Finite Group Databasebeta

← C3×C4×C3⋊AGL(1,ℤ/9) #14518924

Subgroups — C3×C4×C3⋊AGL(1,ℤ/9) #14518924

Order 1944.

822 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×C3×C3⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
2C32×C4×3−1+2C211yesmaximal, characteristic, complemented
3C3×C3⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
4C32×C4×C3⋊D3—31nomaximal
5C4×C3⋊AGL(1,ℤ/9)C313yesmaximal, direct product factor
6C3×C4×C3⋊D9C311yesmaximal, characteristic, complemented
7C3×C4×AGL(1,ℤ/9)—33nomaximal
8C2×C32×3−1+2C2211yescharacteristic
9C3×C3⋊AGL(1,ℤ/9)C412yesdirect product factor
10C2×C32×C3⋊D3—31no—
11C34×C4D311yescharacteristic
12C32×C3⋊Dic3—31no—
13C2×C3⋊AGL(1,ℤ/9)C2×C313yes—
14C2×C3×C3⋊D9C2×C311yescharacteristic
15C2×C3×AGL(1,ℤ/9)—33no—
16C3×C4×3−1+2—23no—
17C3×C4×3−1+2—29no—
18C3×C4×3−1+2—23no—
19C3×C4×3−1+2D313yescomplemented
20C3×C4×3−1+2C2×C313yescomplemented
21C32×C4×C9—21no—
22C32×C4×C9C2×C311yescharacteristic, complemented
23C3⋊(3−1+2⋊C4)C2×C313yescomplemented
24C3×C3⋊Dic9C2×C311yescharacteristic, complemented
25C3×3−1+2⋊C4—33no—
26C32×3−1+2C2×C411yescharacteristic, complemented, Sylow
27C3×C4×C3⋊D3—31no—
28C3×C4×C3⋊D3—33no—
29C32×C4×D3—93no—
30C32×C4×D3—91no—
31C4×C3⋊D9C3211yescharacteristic, complemented
32C4×AGL(1,ℤ/9)—39no—
33C3×C4×D9—33no—
34C2×C34C2×D311yescharacteristic
35C32×C3⋊D3—32no—
36C2×C3×3−1+2—23no—
37C2×C3×3−1+2—29no—
38C2×C3×3−1+2—23no—
39C2×C3×3−1+2C2×D313yes—
40C2×C3×3−1+2C22×C313yes—
41C2×C32×C9—21no—
42C2×C32×C9C22×C311yescharacteristic
43C3⋊AGL(1,ℤ/9)C3×C416yesdirect product factor
44C3×C3⋊D9C3×C412yescomplemented
45C3×AGL(1,ℤ/9)—36no—
46C2×C3×C3⋊D3—31no—
47C2×C3×C3⋊D3—33no—
48C2×C32×D3—93no—
49C2×C32×D3—91no—
50C33×C4—23no—
51C33×C4—21no—
52C33×C4—33no—
53C33×C4—63no—
54C33×C4C3×D311yescharacteristic
55C33×C4C3×D313yes—
56C33×C4C3⋊D311yescharacteristic
57C3×C3⋊Dic3—31no—
58C3×C3⋊Dic3—33no—
59C32×Dic3—93no—
60C32×Dic3—91no—
61C2×C3⋊D9C2×C3211yescharacteristic
62C2×AGL(1,ℤ/9)—39no—
63C2×C3×D9—33no—
64C4×3−1+2—218no—
65C4×3−1+2—29no—
66C4×3−1+2—29no—
67C4×3−1+2C3×D319yescomplemented
68C3×C4×C9—26no—
69C3×C4×C9—23no—
70C3×C4×C9—21no—
71C3×C4×C9—23no—
72C3×C4×C9—23no—
73C3×C4×C9C3×D313yescomplemented
74C3×C4×C9C2×C3211yescharacteristic, complemented
75C3⋊Dic9C2×C3211yescharacteristic, complemented
763−1+2⋊C4—39no—
77C3×Dic9—33no—
78C34C4×D311yescharacteristic
79C3×3−1+2—23no—
80C3×3−1+2—29no—
81C3×3−1+2—23no—
82C3×3−1+2C4×D313yescomplemented
83C3×3−1+2C2×C3×C413yescomplemented
84C32×C9—21no—
85C32×C9C2×C3×C411yescharacteristic, complemented
86C2×C32×C4—271no—
87C4×C3⋊D3—31no—
88C3×C4×D3—99no—
89C3×C4×D3—93no—
90C3×C4×D3—91no—
91C3×C4×D3—93no—
92C4×D9—33no—
93C2×C33—23no—
94C2×C33—21no—
95C2×C33—33no—
96C2×C33—63no—
97C2×C33C2×C3×D311yescharacteristic
98C2×C33C2×C3×D313yes—
99C2×C33C2×C3⋊D311yescharacteristic
100C3×C3⋊D3—32no—
101C3×C3⋊D3—36no—
102C32×D3—96no—
103C32×D3—92no—
104C2×3−1+2—218no—
105C2×3−1+2—29no—
106C2×3−1+2—29no—
107C2×3−1+2C2×C3×D319yes—
108C2×C3×C9—26no—
109C2×C3×C9—23no—
110C2×C3×C9—21no—
111C2×C3×C9—23no—
112C2×C3×C9—23no—
113C2×C3×C9C2×C3×D313yes—
114C2×C3×C9C22×C3211yescharacteristic
115C3⋊D9C32×C412yescomplemented
116AGL(1,ℤ/9)—318no—
117C3×D9—36no—
118C22×C32—271no—
119C2×C3⋊D3—31no—
120C2×C3×D3—99no—
121C2×C3×D3—93no—
122C2×C3×D3—91no—
123C2×C3×D3—93no—
124C32×C4—23no—
125C32×C4—21no—
126C32×C4—23no—
127C32×C4—31no—
128C32×C4—39no—
129C32×C4—61no—
130C32×C4—69no—
131C32×C4—62no—
132C32×C4—61no—
133C32×C4—271no—
134C32×C4AGL(1,ℤ/9)13yescomplemented
135C32×C4C32×D311yescharacteristic
136C32×C4C3×C3⋊D311yescharacteristic
137C32×C4C3×C3⋊D313yes—
138C3⋊Dic3—31no—
139C3×Dic3—99no—
140C3×Dic3—93no—
141C3×Dic3—91no—
142C3×Dic3—93no—
143C2×D9—33no—
144C4×C9—29no—
145C4×C9—23no—
146C4×C9C32×D313yescomplemented
147Dic9—33no—
148C33—23no—
149C33—21no—
150C33—33no—
151C33—63no—
152C33C3×C4×D311yescharacteristic
153C33C3×C4×D313yes—
154C33C4×C3⋊D311yescharacteristic
1553−1+2—218no—
1563−1+2—29no—
1573−1+2—29no—
1583−1+2C3×C4×D319yescomplemented
159C3×C9—26no—
160C3×C9—23no—
161C3×C9—21no—
162C3×C9—23no—
163C3×C9—23no—
164C3×C9C3×C4×D313yescomplemented
165C3×C9C2×C32×C411yescharacteristic, complemented
166C2×C3×C4—273no—
167C2×C3×C4—271no—
168C4×D3—93no—
169C4×D3—91no—
170C2×C32—23no—
171C2×C32—21no—
172C2×C32—23no—
173C2×C32—31no—
174C2×C32—39no—
175C2×C32—61no—
176C2×C32—69no—
177C2×C32—62no—
178C2×C32—61no—
179C2×C32—272no—
180C2×C32C2×AGL(1,ℤ/9)13yes—
181C2×C32C2×C32×D311yescharacteristic
182C2×C32C2×C3×C3⋊D311yescharacteristic
183C2×C32C2×C3×C3⋊D313yes—
184C3⋊D3—32no—
185C3×D3—918no—
186C3×D3—96no—
187C3×D3—92no—
188C3×D3—96no—
189C2×C9—29no—
190C2×C9—23no—
191C2×C9C2×C32×D313yes—
192D9—36no—
193C22×C3—273no—
194C22×C3—271no—
195C2×D3—93no—
196C2×D3—91no—
197C3×C4—23no—
198C3×C4—21no—
199C3×C4—33no—
200C3×C4—63no—
201C3×C4—273no—
202C3×C4—271no—
203C3×C4C3×AGL(1,ℤ/9)13yescomplemented
204C3×C4C3⋊AGL(1,ℤ/9)11yescharacteristic, direct product factor
205C3×C4C32×C3⋊D311yescharacteristic
206Dic3—93no—
207Dic3—91no—
208C32—21no—
209C32—23no—
210C32—23no—
211C32—31no—
212C32—39no—
213C32—61no—
214C32—69no—
215C32—62no—
216C32—61no—
217C32C4×AGL(1,ℤ/9)13yescomplemented
218C32C32×C4×D311yescharacteristic
219C32C3×C4×C3⋊D311yescharacteristic
220C32C3×C4×C3⋊D313yes—
221C9—29no—
222C9—23no—
223C9C32×C4×D313yescomplemented
224C2×C4—271noSylow
225C2×C3—23no—
226C2×C3—21no—
227C2×C3—33no—
228C2×C3—63no—
229C2×C3—276no—
230C2×C3—272no—
231C2×C3C2×C3×AGL(1,ℤ/9)13yes—
232C2×C3C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
233C2×C3C2×C32×C3⋊D311yescharacteristic
234D3—92no—
235D3—96no—
236C22—271no—
237C4—271no—
238C4C3×C3⋊AGL(1,ℤ/9)11yescharacteristic, direct product factor
239C3—23no—
240C3—21no—
241C3—33no—
242C3—63no—
243C3C3×C4×AGL(1,ℤ/9)13yescomplemented
244C3C4×C3⋊AGL(1,ℤ/9)11yescharacteristic, direct product factor
245C3C32×C4×C3⋊D311yescharacteristic
246C2—272no—
247C2C2×C3×C3⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
center, hypercenterC3×C4C3⋊AGL(1,ℤ/9)
commutator, hypocenterC3×C9C2×C32×C4
FrattiniC2×C3C2×C32×C3⋊D3
FittingC32×C4×3−1+2C2
socleC2×C33C2×C3×D3
prime radicalC2×C3×C3⋊AGL(1,ℤ/9)C2
cosocleC2×C3×C9C22×C32
LCSG ▷ C3×C9
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×3−1+2C2semidirect
3. C3×C3⋊(3−1+2⋊C4)C2semidirect
5. C4×C3⋊AGL(1,ℤ/9)C3direct (N × H)
6. C3×C4×C3⋊D9C3semidirect
9. C3×C3⋊AGL(1,ℤ/9)C4direct (N × H)
19. C3×C4×3−1+2D3semidirect
20. C3×C4×3−1+2C2×C3semidirect
22. C32×C4×C9C2×C3semidirect
23. C3⋊(3−1+2⋊C4)C2×C3semidirect
24. C3×C3⋊Dic9C2×C3semidirect
26. C32×3−1+2C2×C4semidirect
31. C4×C3⋊D9C32semidirect
43. C3⋊AGL(1,ℤ/9)C3×C4direct (N × H)
44. C3×C3⋊D9C3×C4semidirect
67. C4×3−1+2C3×D3semidirect
73. C3×C4×C9C3×D3semidirect
74. C3×C4×C9C2×C32semidirect
75. C3⋊Dic9C2×C32semidirect
82. C3×3−1+2C4×D3semidirect
83. C3×3−1+2C2×C3×C4semidirect
85. C32×C9C2×C3×C4semidirect
115. C3⋊D9C32×C4semidirect
134. C32×C4AGL(1,ℤ/9)semidirect
146. C4×C9C32×D3semidirect
158. 3−1+2C3×C4×D3semidirect
164. C3×C9C3×C4×D3semidirect
165. C3×C9C2×C32×C4semidirect
203. C3×C4C3×AGL(1,ℤ/9)semidirect
204. C3×C4C3⋊AGL(1,ℤ/9)direct (N × H)
217. C32C4×AGL(1,ℤ/9)semidirect
223. C9C32×C4×D3semidirect
238. C4C3×C3⋊AGL(1,ℤ/9)direct (N × H)
243. C3C3×C4×AGL(1,ℤ/9)semidirect
244. C3C4×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
C2×C3×C3⋊AGL(1,ℤ/9)2
C32×C4×3−1+22
C3×C3⋊(3−1+2⋊C4)2
C32×C4×C3⋊D33
C4×C3⋊AGL(1,ℤ/9)3
C3×C4×C3⋊D93
C3×C4×AGL(1,ℤ/9)3