Finite Group Databasebeta

← #11723957

Subgroups — #11723957

Order 864.

868 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 209 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#89080C211yesmaximal, characteristic, complemented
2#89078C211yesmaximal, characteristic, complemented
3C33×SD16C211yesmaximal, characteristic, complemented
4#88863C211yesmaximal, characteristic, complemented
5#88860C211yesmaximal, characteristic, complemented
6#88851C211yesmaximal, characteristic, complemented
7C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
8#64125—313nomaximal
9C33×Q8C2211yescharacteristic
10C33×D4C2211yescharacteristic
11#6312—21no—
12C2×C3⋊(C3⋊Dic3)—21no—
13C4⋊(C3⋊(C3⋊D3))—21no—
14C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C4×C3⋊(C3⋊D3)—21no—
16C4×C3⋊(C3⋊D3)C2211yescharacteristic
17#6308C2211yescharacteristic
18C33×C8C2211yescharacteristic
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#3832—313no—
21#3830—313no—
22C32×SD16D3113yescomplemented
23#3756—313no—
24#3753—313no—
25#3744—313no—
26C8×C3⋊D3—313no—
27C22×C33—21no—
28C2×C3⋊(C3⋊D3)—21no—
29C2×C3⋊(C3⋊D3)D411yescharacteristic
30C33×C4—21no—
31C33×C4C2311yescharacteristic
32C3⋊(C3⋊Dic3)—21no—
33C3⋊(C3⋊Dic3)D411yescharacteristic
34#916—913no—
35C32×Q8C2×D3113yes—
36C32×D4C2×D3113yes—
37#638—613no—
38C2×C3⋊Dic3—613no—
39C4⋊(C3⋊D3)—313no—
40C4⋊(C3⋊D3)—613no—
41C4×C3⋊D3—313no—
42C4×C3⋊D3—613no—
43#634—313no—
44C32×C8C2×D3113yes—
45C3⋊(C3⋊C8)—313no—
46C2×C33—41no—
47C2×C33C2×D411yescharacteristic
48C3⋊(C3⋊D3)—21no—
49C3⋊(C3⋊D3)—41no—
50#239—913no—
51#237—913no—
52C3×SD16C3⋊D3113yescomplemented
53#216—913no—
54#213—913no—
55#204—913no—
56D3×C8—913no—
57C22×C32—213no—
58C2×C3⋊D3—313no—
59C2×C3⋊D3—613no—
60C32×C4—213no—
61C32×C4C22×D3113yes—
62C3⋊Dic3—313no—
63C3⋊Dic3—613no—
64#135—271noSylow
65C33#13511yescharacteristic, complemented, Sylow
66C3×Q8C2×C3⋊D3113yes—
67C3×D4C2×C3⋊D3113yes—
68#67—1813no—
69C2×Dic3—1813no—
70D12—913no—
71D12—1813no—
72C4×D3—913no—
73C4×D3—1813no—
74Dic6—913no—
75C3×C8C2×C3⋊D3113yes—
76C3⋊C8—913no—
77C2×C32—413no—
78C2×C32D3×D4113yes—
79C3⋊D3—613no—
80C3⋊D3—1213no—
81#41—271no—
82#41—271no—
83Q16—271no—
84SD16—271no—
85SD16C3⋊(C3⋊D3)11yescharacteristic, complemented
86D8—271no—
87C2×C8—271no—
88C22×C3—213no—
89C2×D3—913no—
90C2×D3—1813no—
91C3×C4—213no—
92C3×C4C22×C3⋊D3113yes—
93Dic3—913no—
94Dic3—1813no—
95C32#916113yescomplemented
96Q8—271no—
97Q8C2×C3⋊(C3⋊D3)11yescharacteristic
98D4—271no—
99D4—541no—
100D4—541no—
101D4C2×C3⋊(C3⋊D3)11yescharacteristic
102C2×C4—271no—
103C2×C4—541no—
104C2×C4—541no—
105C8—271no—
106C8C2×C3⋊(C3⋊D3)11yescharacteristic
107C2×C3—413no—
108C2×C3D4×C3⋊D3113yes—
109D3—1813no—
110D3—3613no—
111C22—21no—
112C22—271no—
113C22—541no—
114C4—21no—
115C4—271no—
116C4—541no—
117C4C22×C3⋊(C3⋊D3)11yescharacteristic
118C3#64125113yescomplemented
119C2—41no—
120C2—541no—
121C2—1081no—
122C2D4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C23
FrattiniC4C22×C3⋊(C3⋊D3)
FittingC33×SD16C2
socleC2×C33C2×D4
hypercenterSD16C3⋊(C3⋊D3)
hypocenterC33#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ SD16

Complements

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

NH ≅ G/Ntype
1. #89080C2semidirect
2. #89078C2semidirect
3. C33×SD16C2semidirect
4. #88863C2semidirect
5. #88860C2semidirect
6. #88851C2semidirect
7. C8×C3⋊(C3⋊D3)C2semidirect
22. C32×SD16D3semidirect
52. C3×SD16C3⋊D3semidirect
65. C33#135semidirect
85. SD16C3⋊(C3⋊D3)semidirect
95. C32#916semidirect
118. C3#64125semidirect

Maximal subgroups

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

subgroupindex
#890802
#890782
C33×SD162
#888632
#888602
#888512
C8×C3⋊(C3⋊D3)2
#641253