Finite Group Databasebeta

← C22×C9×AGL(1,ℤ/9) #14518462

Subgroups — C22×C9×AGL(1,ℤ/9) #14518462

Order 1944.

590 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 225 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×C9×3−1+2C211yesmaximal, characteristic, complemented
2C2×C9×AGL(1,ℤ/9)C216yesmaximal, direct product factor
3C22×C3×AGL(1,ℤ/9)C311yesmaximal, characteristic
4C22×C3×D3×C9—31nomaximal
5C22×D9⋊C9C312yesmaximal, complemented
6C22×C9×D9C311yesmaximal, characteristic, complemented
7C2×C9×3−1+2C2213yescomplemented
8C9×AGL(1,ℤ/9)C2214yesdirect product factor
9C22×C3×3−1+2C2×C311yescharacteristic
10C22×C32×C9D311yescharacteristic
11C2×C3×AGL(1,ℤ/9)C2×C316yes—
12C2×C3×D3×C9—36no—
13C22×C9⋊C9—22no—
14C22×C9⋊C9C2×C312yescomplemented
15C22×C32⋊C9—21no—
16C22×C92—21no—
17C22×C92C2×C311yescharacteristic, complemented
18C2×D9⋊C9C2×C3112yescomplemented
19C2×C9×D9C2×C316yescomplemented
20C9×3−1+2C2311yescharacteristic, complemented, Sylow
21C22×C32×D3—31no—
22C23×C3×C9—91no—
23C22×AGL(1,ℤ/9)C913yesdirect product factor
24C22×D3×C9—31no—
25C22×D3×C9—32no—
26C22×C3×D9C3211yescharacteristic
27C2×C3×3−1+2C22×C313yes—
28C2×C32×C9C2×D313yes—
29C3×AGL(1,ℤ/9)C22×C314yes—
30C3×D3×C9—34no—
31C2×C9⋊C9—26no—
32C2×C9⋊C9C22×C316yescomplemented
33C2×C32⋊C9—23no—
34C2×C92—23no—
35C2×C92C22×C313yescomplemented
36D9⋊C9C22×C318yescomplemented
37C9×D9C22×C314yescomplemented
38C22×C33C3×D311yescharacteristic
39C2×C32×D3—36no—
40C22×3−1+2—23no—
41C22×3−1+2C2×C913yescomplemented
42C22×C3×C9—21no—
43C22×C3×C9—22no—
44C22×C3×C9—21no—
45C22×C3×C9—31no—
46C22×C3×C9—61no—
47C22×C3×C9—96no—
48C22×C3×C9C3×D311yescharacteristic
49C22×C3×C9C3×D312yes—
50C22×C3×C9C2×C3211yescharacteristic
51C2×AGL(1,ℤ/9)C2×C9118yesdirect product factor
52C2×D3×C9—36no—
53C2×D3×C9—312no—
54C2×C3×D9C2×C3216yes—
55C3×3−1+2C23×C311yescharacteristic
56C32×C9C22×D311yescharacteristic
57C9⋊C9—22no—
58C9⋊C9C23×C312yescomplemented
59C32⋊C9—21no—
60C92—21no—
61C92C23×C311yescharacteristic, complemented
62C23×C32—91no—
63C22×C3×D3—33no—
64C22×C3×D3—31no—
65C23×C9—92no—
66C23×C9—91no—
67C22×D9C3×C911yescharacteristic, complemented
68C2×C33C2×C3×D313yes—
69C32×D3—34no—
70C2×3−1+2—29no—
71C2×3−1+2C22×C919yescomplemented
72C2×C3×C9—23no—
73C2×C3×C9—26no—
74C2×C3×C9—23no—
75C2×C3×C9—33no—
76C2×C3×C9—63no—
77C2×C3×C9—94no—
78C2×C3×C9C2×C3×D313yes—
79C2×C3×C9C2×C3×D316yes—
80C2×C3×C9C22×C3213yes—
81AGL(1,ℤ/9)C22×C9112yesdirect product factor
82D3×C9—34no—
83D3×C9—38no—
84C3×D9C22×C3214yes—
85C22×C32—31no—
86C22×C32—61no—
87C22×C32—96no—
88C22×C32D3×C913yes—
89C22×C32C32×D311yescharacteristic
90C2×C3×D3—318no—
91C2×C3×D3—36no—
92C22×C9—23no—
93C22×C9—21no—
94C22×C9—21no—
95C22×C9—32no—
96C22×C9—62no—
97C22×C9—61no—
98C22×C9—912no—
99C22×C9—96no—
100C22×C9AGL(1,ℤ/9)11yescharacteristic, direct product factor
101C22×C9C2×C3×C911yescharacteristic, complemented
102C2×D9C2×C3×C916yescomplemented
103C33C22×C3×D311yescharacteristic
1043−1+2—23no—
1053−1+2C23×C913yescomplemented
106C3×C9—21no—
107C3×C9—22no—
108C3×C9—21no—
109C3×C9—31no—
110C3×C9—61no—
111C3×C9C22×C3×D311yescharacteristic
112C3×C9C22×C3×D312yes—
113C3×C9C23×C3211yescharacteristic
114C23×C3—93no—
115C23×C3—91no—
116C22×D3—31no—
117C2×C32—33no—
118C2×C32—63no—
119C2×C32—94no—
120C2×C32C2×D3×C919yes—
121C2×C32C2×C32×D313yes—
122C3×D3—312no—
123C3×D3—34no—
124C2×C9—29no—
125C2×C9—23no—
126C2×C9—23no—
127C2×C9—36no—
128C2×C9—66no—
129C2×C9—63no—
130C2×C9—98no—
131C2×C9—94no—
132C2×C9C2×AGL(1,ℤ/9)13yesdirect product factor
133C2×C9C22×C3×C913yescomplemented
134D9C22×C3×C914yescomplemented
135C22×C3—21no—
136C22×C3—33no—
137C22×C3—918no—
138C22×C3—96no—
139C22×C3C3×D3×C911yescharacteristic
140C22×C3C3×AGL(1,ℤ/9)11yescharacteristic
141C2×D3—36no—
142C32—31no—
143C32—61no—
144C32C22×D3×C913yes—
145C32C22×C32×D311yescharacteristic
146C9—23no—
147C9—21no—
148C9—21no—
149C9—32no—
150C9—62no—
151C9—61no—
152C9C22×AGL(1,ℤ/9)11yescharacteristic, direct product factor
153C9C23×C3×C911yescharacteristic, complemented
154C23—91noSylow
155C2×C3—23no—
156C2×C3—39no—
157C2×C3—912no—
158C2×C3—94no—
159C2×C3C2×C3×D3×C913yes—
160C2×C3C2×C3×AGL(1,ℤ/9)13yes—
161D3—34no—
162C22—96no—
163C22C9×AGL(1,ℤ/9)11yescharacteristic, direct product factor
164C3—21no—
165C3—33no—
166C3C22×C3×D3×C911yescharacteristic
167C3C22×C3×AGL(1,ℤ/9)11yescharacteristic
168C2—94no—
169C2C2×C9×AGL(1,ℤ/9)13yesdirect product factor
subgroupG/H
center, hypercenterC22×C9AGL(1,ℤ/9)
commutator, hypocenterC9C23×C3×C9
FrattiniC32C22×C32×D3
FittingC22×C9×3−1+2C2
socleC22×C32C32×D3
prime radicalC22×C3×AGL(1,ℤ/9)C3
cosocleC3×C9C23×C32
LCSG ▷ C9
UCSC1 ◁ C22×C9

Complements

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

NH ≅ G/Ntype
1. C22×C9×3−1+2C2semidirect
2. C2×C9×AGL(1,ℤ/9)C2direct (N × H)
5. C22×D9⋊C9C3semidirect
6. C22×C9×D9C3semidirect
7. C2×C9×3−1+2C22semidirect
8. C9×AGL(1,ℤ/9)C22direct (N × H)
14. C22×C9⋊C9C2×C3semidirect
17. C22×C92C2×C3semidirect
18. C2×D9⋊C9C2×C3semidirect
19. C2×C9×D9C2×C3semidirect
20. C9×3−1+2C23semidirect
23. C22×AGL(1,ℤ/9)C9direct (N × H)
32. C2×C9⋊C9C22×C3semidirect
35. C2×C92C22×C3semidirect
36. D9⋊C9C22×C3semidirect
37. C9×D9C22×C3semidirect
41. C22×3−1+2C2×C9semidirect
51. C2×AGL(1,ℤ/9)C2×C9direct (N × H)
58. C9⋊C9C23×C3semidirect
61. C92C23×C3semidirect
67. C22×D9C3×C9semidirect
71. C2×3−1+2C22×C9semidirect
81. AGL(1,ℤ/9)C22×C9direct (N × H)
100. C22×C9AGL(1,ℤ/9)direct (N × H)
101. C22×C9C2×C3×C9semidirect
102. C2×D9C2×C3×C9semidirect
105. 3−1+2C23×C9semidirect
132. C2×C9C2×AGL(1,ℤ/9)direct (N × H)
133. C2×C9C22×C3×C9semidirect
134. D9C22×C3×C9semidirect
152. C9C22×AGL(1,ℤ/9)direct (N × H)
153. C9C23×C3×C9semidirect
163. C22C9×AGL(1,ℤ/9)direct (N × H)
169. C2C2×C9×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×C9×3−1+22
C2×C9×AGL(1,ℤ/9)2
C22×C3×AGL(1,ℤ/9)3
C22×C3×D3×C93
C22×D9⋊C93
C22×C9×D93