Finite Group Databasebeta

← #89020

Subgroups — #89020

Order 432.

304 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×C4×D3C211yesmaximal, characteristic, complemented
4#6293C211yesmaximal, characteristic, complemented
5#6292C211yesmaximal, characteristic, complemented
6#6291C211yesmaximal, characteristic, complemented
7#6290C211yesmaximal, characteristic, complemented
8#3828—31nomaximal
9#3798—34nomaximal
10C2×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C3⋊D3C2211yescharacteristic
12C2×C32×D3C2211yescharacteristic
13C33×C4C2211yescharacteristic
14C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C3⋊Dic3C2211yescharacteristic
16C32×Dic3C2211yescharacteristic
17C2×C32×C4—31no—
18#638—31no—
19#638—31no—
20C4⋊(C3⋊D3)—31no—
21C4×C3⋊D3—31no—
22C4×C3⋊D3—34no—
23C4×C3⋊D3—64no—
24C4×C3⋊D3D311yescharacteristic, complemented
25#634—31no—
26C3×C4×D3—34no—
27C3×C4×D3D314yescomplemented
28#627—34no—
29#626—34no—
30#626—34no—
31#625—34no—
32C2×C33C2311yescharacteristic
33C3⋊(C3⋊D3)—21no—
34C3×C3⋊D3—21no—
35C32×D3—21no—
36#235—94no—
37#235—91no—
38C22×C32—31no—
39C2×C3⋊D3—31no—
40C2×C3⋊D3—34no—
41C2×C3⋊D3—64no—
42C2×C3⋊D3C2×D311yescharacteristic
43C2×C3×D3—34no—
44C2×C3×D3C2×D314yes—
45C32×C4—24no—
46C32×C4—31no—
47C32×C4C2×D314yes—
48C32×C4C2×D311yescharacteristic
49C3⋊Dic3—31no—
50C3⋊Dic3—34no—
51C3⋊Dic3—64no—
52C3⋊Dic3C2×D311yescharacteristic
53C3×Dic3—34no—
54C3×Dic3C2×D314yes—
55C33#4111yescharacteristic, complemented, Sylow
56C2×C3×C4—34no—
57C2×C3×C4—91no—
58#67—94no—
59#67—94no—
60#67—91no—
61#67—91no—
62D12—94no—
63D12—91no—
64C4×D3—34no—
65C4×D3—94no—
66C4×D3—91no—
67C4×D3—184no—
68C4×D3C3⋊D311yescharacteristic, complemented
69Dic6—94no—
70Dic6—91no—
71C2×C32—24no—
72C2×C32—61no—
73C2×C32C22×D314yes—
74C2×C32C22×D311yescharacteristic
75C3⋊D3—21no—
76C3⋊D3—68no—
77C3⋊D3—64no—
78C3⋊D3—61no—
79C3×D3—24no—
80C3×D3—64no—
81#41—271noSylow
82C22×C3—34no—
83C22×C3—91no—
84C2×D3—34no—
85C2×D3—94no—
86C2×D3—91no—
87C2×D3—184no—
88C2×D3C2×C3⋊D311yescharacteristic
89C3×C4—24no—
90C3×C4—34no—
91C3×C4—91no—
92C3×C4D3214yes—
93C3×C4C2×C3⋊D311yescharacteristic
94Dic3—34no—
95Dic3—94no—
96Dic3—91no—
97Dic3—184no—
98Dic3C2×C3⋊D311yescharacteristic
99C32—24no—
100C32#23514yescomplemented
101C32#23511yescharacteristic, complemented
102Q8—271no—
103D4—271no—
104D4—271no—
105D4—271no—
106C2×C4—31no—
107C2×C4—91no—
108C2×C4—271no—
109C2×C3—24no—
110C2×C3—64no—
111C2×C3—181no—
112C2×C3C2×D3214yes—
113C2×C3C22×C3⋊D311yescharacteristic
114D3—21no—
115D3—64no—
116D3—188no—
117D3—184no—
118D3—181no—
119C22—31no—
120C22—91no—
121C22—271no—
122C4—31no—
123C4—91no—
124C4—271no—
125C4D3×C3⋊D311yescharacteristic
126C3—24no—
127C3#379814yescomplemented
128C3#382811yescharacteristic, complemented
129C2—61no—
130C2—181no—
131C2—541no—
132C2C2×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC4D3×C3⋊D3
commutator, socle, cosocleC2×C33C23
FrattiniC2C2×D3×C3⋊D3
FittingC33×C4C22
hypocenterC33#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊(C3⋊D3)C2semidirect
2. C3×C4×C3⋊D3C2semidirect
3. C32×C4×D3C2semidirect
4. #6293C2semidirect
5. #6292C2semidirect
6. #6291C2semidirect
7. #6290C2semidirect
24. C4×C3⋊D3D3semidirect
27. C3×C4×D3D3semidirect
55. C33#41semidirect
68. C4×D3C3⋊D3semidirect
100. C32#235semidirect
101. C32#235semidirect
127. C3#3798semidirect
128. C3#3828semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(C3⋊D3)2
C3×C4×C3⋊D32
C32×C4×D32
#62932
#62922
#62912
#62902
#38283
#37983