Finite Group Databasebeta

← #30682104

Subgroups — #30682104

Order 16832.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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#22554516C211yesmaximal, characteristic, complemented
2#22554496C211yesmaximal, characteristic, complemented
3#22554466C211yesmaximal, characteristic, complemented
4#22554397C211yesmaximal, characteristic, complemented
5#22554396C211yesmaximal, characteristic, complemented
6#22554386C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C263C211yesmaximal, characteristic, complemented
8#16862377—21no—
9#16862376—21no—
10#16862376C2211yescharacteristic
11C2×D1052—21no—
12C2×D1052C2211yescharacteristic
13C2×C22⋊D263—22no—
14D4×D263—22no—
15D4×D263—21no—
16#16862364—21no—
17#16862361—21no—
18#16862360C2211yescharacteristic, complemented
19C4×Dic263C2211yescharacteristic, complemented
20#41×C263C2211yescharacteristic
21C2×D4×C263C2211yescharacteristic, complemented
22SD16×C263—21no—
23D8×C263—21no—
24M16×C263C2211yescharacteristic, complemented
25C22×D263—22no—
26C22×D263D412yes—
27Dic526D411yescharacteristic, complemented
28D1052—22no—
29D1052—21no—
30D1052—21no—
31D1052D411yescharacteristic, complemented
32C22⋊D263—22no—
33C22⋊D263—21no—
34C22⋊D263—44no—
35C2×Dic263—21no—
36C4×D263—22no—
37C4×D263—21no—
38C263⋊C8—21no—
39C23×C263—21no—
40Q8×C263D411yescharacteristic, complemented
41D4×C263—21no—
42D4×C263—21no—
43D4×C263—21no—
44D4×C263D411yescharacteristic, complemented
45C2×C4×C263—21no—
46C2×C4×C263C2311yescharacteristic
47C8×C263—21no—
48C22×C263—21no—
49C22×C263—42no—
50C22×C263—41no—
51C22×C263C2×D411yescharacteristic
52C2×D263—22no—
53C2×D263—21no—
54C2×D263—42no—
55C2×D263—82no—
56C4×C263—21no—
57C4×C263C2×D411yescharacteristic
58C4×C263C2×D411yescharacteristic
59Dic263—22no—
60Dic263—21no—
61C2×C263—21no—
62C2×C263—41no—
63C2×C263—81no—
64C2×C263#12011yescharacteristic
65D263—42no—
66D263—41no—
67C263#45311yescharacteristic, complemented, Sylow
68#453—2631nomaximal, Sylow
69H(5,𝔽2)—2631no—
70AGL(1,ℤ/8)—2631no—
71AGL(1,ℤ/8)D26311yescharacteristic, complemented
72#127—2631no—
73#104—2631no—
74#104—2631no—
75#100—2631no—
76#41—2631no—
77#41—5261no—
78#41—5261no—
79#41C2×D26311yescharacteristic
80C2×D4—2631no—
81C2×D4—5261no—
82C2×D4—5262no—
83C2×D4—5261no—
84C2×D4—5262no—
85C2×D4C2×D26311yescharacteristic, complemented
86SD16—21no—
87SD16—5261no—
88D8—21no—
89D8—5261no—
90M16—2631no—
91M16C2×D26311yescharacteristic, complemented
92C42—2631no—
93C23—21no—
94C23—2632no—
95C23—5262no—
96Q8—2631no—
97Q8C22⋊D26311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—2631no—
102D4—5262no—
103D4—5262no—
104D4—5261no—
105D4—5261no—
106D4—5261no—
107D4—10524no—
108D4C22⋊D26311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—5262no—
111C2×C4—5261no—
112C2×C4—5261no—
113C2×C4C22×D26311yescharacteristic
114C8—21no—
115C8—5261no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—5261no—
120C22—5262no—
121C22—10522no—
122C22—21042no—
123C22D4×D26311yescharacteristic
124C4—21no—
125C4—5262no—
126C4—5261no—
127C4D4×D26311yescharacteristic
128C4C2×C22⋊D26311yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—10522no—
133C2—10521no—
134C2#2255443811yescharacteristic
subgroupG/H
centerC2#22554438
commutator, cosocleC2×C4×C263C23
FrattiniC2×C4C22×D263
FittingAGL(1,ℤ/8)×C263C2
socleC2×C263#120
hypercenterAGL(1,ℤ/8)D263
hypocenterC263#453
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22554516C2semidirect
2. #22554496C2semidirect
3. #22554466C2semidirect
4. #22554397C2semidirect
5. #22554396C2semidirect
6. #22554386C2semidirect
7. AGL(1,ℤ/8)×C263C2semidirect
18. #16862360C22semidirect
19. C4×Dic263C22semidirect
21. C2×D4×C263C22semidirect
24. M16×C263C22semidirect
27. Dic526D4semidirect
31. D1052D4semidirect
40. Q8×C263D4semidirect
44. D4×C263D4semidirect
67. C263#453semidirect
71. AGL(1,ℤ/8)D263semidirect
85. C2×D4C2×D263semidirect
91. M16C2×D263semidirect
97. Q8C22⋊D263semidirect
108. D4C22⋊D263semidirect

Maximal subgroups

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

subgroupindex
#225545162
#225544962
#225544662
#225543972
#225543962
#225543862
AGL(1,ℤ/8)×C2632
#453263