Finite Group Databasebeta

← C2×C3×C4×AGL(1,ℤ/9) #13048839

Subgroups — C2×C3×C4×AGL(1,ℤ/9) #13048839

Order 1296.

566 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 226 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)C211yesmaximal, characteristic
2C2×C3×C4×3−1+2C211yesmaximal, characteristic, complemented
3C2×C3×3−1+2⋊C4C211yesmaximal, characteristic, complemented
4C3×C4×AGL(1,ℤ/9)C214yesmaximal, direct product factor
5C2×C32×C4×D3—31nomaximal
6C2×C4×AGL(1,ℤ/9)C313yesmaximal, direct product factor
7C2×C3×C4×D9C311yesmaximal, characteristic, complemented
8C22×C3×3−1+2C2211yescharacteristic
9C2×C3×AGL(1,ℤ/9)C414yesdirect product factor
10C2×C3×AGL(1,ℤ/9)C2212yes—
11C3×C4×3−1+2C2212yescomplemented
12C3×3−1+2⋊C4C2212yescomplemented
13C22×C32×D3—31no—
14C2×C33×C4D311yescharacteristic
15C2×C32×Dic3—31no—
16C32×C4×D3—34no—
17C22×AGL(1,ℤ/9)C2×C313yes—
18C22×C3×D9C2×C311yescharacteristic
19C2×C4×3−1+2—23no—
20C2×C4×3−1+2C2×C313yescomplemented
21C2×C3×C4×C9—21no—
22C2×C3×C4×C9C2×C311yescharacteristic, complemented
23C2×3−1+2⋊C4C2×C313yescomplemented
24C2×C3×Dic9C2×C311yescharacteristic, complemented
25C4×AGL(1,ℤ/9)C2×C3112yesdirect product factor
26C3×C4×D9C2×C314yescomplemented
27C2×C3×3−1+2C2×C412yescomplemented
28C2×C3×3−1+2C2311yescharacteristic
29C3×AGL(1,ℤ/9)C2×C414yesdirect product factor
30C22×C32×C4—91no—
31C2×C3×C4×D3—31no—
32C2×C3×C4×D3—33no—
33C2×C4×D9C3211yescharacteristic, complemented
34C22×C33C2×D311yescharacteristic
35C2×C32×D3—34no—
36C2×C32×D3—32no—
37C33×C4C2×D312yes—
38C32×Dic3—32no—
39C22×3−1+2—23no—
40C22×3−1+2C22×C313yes—
41C22×C3×C9—21no—
42C22×C3×C9C22×C311yescharacteristic
43C2×AGL(1,ℤ/9)C3×C4112yesdirect product factor
44C2×AGL(1,ℤ/9)C22×C316yes—
45C2×C3×D9C3×C414yescomplemented
46C2×C3×D9C22×C312yes—
47C4×3−1+2—26no—
48C4×3−1+2C22×C316yescomplemented
49C3×C4×C9—22no—
50C3×C4×C9C22×C312yescomplemented
513−1+2⋊C4C22×C316yescomplemented
52C3×Dic9C22×C312yescomplemented
53C3×3−1+2C22×C411yescharacteristic, complemented, Sylow
54C23×C32—91no—
55C22×C3×D3—31no—
56C22×C3×D3—33no—
57C2×C32×C4—31no—
58C2×C32×C4—61no—
59C2×C32×C4—94no—
60C2×C32×C4—91no—
61C2×C32×C4C3×D311yescharacteristic
62C2×C32×C4C3×D313yes—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—33no—
65C3×C4×D3—34no—
66C3×C4×D3—312no—
67C22×D9C2×C3211yescharacteristic
68C2×C4×C9—23no—
69C2×C4×C9—21no—
70C2×C4×C9C2×C3211yescharacteristic, complemented
71C2×Dic9C2×C3211yescharacteristic, complemented
72C4×D9C2×C3214yescomplemented
73C2×C33C4×D312yes—
74C2×C33C22×D311yescharacteristic
75C32×D3—34no—
76C2×3−1+2—26no—
77C2×3−1+2—23no—
78C2×3−1+2C2×C3×C416yescomplemented
79C2×3−1+2C23×C313yes—
80C2×C3×C9—22no—
81C2×C3×C9—21no—
82C2×C3×C9C2×C3×C412yescomplemented
83C2×C3×C9C23×C311yescharacteristic
84AGL(1,ℤ/9)C2×C3×C4112yesdirect product factor
85C3×D9C2×C3×C414yescomplemented
86C22×C3×C4—93no—
87C22×C3×C4—91no—
88C2×C4×D3—31no—
89C22×C32—31no—
90C22×C32—61no—
91C22×C32—92no—
92C22×C32—94no—
93C22×C32C2×C3×D311yescharacteristic
94C22×C32C2×C3×D313yes—
95C2×C3×D3—34no—
96C2×C3×D3—312no—
97C2×C3×D3—32no—
98C2×C3×D3—36no—
99C32×C4—32no—
100C32×C4—62no—
101C32×C4—92no—
102C32×C4C2×C3×D312yes—
103C32×C4C2×C3×D316yes—
104C3×Dic3—32no—
105C3×Dic3—36no—
106C22×C9—23no—
107C22×C9—21no—
108C22×C9C22×C3211yescharacteristic
109C2×D9C32×C414yescomplemented
110C2×D9C22×C3212yes—
111C4×C9—26no—
112C4×C9—22no—
113C4×C9C22×C3212yescomplemented
114Dic9C22×C3212yescomplemented
115C33C2×C4×D311yescharacteristic
1163−1+2—23no—
1173−1+2C22×C3×C413yescomplemented
118C3×C9—21no—
119C3×C9C22×C3×C411yescharacteristic, complemented
120C23×C3—93no—
121C23×C3—91no—
122C22×D3—31no—
123C2×C3×C4—21no—
124C2×C3×C4—33no—
125C2×C3×C4—912no—
126C2×C3×C4—93no—
127C2×C3×C4—91no—
128C2×C3×C4—94no—
129C2×C3×C4AGL(1,ℤ/9)11yescharacteristic, direct product factor
130C2×C3×C4C32×D311yescharacteristic
131C2×Dic3—31no—
132C4×D3—34no—
133C2×C32—32no—
134C2×C32—31no—
135C2×C32—62no—
136C2×C32—61no—
137C2×C32—94no—
138C2×C32C3×C4×D312yes—
139C2×C32C3×C4×D316yes—
140C2×C32C22×C3×D311yescharacteristic
141C2×C32C22×C3×D313yes—
142C3×D3—34no—
143C3×D3—312no—
144C2×C9—26no—
145C2×C9—22no—
146C2×C9—23no—
147C2×C9—21no—
148C2×C9C2×C32×C412yescomplemented
149C2×C9C23×C3211yescharacteristic
150D9C2×C32×C414yescomplemented
151C22×C4—91noSylow
152C22×C3—21no—
153C22×C3—33no—
154C22×C3—96no—
155C22×C3—912no—
156C22×C3—94no—
157C22×C3—92no—
158C22×C3C2×AGL(1,ℤ/9)11yescharacteristic
159C22×C3C2×C32×D311yescharacteristic
160C2×D3—34no—
161C2×D3—32no—
162C3×C4—22no—
163C3×C4—36no—
164C3×C4—96no—
165C3×C4—92no—
166C3×C4C2×AGL(1,ℤ/9)12yesdirect product factor
167C3×C4C2×C32×D312yes—
168Dic3—32no—
169C32—31no—
170C32—61no—
171C32C2×C3×C4×D311yescharacteristic
172C32C2×C3×C4×D313yes—
173C9—23no—
174C9—21no—
175C9C22×C32×C411yescharacteristic, complemented
176C23—91no—
177C2×C4—91no—
178C2×C4—94no—
179C2×C4C3×AGL(1,ℤ/9)11yescharacteristic, direct product factor
180C2×C3—22no—
181C2×C3—21no—
182C2×C3—36no—
183C2×C3—33no—
184C2×C3—912no—
185C2×C3—94no—
186C2×C3C4×AGL(1,ℤ/9)12yesdirect product factor
187C2×C3C22×AGL(1,ℤ/9)11yescharacteristic
188C2×C3C32×C4×D312yes—
189C2×C3C22×C32×D311yescharacteristic
190D3—34no—
191C22—94no—
192C22—92no—
193C22C2×C3×AGL(1,ℤ/9)11yescharacteristic
194C4—92no—
195C4C2×C3×AGL(1,ℤ/9)12yesdirect product factor
196C3—21no—
197C3—33no—
198C3C2×C4×AGL(1,ℤ/9)11yescharacteristic, direct product factor
199C3C2×C32×C4×D311yescharacteristic
200C2—94no—
201C2C3×C4×AGL(1,ℤ/9)12yesdirect product factor
202C2C22×C3×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4AGL(1,ℤ/9)
commutator, hypocenterC9C22×C32×C4
FrattiniC2×C3C22×C32×D3
FittingC2×C3×C4×3−1+2C2
socleC22×C32C2×C3×D3
prime radicalC22×C3×AGL(1,ℤ/9)C2
cosocleC2×C9C23×C32
LCSG ▷ C9
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×3−1+2C2semidirect
3. C2×C3×3−1+2⋊C4C2semidirect
4. C3×C4×AGL(1,ℤ/9)C2direct (N × H)
6. C2×C4×AGL(1,ℤ/9)C3direct (N × H)
7. C2×C3×C4×D9C3semidirect
9. C2×C3×AGL(1,ℤ/9)C4direct (N × H)
11. C3×C4×3−1+2C22semidirect
12. C3×3−1+2⋊C4C22semidirect
20. C2×C4×3−1+2C2×C3semidirect
22. C2×C3×C4×C9C2×C3semidirect
23. C2×3−1+2⋊C4C2×C3semidirect
24. C2×C3×Dic9C2×C3semidirect
25. C4×AGL(1,ℤ/9)C2×C3direct (N × H)
26. C3×C4×D9C2×C3semidirect
27. C2×C3×3−1+2C2×C4semidirect
29. C3×AGL(1,ℤ/9)C2×C4direct (N × H)
33. C2×C4×D9C32semidirect
43. C2×AGL(1,ℤ/9)C3×C4direct (N × H)
45. C2×C3×D9C3×C4semidirect
48. C4×3−1+2C22×C3semidirect
50. C3×C4×C9C22×C3semidirect
51. 3−1+2⋊C4C22×C3semidirect
52. C3×Dic9C22×C3semidirect
53. C3×3−1+2C22×C4semidirect
70. C2×C4×C9C2×C32semidirect
71. C2×Dic9C2×C32semidirect
72. C4×D9C2×C32semidirect
78. C2×3−1+2C2×C3×C4semidirect
82. C2×C3×C9C2×C3×C4semidirect
84. AGL(1,ℤ/9)C2×C3×C4direct (N × H)
85. C3×D9C2×C3×C4semidirect
109. C2×D9C32×C4semidirect
113. C4×C9C22×C32semidirect
114. Dic9C22×C32semidirect
117. 3−1+2C22×C3×C4semidirect
119. C3×C9C22×C3×C4semidirect
129. C2×C3×C4AGL(1,ℤ/9)direct (N × H)
148. C2×C9C2×C32×C4semidirect
150. D9C2×C32×C4semidirect
166. C3×C4C2×AGL(1,ℤ/9)direct (N × H)
175. C9C22×C32×C4semidirect
179. C2×C4C3×AGL(1,ℤ/9)direct (N × H)
186. C2×C3C4×AGL(1,ℤ/9)direct (N × H)
195. C4C2×C3×AGL(1,ℤ/9)direct (N × H)
198. C3C2×C4×AGL(1,ℤ/9)direct (N × H)
201. C2C3×C4×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
C2×C3×3−1+2⋊C42
C3×C4×AGL(1,ℤ/9)2
C2×C32×C4×D33
C2×C4×AGL(1,ℤ/9)3
C2×C3×C4×D93