Finite Group Databasebeta

← C2×#10620147 #13049680

Subgroups — C2×#10620147 #13049680

Order 1296.

2760 conjugacy classes of subgroups; only the 378 normal subgroups are shown.

normal subgroups only

This group has 2760 conjugacy classes of subgroups — too many to draw. Only its 378 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
1C22×C3⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×C33×Dic3C211yesmaximal, characteristic, complemented
4#10620147C214yesmaximal, direct product factor
5C2×C4⋊(C3⋊(C3⋊D3))—31nomaximal
6C2×#6292—313nomaximal
7C22×C34C2211yescharacteristic
8C2×C3⋊(C3⋊(C3⋊D3))—22no—
9C2×C3⋊(C3⋊(C3⋊D3))C2212yescomplemented
10C2×C3×C3⋊(C3⋊D3)—22no—
11C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
12C33×Dic3C2212yescomplemented
13C22×C3⋊(C3⋊D3)—31no—
14C22×C3⋊(C3⋊D3)—313no—
15C22×C3⋊(C3⋊D3)—613no—
16C22×C3⋊(C3⋊D3)D311yescharacteristic, complemented
17C22×C3×C3⋊D3—313no—
18C2×C33×C4—31no—
19C4⋊(C3⋊(C3⋊D3))—34no—
20C2×C32×Dic3D3113yescomplemented
21#6292—352no—
22C2×C34D412yescomplemented
23C2×C34C2311yescharacteristic
24C3⋊(C3⋊(C3⋊D3))—22no—
25C3×C3⋊(C3⋊D3)—22no—
26C2×C4⋊(C3⋊D3)—913no—
27C2×#626—913no—
28C22×C33—213no—
29C22×C33C2×D3113yes—
30C22×C33C2×D311yescharacteristic
31C2×C3⋊(C3⋊D3)—22no—
32C2×C3⋊(C3⋊D3)—32no—
33C2×C3⋊(C3⋊D3)—326no—
34C2×C3⋊(C3⋊D3)—626no—
35C2×C3⋊(C3⋊D3)—652no—
36C2×C3⋊(C3⋊D3)—626no—
37C2×C3⋊(C3⋊D3)—62no—
38C2×C3⋊(C3⋊D3)C2×D312yescomplemented
39C2×C3×C3⋊D3—326no—
40C2×C3×C3⋊D3—626no—
41C33×C4—32no—
42C32×Dic3C2×D3126yescomplemented
43C34C2×D411yescharacteristic, complemented, Sylow
44C22×C3⋊D3—313no—
45C22×C3⋊D3—913no—
46C22×C3⋊D3—913no—
47C22×C3⋊D3—1852no—
48C22×C3×D3—913no—
49C2×C32×C4—313no—
50C4⋊(C3⋊D3)—952no—
51C2×C3×Dic3C3⋊D3113yescomplemented
52#626—952no—
53C2×C33—226no—
54C2×C33—213no—
55C2×C33D12126yescomplemented
56C2×C33#6712yescomplemented
57C2×C33C22×D3113yes—
58C2×C33C22×D311yescharacteristic
59C3⋊(C3⋊D3)—22no—
60C3⋊(C3⋊D3)—652no—
61C3⋊(C3⋊D3)—626no—
62C3⋊(C3⋊D3)—62no—
63C3×C3⋊D3—626no—
64C2×#67—271no—
65C2×D12—2713no—
66C22×C32—252no—
67C22×C32D32113yes—
68C22×C32C2×C3⋊D3113yes—
69C2×C3⋊D3—326no—
70C2×C3⋊D3—626no—
71C2×C3⋊D3—926no—
72C2×C3⋊D3—926no—
73C2×C3⋊D3—18104no—
74C2×C3⋊D3—18208no—
75C2×C3⋊D3—1826no—
76C2×C3⋊D3—1826no—
77C2×C3×D3—926no—
78C2×C3×D3—1826no—
79C32×C4—326no—
80C3×Dic3C2×C3⋊D3126yescomplemented
81C33—213no—
82C33C2×D12113yescomplemented
83C33C2×#6711yescharacteristic, complemented
84C23×C3—271no—
85C22×D3—913no—
86C22×D3—2713no—
87C22×D3—271no—
88C22×D3—5413no—
89C2×C3×C4—313no—
90#67—274no—
91C2×Dic3C3⋊(C3⋊D3)11yescharacteristic, complemented
92D12—2752no—
93C2×C32—2104no—
94C2×C32—252no—
95C2×C32#626126yescomplemented
96C2×C32C4⋊(C3⋊D3)126yescomplemented
97C2×C32C2×D32113yes—
98C2×C32C22×C3⋊D3113yes—
99C3⋊D3—626no—
100C3⋊D3—18208no—
101C3⋊D3—1826no—
102C3⋊D3—1826no—
103C3×D3—1826no—
104C2×D4—811noSylow
105C22×C3—213no—
106C22×C3—272no—
107C22×C3—542no—
108C22×C3D3×C3⋊D3113yes—
109C22×C3C2×C3⋊(C3⋊D3)11yescharacteristic
110C2×D3—926no—
111C2×D3—1826no—
112C2×D3—2726no—
113C2×D3—272no—
114C2×D3—5426no—
115C2×D3—5452no—
116C2×D3—5426no—
117C2×D3—542no—
118C3×C4—326no—
119Dic3C2×C3⋊(C3⋊D3)12yescomplemented
120C32—252no—
121C32C2×#626113yescomplemented
122C32C2×C4⋊(C3⋊D3)113yescomplemented
123C23—271no—
124C23—811no—
125D4—814no—
126C2×C4—31no—
127C2×C3—213no—
128C2×C3—226no—
129C2×C3—542no—
130C2×C3#6292126yescomplemented
131C2×C3C4⋊(C3⋊(C3⋊D3))12yescomplemented
132C2×C3C2×D3×C3⋊D3113yes—
133C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
134D3—1826no—
135D3—5452no—
136D3—5426no—
137D3—542no—
138C22—272no—
139C22—542no—
140C22—812no—
141C22—1622no—
142C22D3×C3⋊(C3⋊D3)11yescharacteristic
143C4—32no—
144C3—213no—
145C3C2×#6292113yescomplemented
146C3C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
147C2—542no—
148C2—1622no—
149C2#1062014712yesdirect product factor
150C2C2×D3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊(C3⋊D3)
commutator, cosocleC2×C34C23
FrattiniC2C2×D3×C3⋊(C3⋊D3)
Fitting, socleC22×C34C22
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(C3⋊(C3⋊D3))C2semidirect
2. C22×C3×C3⋊(C3⋊D3)C2semidirect
3. C2×C33×Dic3C2semidirect
4. #10620147C2direct (N × H)
9. C2×C3⋊(C3⋊(C3⋊D3))C22semidirect
11. C2×C3×C3⋊(C3⋊D3)C22semidirect
12. C33×Dic3C22semidirect
16. C22×C3⋊(C3⋊D3)D3semidirect
20. C2×C32×Dic3D3semidirect
22. C2×C34D4semidirect
38. C2×C3⋊(C3⋊D3)C2×D3semidirect
42. C32×Dic3C2×D3semidirect
43. C34C2×D4semidirect
51. C2×C3×Dic3C3⋊D3semidirect
55. C2×C33D12semidirect
56. C2×C33#67semidirect
80. C3×Dic3C2×C3⋊D3semidirect
82. C33C2×D12semidirect
83. C33C2×#67semidirect
91. C2×Dic3C3⋊(C3⋊D3)semidirect
95. C2×C32#626semidirect
96. C2×C32C4⋊(C3⋊D3)semidirect
119. Dic3C2×C3⋊(C3⋊D3)semidirect
121. C32C2×#626semidirect
122. C32C2×C4⋊(C3⋊D3)semidirect
130. C2×C3#6292semidirect
131. C2×C3C4⋊(C3⋊(C3⋊D3))semidirect
145. C3C2×#6292semidirect
146. C3C2×C4⋊(C3⋊(C3⋊D3))semidirect
149. C2#10620147direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊(C3⋊D3))2
C22×C3×C3⋊(C3⋊D3)2
C2×C33×Dic32
#106201472
C2×C4⋊(C3⋊(C3⋊D3))3
C2×#62923