C2212yescomplemented
22C16×D101C2212yescomplemented
23#135×C101C2212yescomplemented
24Q32×C101C2211yescharacteristic, complemented
25SD32×C101C2212yescomplemented
26D16×C101C2211yescharacteristic, complemented
27C2×C16×C101C2211yescharacteristic, complemented
28#13108075—24no—
29#13108074—42no—
30#13108074D411yescharacteristic
31C2×D404—22no—
32C2×D404—42no—
33D4×D101—24no—
34D4×D101—44no—
35#13108061—24no—
36D808—22no—
37D808C2314yes—
38#13108059—24no—
39#13108058D411yescharacteristic
40#13108057D412yes—
41C8×D101C2312yes—
42#13108049C2312yes—
43#41×C101—22no—
44Q16×C101C2312yes—
45SD16×C101—22no—
46D8×C101C2312yes—
47C2×C8×C101C2311yescharacteristic
48C16×C101C2311yescharacteristic
49C16×C101C2311yescharacteristic
50C22×D101—42no—
51C22×D101—44no—
52Q8×C101—22no—
53D4×C101—22no—
54D4×C101—42no—
55C2×C4×C101—42no—
56C2×C4×C101C2×D411yescharacteristic
57C22⋊D101—44no—
58C22⋊D101D812yes—
59D404—24no—
60D404—44no—
61D404—42no—
62D404D811yescharacteristic
63C4×D101—44no—
64C4×D101C2×D412yes—
65Dic202D811yescharacteristic
66C8×C101C2×D411yescharacteristic
67C8×C101C2411yescharacteristic
68#11717579C2×D412yes—
69C22×C101—42no—
70C22×C101C2×D811yescharacteristic
71C2×D101—44no—
72C2×D101—164no—
73C2×D101—162no—
74C2×D101C2×D812yes—
75C4×C101—42no—
76C4×C101C2×D811yescharacteristic
77C4×C101C22×D411yescharacteristic
78Dic101C2×D812yes—
79C2×C101—21no—
80C2×C101—82no—
81C2×C101C22×D811yescharacteristic
82D101—22no—
83D101—84no—
84#3415—1011nomaximal, Sylow
85C101#341511yescharacteristic, complemented, Sylow
86#576—1012no—
87#509—1012no—
88#509—1014no—
89#508—1012no—
90#508D10111yescharacteristic, complemented
91C2×D16—1012no—
92C2×D16—1011no—
93#504—1011no—
94H(5,𝔽2)—2022no—
95AGL(1,ℤ/8)—2024no—
96AGL(1,ℤ/8)—2022no—
97#135—1014no—
98#135C2×D10112yescomplemented
99C2×D8—1014no—
100C2×D8—1012no—
101#131—1011no—
102Q32C2×D10111yescharacteristic, complemented
103SD32—1014no—
104SD32C2×D10112yescomplemented
105D16—1014no—
106D16—1012no—
107D16—1012no—
108D16C2×D10111yescharacteristic, complemented
109M32—1011no—
110M32—1012no—
111C2×C16—1012no—
112C2×C16C2×D10111yescharacteristic, complemented
113#41—22no—
114#41—1011no—
115#41—4042no—
116#41—2024no—
117C2×D4—4042no—
118C2×D4—4044no—
119C2×D4—2024no—
120C2×D4—2022no—
121Q16C22×D10112yes—
122SD16—22no—
123SD16—2024no—
124D8—1014no—
125D8—2024no—
126D8—2022no—
127D8C22×D10112yes—
128M16—1011no—
129M16—1012no—
130C2×C8—1012no—
131C2×C8C22×D10111yescharacteristic
132C16—1012no—
133C16C22×D10111yescharacteristic
134C16C22×D10111yescharacteristic
135C23—4044no—
136C23—4042no—
137Q8—22no—
138Q8—1011no—
139D4—22no—
140D4—42no—
141D4—1012no—
142D4—1011no—
143D4—4044no—
144D4—4044no—
145D4—4042no—
146D4—2024no—
147C2×C4—42no—
148C2×C4—1012no—
149C2×C4—4044no—
150C2×C4D4×D10111yescharacteristic
151C8—1012no—
152C8D4×D10111yescharacteristic
153C8C23×D10111yescharacteristic
154C22—42no—
155C22—16162no—
156C22—16164no—
157C22—1012no—
158C22—4044no—
159C22D8×D10111yescharacteristic
160C4—42no—
161C4—1012no—
162C4D8×D10111yescharacteristic
163C4C2×D4×D10111yescharacteristic
164C2—21no—
165C2—82no—
166C2—8084no—
167C2—2022no—
168C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#508×C101C2
socleC2×C101C22×D8
hypercenter#508D101
hypocenterC101#3415
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #20265979C2semidirect
2. #20265614C2semidirect
3. #20265612C2semidirect
4. #20265610C2semidirect
5. C2×D1616C2semidirect
6. D16×D101C2semidirect
7. #20265593C2semidirect
8. #508×C101C2semidirect
12. #15692278C22semidirect
13. C2×D808C22semidirect
14. D8×D101C22semidirect
16. #15692198C22semidirect
17. D1616C22semidirect
18. D1616C22semidirect
19. #15692196C22semidirect
20. #15692195C22semidirect
21. #15692194C22semidirect
22. C16×D101C22semidirect
23. #135×C101C22semidirect
24. Q32×C101C22semidirect
25. SD32×C101C22semidirect
26. D16×C101C22semidirect
27. C2×C16×C101C22semidirect
85. C101#3415semidirect
90. #508D101semidirect
98. #135C2×D101semidirect
102. Q32C2×D101semidirect
104. SD32C2×D101semidirect
108. D16C2×D101semidirect
112. C2×C16C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659792
#202656142
#202656122
#202656102
C2×D16162
D16×D1012
#202655932
#508×C1012
#3415101