Finite Group Databasebeta

← C2×C4×#4184 #13047430

Subgroups — C2×C4×#4184 #13047430

Order 1296.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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×#4184C211yesmaximal, characteristic
2C2×C4×#731C211yesmaximal, characteristic, complemented
3C2×#66331C211yesmaximal, characteristic, complemented
4C4×#4184C214yesmaximal, direct product factor
5C2×C4×AGL(1,ℤ/9)—32nomaximal
6C2×C4×(C3⋊D3)⋊C3—31nomaximal
7C2×C3×C4×D9—31nomaximal
8C22×#731C2211yescharacteristic
9C2×#4184C414yesdirect product factor
10C2×#4184C2212yes—
11C4×#731C2212yescomplemented
12#66331C2212yescomplemented
13C22×AGL(1,ℤ/9)—32no—
14C22×(C3⋊D3)⋊C3—31no—
15C22×C3×D9—31no—
16C2×C4×3−1+2D312yescomplemented
17C2×C4×H(3,𝔽3)D311yescharacteristic
18C2×C3×C4×C9D311yescharacteristic, complemented
19C2×3−1+2⋊C4—32no—
20C2×(C3⋊Dic3)⋊C3—31no—
21C2×C3×Dic9—31no—
22C4×AGL(1,ℤ/9)—38no—
23C4×(C3⋊D3)⋊C3—34no—
24C3×C4×D9—34no—
25C2×#731C2×C412yescomplemented
26C2×#731C2311yescharacteristic
27#4184C2×C414yesdirect product factor
28C2×C4×C3⋊D3—31no—
29C2×C3×C4×D3—91no—
30C2×C4×D9—31no—
31C2×C4×D9—32no—
32C22×3−1+2C2×D312yes—
33C22×H(3,𝔽3)C2×D311yescharacteristic
34C22×C3×C9C2×D311yescharacteristic
35C2×AGL(1,ℤ/9)—38no—
36C2×AGL(1,ℤ/9)—34no—
37C2×(C3⋊D3)⋊C3—34no—
38C2×(C3⋊D3)⋊C3—32no—
39C2×C3×D9—34no—
40C2×C3×D9—32no—
41C4×3−1+2C2×D314yescomplemented
42C4×H(3,𝔽3)C2×D312yes—
43C3×C4×C9C2×D312yescomplemented
443−1+2⋊C4—34no—
45(C3⋊Dic3)⋊C3—32no—
46C3×Dic9—32no—
47#731C22×C411yescharacteristic, complemented, Sylow
48C22×C3⋊D3—31no—
49C22×C3×D3—91no—
50C2×C32×C4—31no—
51C2×C32×C4C3⋊D311yescharacteristic
52C2×C3⋊Dic3—31no—
53C4×C3⋊D3—34no—
54C2×C3×Dic3—91no—
55C3×C4×D3—94no—
56C22×D9—31no—
57C22×D9—32no—
58C2×C4×C9—31no—
59C2×C4×C9—32no—
60C2×Dic9—31no—
61C2×Dic9—32no—
62C4×D9—34no—
63C4×D9—38no—
64C2×3−1+2C4×D314yescomplemented
65C2×3−1+2C22×D312yes—
66C2×H(3,𝔽3)C4×D312yes—
67C2×H(3,𝔽3)C22×D311yescharacteristic
68C2×C3×C9C4×D312yescomplemented
69C2×C3×C9C22×D311yescharacteristic
70AGL(1,ℤ/9)—38no—
71(C3⋊D3)⋊C3—34no—
72C3×D9—34no—
73C22×C3×C4—271no—
74C2×C4×D3—91no—
75C2×C4×D3—271no—
76C22×C32—31no—
77C22×C32C2×C3⋊D311yescharacteristic
78C2×C3⋊D3—32no—
79C2×C3⋊D3—34no—
80C2×C3×D3—94no—
81C2×C3×D3—92no—
82C32×C4—32no—
83C32×C4C2×C3⋊D312yes—
84C3⋊Dic3—32no—
85C3×Dic3—92no—
86C22×C9—31no—
87C22×C9—32no—
88C2×D9—32no—
89C2×D9—34no—
90C2×D9—34no—
91C2×D9—38no—
92C4×C9—32no—
93C4×C9—34no—
94Dic9—32no—
95Dic9—34no—
963−1+2C2×C4×D312yescomplemented
97H(3,𝔽3)C2×C4×D311yescharacteristic
98C3×C9C2×C4×D311yescharacteristic, complemented
99C23×C3—271no—
100C22×D3—91no—
101C22×D3—271no—
102C2×C3×C4—31no—
103C2×C3×C4—91no—
104C2×C3×C4—274no—
105C2×C3×C4—271no—
106C2×C3×C4#27211yescharacteristic
107C2×Dic3—91no—
108C2×Dic3—271no—
109C4×D3—94no—
110C4×D3—274no—
111C2×C32—32no—
112C2×C32—31no—
113C2×C32C4×C3⋊D312yes—
114C2×C32C22×C3⋊D311yescharacteristic
115C3⋊D3—34no—
116C3×D3—94no—
117C2×C9—32no—
118C2×C9—34no—
119C2×C9—31no—
120C2×C9—32no—
121D9—34no—
122D9—38no—
123C22×C4—271noSylow
124C22×C3—31no—
125C22×C3—91no—
126C22×C3—272no—
127C22×C3—274no—
128C22×C3C2×#27211yescharacteristic
129C2×D3—94no—
130C2×D3—92no—
131C2×D3—272no—
132C2×D3—274no—
133C3×C4—32no—
134C3×C4—92no—
135C3×C4—272no—
136C3×C4C2×#27212yes—
137Dic3—92no—
138Dic3—272no—
139C32—31no—
140C32C2×C4×C3⋊D311yescharacteristic
141C9—31no—
142C9—32no—
143C23—271no—
144C2×C4—271no—
145C2×C4—274no—
146C2×C4#418411yescharacteristic, direct product factor
147C2×C3—32no—
148C2×C3—31no—
149C2×C3—92no—
150C2×C3—91no—
151C2×C3—274no—
152C2×C3C4×#27212yes—
153C2×C3C22×#27211yescharacteristic
154D3—94no—
155D3—274no—
156C22—274no—
157C22—272no—
158C22C2×#418411yescharacteristic
159C4—272no—
160C4C2×#418412yesdirect product factor
161C3—31no—
162C3—91no—
163C3C2×C4×#27211yescharacteristic
164C2—274no—
165C2C4×#418412yesdirect product factor
166C2C22×#418411yescharacteristic
subgroupG/H
center, hypercenterC2×C4#4184
commutator, hypocenter#731C22×C4
FrattiniC2×C32C22×C3⋊D3
FittingC2×C4×#731C2
socleC22×C3C2×#272
prime radicalC22×#4184C2
cosocleC2×#731C23
LCSG ▷ #731
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#731C2semidirect
3. C2×#66331C2semidirect
4. C4×#4184C2direct (N × H)
9. C2×#4184C4direct (N × H)
11. C4×#731C22semidirect
12. #66331C22semidirect
16. C2×C4×3−1+2D3semidirect
18. C2×C3×C4×C9D3semidirect
25. C2×#731C2×C4semidirect
27. #4184C2×C4direct (N × H)
41. C4×3−1+2C2×D3semidirect
43. C3×C4×C9C2×D3semidirect
47. #731C22×C4semidirect
64. C2×3−1+2C4×D3semidirect
68. C2×C3×C9C4×D3semidirect
96. 3−1+2C2×C4×D3semidirect
98. C3×C9C2×C4×D3semidirect
146. C2×C4#4184direct (N × H)
160. C4C2×#4184direct (N × H)
165. C2C4×#4184direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#41842
C2×C4×#7312
C2×#663312
C4×#41842
C2×C4×AGL(1,ℤ/9)3
C2×C4×(C3⋊D3)⋊C33
C2×C3×C4×D93