Finite Group Databasebeta

← #89031

Subgroups — #89031

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
1#6312C211yesmaximal, characteristic, complemented
2C3×#638C211yesmaximal, characteristic, complemented
3C2×C32×Dic3C211yesmaximal, characteristic, complemented
4#6293C211yesmaximal, characteristic, complemented
5#6292C211yesmaximal, characteristic, complemented
6#6289C211yesmaximal, characteristic, complemented
7Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
8#3828—31nomaximal
9#3804—34nomaximal
10C22×C33C2211yescharacteristic
11C2×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C3⋊D3C2211yescharacteristic
13C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C3⋊Dic3C2211yescharacteristic
15C32×Dic3C2211yescharacteristic
16C32×Dic3C2211yescharacteristic
17C2×C32×C4—31no—
18#638—31no—
19#638—34no—
20#638—64no—
21#638D311yescharacteristic, complemented
22C4⋊(C3⋊D3)—31no—
23C4×C3⋊D3—31no—
24C4×C3⋊D3—31no—
25#634—31no—
26C3×#67—34no—
27C2×C3×Dic3D314yescomplemented
28#627—34no—
29#626—34no—
30#624—34no—
31D3×Dic3—34no—
32C2×C33—21no—
33C2×C33C2311yescharacteristic
34C3⋊(C3⋊D3)—21no—
35C3×C3⋊D3—21no—
36#237—91no—
37#235—94no—
38C22×C32—24no—
39C22×C32C2×D314yes—
40C22×C32C2×D311yescharacteristic
41C2×C3⋊D3—31no—
42C2×C3⋊D3—34no—
43C2×C3⋊D3—64no—
44C2×C3⋊D3C2×D311yescharacteristic
45C2×C3×D3—34no—
46C32×C4—31no—
47C32×C4—31no—
48C3⋊Dic3—31no—
49C3⋊Dic3—34no—
50C3⋊Dic3—64no—
51C3⋊Dic3C2×D311yescharacteristic
52C3×Dic3—34no—
53C3×Dic3C2×D314yes—
54C3×Dic3C2×D314yes—
55C33#4111yescharacteristic, complemented, Sylow
56C3×D4—91no—
57C2×C3×C4—34no—
58#67—34no—
59#67—94no—
60#67—91no—
61#67—91no—
62#67—184no—
63C2×Dic3—91no—
64C2×Dic3C3⋊D311yescharacteristic, complemented
65D12—94no—
66C4×D3—94no—
67C4×D3—94no—
68C4×D3—91no—
69Dic6—94no—
70Dic6—91no—
71C2×C32—21no—
72C2×C32—24no—
73C2×C32—24no—
74C2×C32—44no—
75C2×C32C22×D314yes—
76C2×C32C22×D311yescharacteristic
77C3⋊D3—21no—
78C3⋊D3—61no—
79C3⋊D3—64no—
80C3⋊D3—124no—
81C3×D3—64no—
82#41—271noSylow
83C22×C3—24no—
84C22×C3—91no—
85C22×C3D3214yes—
86C22×C3C2×C3⋊D311yescharacteristic
87C2×D3—34no—
88C2×D3—94no—
89C2×D3—91no—
90C2×D3—184no—
91C3×C4—34no—
92C3×C4—34no—
93C3×C4—91no—
94Dic3—34no—
95Dic3—94no—
96Dic3—91no—
97Dic3—184no—
98Dic3C2×C3⋊D311yescharacteristic
99Dic3C2×C3⋊D311yescharacteristic
100C32—24no—
101C32#23514yescomplemented
102C32#23711yescharacteristic, complemented
103Q8—271no—
104D4—91no—
105D4—271no—
106D4—271no—
107C2×C4—31no—
108C2×C4—271no—
109C2×C4—271no—
110C2×C3—24no—
111C2×C3—21no—
112C2×C3—24no—
113C2×C3—44no—
114C2×C3—181no—
115C2×C3C2×D3214yes—
116C2×C3C22×C3⋊D311yescharacteristic
117D3—64no—
118D3—184no—
119D3—181no—
120D3—364no—
121C22—91no—
122C22—271no—
123C22D3×C3⋊D311yescharacteristic
124C4—31no—
125C4—31no—
126C4—91no—
127C4—271no—
128C3—24no—
129C3#380414yescomplemented
130C3#382811yescharacteristic, complemented
131C2—21no—
132C2—181no—
133C2—541no—
134C2C2×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊D3
commutator, socle, cosocleC2×C33C23
FittingC22×C33C22
hypercenterC22D3×C3⋊D3
hypocenterC33#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #6312C2semidirect
2. C3×#638C2semidirect
3. C2×C32×Dic3C2semidirect
4. #6293C2semidirect
5. #6292C2semidirect
6. #6289C2semidirect
7. Dic3×C3⋊D3C2semidirect
21. #638D3semidirect
27. C2×C3×Dic3D3semidirect
55. C33#41semidirect
64. C2×Dic3C3⋊D3semidirect
101. C32#235semidirect
102. C32#237semidirect
129. C3#3804semidirect
130. C3#3828semidirect

Maximal subgroups

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

subgroupindex
#63122
C3×#6382
C2×C32×Dic32
#62932
#62922
#62892
Dic3×C3⋊D32
#38283
#38043