Finite Group Databasebeta

← C2×C4⋊(C3⋊AGL(1,ℤ/9)) #13048899

Subgroups — C2×C4⋊(C3⋊AGL(1,ℤ/9)) #13048899

Order 1296.

612 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 150 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⋊AGL(1,ℤ/9)C212yesmaximal, complemented
2C2×C3×C4×3−1+2C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊AGL(1,ℤ/9))C214yesmaximal, direct product factor
4C2×C3×C4⋊(C3⋊D3)—31nomaximal
5C2×C4⋊(C3⋊D9)C311yesmaximal, characteristic, complemented
6C2×C4⋊AGL(1,ℤ/9)—33nomaximal
7C22×C3×3−1+2C2211yescharacteristic
8C2×C3⋊AGL(1,ℤ/9)—24no—
9C2×C3⋊AGL(1,ℤ/9)C2214yescomplemented
10C3×C4×3−1+2C2212yescomplemented
11C22×C3×C3⋊D3—32no—
12C2×C33×C4D311yescharacteristic
13C3×C4⋊(C3⋊D3)—34no—
14C22×C3⋊D9C2×C312yescomplemented
15C22×AGL(1,ℤ/9)—36no—
16C2×C4×3−1+2—23no—
17C2×C4×3−1+2D313yescomplemented
18C2×C3×C4×C9—21no—
19C2×C3×C4×C9C2×C311yescharacteristic, complemented
20C4⋊(C3⋊D9)C2×C314yescomplemented
21C4⋊AGL(1,ℤ/9)—312no—
22C2×C3×3−1+2D412yescomplemented
23C2×C3×3−1+2C2311yescharacteristic
24C3⋊AGL(1,ℤ/9)—24no—
25C2×C4⋊(C3⋊D3)—31no—
26C2×C3×D12—93no—
27C2×C3×D12—91no—
28C2×D36—33no—
29C22×C33C2×D311yescharacteristic
30C2×C3×C3⋊D3—34no—
31C2×C3×C3⋊D3—64no—
32C33×C4C2×D312yes—
33C22×3−1+2—23no—
34C22×3−1+2C2×D313yes—
35C22×C3×C9—21no—
36C22×C3×C9C22×C311yescharacteristic
37C2×C3⋊D9—24no—
38C2×C3⋊D9C22×C314yescomplemented
39C2×AGL(1,ℤ/9)—312no—
40C2×AGL(1,ℤ/9)—612no—
41C4×3−1+2—26no—
42C4×3−1+2C2×D316yescomplemented
43C3×C4×C9—22no—
44C3×C4×C9C22×C312yescomplemented
45C3×3−1+2C2×D411yescharacteristic, complemented, Sylow
46C22×C3⋊D3—32no—
47C22×C3×D3—96no—
48C22×C3×D3—92no—
49C2×C32×C4—21no—
50C2×C32×C4—33no—
51C2×C32×C4C3×D311yescharacteristic
52C2×C32×C4C3⋊D311yescharacteristic
53C4⋊(C3⋊D3)—34no—
54C3×D12—912no—
55C3×D12—94no—
56C22×D9—36no—
57C2×C4×C9—23no—
58C2×C4×C9C3×D313yescomplemented
59D36—312no—
60C2×C33D1212yes—
61C2×C33C22×D311yescharacteristic
62C3×C3⋊D3—64no—
63C2×3−1+2—26no—
64C2×3−1+2—23no—
65C2×3−1+2D1216yescomplemented
66C2×3−1+2C22×D313yes—
67C2×C3×C9—22no—
68C2×C3×C9—21no—
69C2×C3×C9C3×D412yescomplemented
70C2×C3×C9C23×C311yescharacteristic
71C3⋊D9—24no—
72AGL(1,ℤ/9)—612no—
73C2×C3×D4—271no—
74C2×D12—93no—
75C2×D12—91no—
76C22×C32—21no—
77C22×C32—33no—
78C22×C32C2×C3×D311yescharacteristic
79C22×C32C2×C3⋊D311yescharacteristic
80C2×C3⋊D3—34no—
81C2×C3⋊D3—64no—
82C2×C3×D3—912no—
83C2×C3×D3—94no—
84C2×C3×D3—1812no—
85C2×C3×D3—184no—
86C32×C4—22no—
87C32×C4—36no—
88C32×C4C2×C3×D312yes—
89C32×C4C2×C3⋊D312yes—
90C22×C9—23no—
91C22×C9C2×C3×D313yes—
92C2×D9—312no—
93C2×D9—612no—
94C4×C9—26no—
95C4×C9C2×C3×D316yescomplemented
96C33C2×D1211yescharacteristic
973−1+2—23no—
983−1+2C2×D1213yescomplemented
99C3×C9—21no—
100C3×C9C2×C3×D411yescharacteristic, complemented
101C23×C3—272no—
102C22×D3—96no—
103C22×D3—92no—
104C3×D4—274no—
105C2×C3×C4—31no—
106C2×C3×C4—61no—
107C2×C3×C4AGL(1,ℤ/9)13yescomplemented
108C2×C3×C4C3×C3⋊D311yescharacteristic
109D12—912no—
110D12—94no—
111C2×C32—22no—
112C2×C32—21no—
113C2×C32—36no—
114C2×C32—33no—
115C2×C32C3×D1212yes—
116C2×C32C4⋊(C3⋊D3)12yes—
117C2×C32C22×C3×D311yescharacteristic
118C2×C32C22×C3⋊D311yescharacteristic
119C3⋊D3—64no—
120C3×D3—1812no—
121C3×D3—184no—
122C2×C9—26no—
123C2×C9—23no—
124C2×C9C3×D1216yescomplemented
125C2×C9C22×C3×D313yes—
126D9—612no—
127C2×D4—271noSylow
128C22×C3—31no—
129C22×C3—61no—
130C22×C3—274no—
131C22×C3—544no—
132C22×C3C2×AGL(1,ℤ/9)13yes—
133C22×C3C2×C3×C3⋊D311yescharacteristic
134C2×D3—912no—
135C2×D3—94no—
136C2×D3—1812no—
137C2×D3—184no—
138C3×C4—32no—
139C3×C4—62no—
140C3×C4C2×AGL(1,ℤ/9)16yescomplemented
141C3×C4C2×C3×C3⋊D312yes—
142C32—21no—
143C32—33no—
144C32C2×C3×D1211yescharacteristic
145C32C2×C4⋊(C3⋊D3)11yescharacteristic
146C9—23no—
147C9C2×C3×D1213yescomplemented
148C23—272no—
149D4—274no—
150C2×C4C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
151C2×C3—32no—
152C2×C3—31no—
153C2×C3—62no—
154C2×C3—61no—
155C2×C3—544no—
156C2×C3C4⋊AGL(1,ℤ/9)16yescomplemented
157C2×C3C22×AGL(1,ℤ/9)13yes—
158C2×C3C3×C4⋊(C3⋊D3)12yes—
159C2×C3C22×C3×C3⋊D311yescharacteristic
160D3—1812no—
161D3—184no—
162C22—274no—
163C22—544no—
164C22C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
165C4C2×C3⋊AGL(1,ℤ/9)12yescomplemented
166C3—31no—
167C3—61no—
168C3C2×C4⋊AGL(1,ℤ/9)13yescomplemented
169C3C2×C3×C4⋊(C3⋊D3)11yescharacteristic
170C2—544no—
171C2C4⋊(C3⋊AGL(1,ℤ/9))12yesdirect product factor
172C2C22×C3⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊AGL(1,ℤ/9)
commutator, cosocleC2×C3×C9C23×C3
FrattiniC2×C3C22×C3×C3⋊D3
FittingC2×C3×C4×3−1+2C2
socleC22×C32C2×C3×D3
hypercenterC2×C4C3⋊AGL(1,ℤ/9)
hypocenterC3×C9C2×C3×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊AGL(1,ℤ/9)C2semidirect
2. C2×C3×C4×3−1+2C2semidirect
3. C4⋊(C3⋊AGL(1,ℤ/9))C2direct (N × H)
5. C2×C4⋊(C3⋊D9)C3semidirect
9. C2×C3⋊AGL(1,ℤ/9)C22semidirect
10. C3×C4×3−1+2C22semidirect
14. C22×C3⋊D9C2×C3semidirect
17. C2×C4×3−1+2D3semidirect
19. C2×C3×C4×C9C2×C3semidirect
20. C4⋊(C3⋊D9)C2×C3semidirect
22. C2×C3×3−1+2D4semidirect
38. C2×C3⋊D9C22×C3semidirect
42. C4×3−1+2C2×D3semidirect
44. C3×C4×C9C22×C3semidirect
45. C3×3−1+2C2×D4semidirect
58. C2×C4×C9C3×D3semidirect
65. C2×3−1+2D12semidirect
69. C2×C3×C9C3×D4semidirect
95. C4×C9C2×C3×D3semidirect
98. 3−1+2C2×D12semidirect
100. C3×C9C2×C3×D4semidirect
107. C2×C3×C4AGL(1,ℤ/9)semidirect
124. C2×C9C3×D12semidirect
140. C3×C4C2×AGL(1,ℤ/9)semidirect
147. C9C2×C3×D12semidirect
150. C2×C4C3⋊AGL(1,ℤ/9)semidirect
156. C2×C3C4⋊AGL(1,ℤ/9)semidirect
165. C4C2×C3⋊AGL(1,ℤ/9)semidirect
168. C3C2×C4⋊AGL(1,ℤ/9)semidirect
171. C2C4⋊(C3⋊AGL(1,ℤ/9))direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊AGL(1,ℤ/9)2
C2×C3×C4×3−1+22
C4⋊(C3⋊AGL(1,ℤ/9))2
C2×C3×C4⋊(C3⋊D3)3
C2×C4⋊(C3⋊D9)3
C2×C4⋊AGL(1,ℤ/9)3