Finite Group Databasebeta

← C2×C3×#6312 #13049754

Subgroups — C2×C3×#6312 #13049754

Order 1296.

2760 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

normal subgroups only

This group has 2760 conjugacy classes of subgroups — too many to draw. Only its 470 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
1C23×C34C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#6312C214yesmaximal, direct product factor
4C2×C3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C2×#6312C311yesmaximal, characteristic, direct product factor
6C2×C3×#638—313nomaximal
7C22×C34—22no—
8C22×C34C2211yescharacteristic
9C22×C34C2212yescomplemented
10C2×C3×C3⋊(C3⋊D3)—22no—
11C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
12C3×C3⋊(C3⋊Dic3)C2212yescomplemented
13C23×C33—213no—
14C23×C33D3113yescomplemented
15C23×C33C2×C311yescharacteristic, complemented
16C22×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
17C22×C3×C3⋊D3—313no—
18#6312C2×C314yesdirect product factor
19C2×C3⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
20C3×#638—352no—
21C2×C3×C3⋊Dic3—313no—
22C2×C34—22no—
23C2×C34D412yescomplemented
24C2×C34C2311yescharacteristic
25C3×C3⋊(C3⋊D3)—22no—
26C2×#638—313no—
27C2×C3×#67—913no—
28C22×C33—252no—
29C22×C33—213no—
30C22×C33—226no—
31C22×C33—226no—
32C22×C33—22no—
33C22×C33C2×D3113yes—
34C22×C33C2×D3126yescomplemented
35C22×C33C22×C311yescharacteristic
36C22×C33C22×C312yescomplemented
37C2×C3⋊(C3⋊D3)—22no—
38C2×C3⋊(C3⋊D3)C22×C312yescomplemented
39C2×C3×C3⋊D3—326no—
40C2×C3×C3⋊D3—626no—
41C3⋊(C3⋊Dic3)C22×C312yescomplemented
42C3×C3⋊Dic3—326no—
43C34C2×D411yescharacteristic, complemented, Sylow
44C23×C32—252no—
45C23×C32C3×D3113yescomplemented
46C23×C32C3⋊D3113yescomplemented
47C22×C3⋊D3—313no—
48C22×C3×D3—913no—
49#638—352no—
50C2×C3⋊Dic3—313no—
51C3×#67—952no—
52C2×C3×Dic3—913no—
53C2×C33—252no—
54C2×C33—226no—
55C2×C33—22no—
56C2×C33—226no—
57C2×C33—213no—
58C2×C33#67126yescomplemented
59C2×C33C3×D412yescomplemented
60C2×C33C22×D3113yes—
61C2×C33C23×C311yescharacteristic
62C3⋊(C3⋊D3)—22no—
63C3×C3⋊D3—626no—
64C2×C3×D4—271no—
65C2×#67—913no—
66C22×C32—2208no—
67C22×C32—226no—
68C22×C32—226no—
69C22×C32—252no—
70C22×C32—2104no—
71C22×C32C2×C3×D3113yes—
72C22×C32C2×C3×D3126yescomplemented
73C22×C32C2×C3⋊D3113yes—
74C22×C32C2×C3⋊D3126yescomplemented
75C2×C3⋊D3—326no—
76C2×C3⋊D3—626no—
77C2×C3×D3—926no—
78C2×C3×D3—1826no—
79C3⋊Dic3—326no—
80C3×Dic3—926no—
81C33—213no—
82C33C2×#67113yescomplemented
83C33C2×C3×D411yescharacteristic, complemented
84C23×C3—213no—
85C23×C3—271no—
86C23×C3C3×C3⋊D3113yescomplemented
87C23×C3C3⋊(C3⋊D3)11yescharacteristic, complemented
88C22×D3—913no—
89C3×D4—274no—
90C2×C3×C4—271no—
91#67—952no—
92C2×Dic3—913no—
93C2×C32—2208no—
94C2×C32—226no—
95C2×C32—226no—
96C2×C32—2104no—
97C2×C32—252no—
98C2×C32C3×#67126yescomplemented
99C2×C32#638126yescomplemented
100C2×C32C22×C3×D3113yes—
101C2×C32C22×C3⋊D3113yes—
102C3⋊D3—626no—
103C3×D3—1826no—
104C2×D4—271noSylow
105C22×C3—252no—
106C22×C3—226no—
107C22×C3—22no—
108C22×C3—213no—
109C22×C3—226no—
110C22×C3—272no—
111C22×C3—542no—
112C22×C3C2×C3×C3⋊D3113yes—
113C22×C3C2×C3×C3⋊D3126yescomplemented
114C22×C3C2×C3⋊(C3⋊D3)11yescharacteristic
115C22×C3C2×C3⋊(C3⋊D3)12yescomplemented
116C2×D3—926no—
117C2×D3—1826no—
118C3×C4—272no—
119Dic3—926no—
120C32—252no—
121C32C2×C3×#67113yescomplemented
122C32C2×#638113yescomplemented
123C23—271no—
124C23C3×C3⋊(C3⋊D3)11yescharacteristic, complemented
125D4—274no—
126C2×C4—271no—
127C2×C3—252no—
128C2×C3—226no—
129C2×C3—22no—
130C2×C3—226no—
131C2×C3—213no—
132C2×C3—542no—
133C2×C3C3×#638126yescomplemented
134C2×C3#631212yesdirect product factor
135C2×C3C22×C3×C3⋊D3113yes—
136C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
137D3—1826no—
138C22—22no—
139C22—272no—
140C22—542no—
141C22C2×C3×C3⋊(C3⋊D3)11yescharacteristic
142C22C2×C3×C3⋊(C3⋊D3)12yescomplemented
143C4—272no—
144C3—213no—
145C3C2×C3×#638113yescomplemented
146C3C2×#631211yescharacteristic, direct product factor
147C2—22no—
148C2—542no—
149C2C3×#631212yesdirect product factor
150C2C22×C3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊(C3⋊D3)
commutator, cosocleC2×C33C23×C3
FrattiniC2C22×C3×C3⋊(C3⋊D3)
FittingC23×C34C2
socleC22×C34C22
hypercenterC23×C3C3⋊(C3⋊D3)
hypocenterC33C2×C3×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C34C2semidirect
2. C22×C3×C3⋊(C3⋊D3)C2semidirect
3. C3×#6312C2direct (N × H)
4. C2×C3×C3⋊(C3⋊Dic3)C2semidirect
5. C2×#6312C3direct (N × H)
9. C22×C34C22semidirect
11. C2×C3×C3⋊(C3⋊D3)C22semidirect
12. C3×C3⋊(C3⋊Dic3)C22semidirect
14. C23×C33D3semidirect
15. C23×C33C2×C3semidirect
16. C22×C3⋊(C3⋊D3)C2×C3semidirect
18. #6312C2×C3direct (N × H)
19. C2×C3⋊(C3⋊Dic3)C2×C3semidirect
23. C2×C34D4semidirect
34. C22×C33C2×D3semidirect
36. C22×C33C22×C3semidirect
38. C2×C3⋊(C3⋊D3)C22×C3semidirect
41. C3⋊(C3⋊Dic3)C22×C3semidirect
43. C34C2×D4semidirect
45. C23×C32C3×D3semidirect
46. C23×C32C3⋊D3semidirect
58. C2×C33#67semidirect
59. C2×C33C3×D4semidirect
72. C22×C32C2×C3×D3semidirect
74. C22×C32C2×C3⋊D3semidirect
82. C33C2×#67semidirect
83. C33C2×C3×D4semidirect
86. C23×C3C3×C3⋊D3semidirect
87. C23×C3C3⋊(C3⋊D3)semidirect
98. C2×C32C3×#67semidirect
99. C2×C32#638semidirect
113. C22×C3C2×C3×C3⋊D3semidirect
115. C22×C3C2×C3⋊(C3⋊D3)semidirect
121. C32C2×C3×#67semidirect
122. C32C2×#638semidirect
124. C23C3×C3⋊(C3⋊D3)semidirect
133. C2×C3C3×#638semidirect
134. C2×C3#6312direct (N × H)
142. C22C2×C3×C3⋊(C3⋊D3)semidirect
145. C3C2×C3×#638semidirect
146. C3C2×#6312direct (N × H)
149. C2C3×#6312direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C342
C22×C3×C3⋊(C3⋊D3)2
C3×#63122
C2×C3×C3⋊(C3⋊Dic3)2
C2×#63123
C2×C3×#6383