Finite Group Databasebeta

← C3×Dic3×AGL(1,ℤ/9) #14518633

Subgroups — C3×Dic3×AGL(1,ℤ/9) #14518633

Order 1944.

574 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 118 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×C32×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C3×C3⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
3C3×Dic3×3−1+2C211yesmaximal, characteristic, complemented
4C32×D3×Dic3—31nomaximal
5C3×C4×AGL(1,ℤ/9)—31nomaximal
6Dic3×AGL(1,ℤ/9)C313yesmaximal, direct product factor
7C3×Dic3×D9C311yesmaximal, characteristic, complemented
8C2×C32×3−1+2C2211yescharacteristic
9C32×AGL(1,ℤ/9)C412yescomplemented
10C2×C33×D3—31no—
11C32×C3⋊Dic3—31no—
12C33×Dic3D311yescharacteristic
13C2×C3×AGL(1,ℤ/9)—23no—
14C2×C3×AGL(1,ℤ/9)—21no—
15C2×C3×AGL(1,ℤ/9)D311yescharacteristic
16C2×C3×AGL(1,ℤ/9)C2×C313yes—
17C2×C32×D9C2×C311yescharacteristic
18C3×C4×3−1+2—31no—
19C3⋊(3−1+2⋊C4)C2×C313yescomplemented
20C3×C3⋊Dic9C2×C311yescharacteristic, complemented
21Dic3×3−1+2—23no—
22Dic3×3−1+2C2×C313yescomplemented
23C3×3−1+2⋊C4—31no—
24C3×3−1+2⋊C4—61no—
25C3×C9×Dic3—21no—
26C3×C9×Dic3C2×C311yescharacteristic, complemented
27C32×3−1+2C2×C411yescharacteristic, complemented, Sylow
28C2×C32×Dic3—91no—
29C32×C4×D3—91no—
30C3×D3×Dic3—31no—
31C3×D3×Dic3—33no—
32C4×AGL(1,ℤ/9)—33no—
33C3×C4×D9—31no—
34Dic3×D9C3211yescharacteristic, complemented
35C2×C34C2×D311yescharacteristic
36C33×D3—32no—
37C2×C3×3−1+2—23no—
38C2×C3×3−1+2—21no—
39C2×C3×3−1+2—23no—
40C2×C3×3−1+2—21no—
41C2×C3×3−1+2—41no—
42C2×C3×3−1+2—43no—
43C2×C3×3−1+2C2×D311yescharacteristic
44C2×C3×3−1+2C22×C313yes—
45C2×C32×C9—21no—
46C2×C32×C9C22×C311yescharacteristic
47C3×AGL(1,ℤ/9)—26no—
48C3×AGL(1,ℤ/9)—22no—
49C3×AGL(1,ℤ/9)Dic312yesdirect product factor
50C3×AGL(1,ℤ/9)C3×C416yescomplemented
51C32×D9C3×C412yescomplemented
52C22×C33—91no—
53C2×C32×D3—31no—
54C2×C32×D3—31no—
55C2×C32×D3—33no—
56C2×C32×D3—61no—
57C2×C32×D3—63no—
58C33×C4—31no—
59C3×C3⋊Dic3—31no—
60C3×C3⋊Dic3—33no—
61C32×Dic3—31no—
62C32×Dic3—61no—
63C32×Dic3—91no—
64C32×Dic3—91no—
65C32×Dic3—181no—
66C32×Dic3C3×D311yescharacteristic
67C32×Dic3C3×D313yes—
68C2×AGL(1,ℤ/9)—23no—
69C2×AGL(1,ℤ/9)C3×D313yes—
70C2×C3×D9—21no—
71C2×C3×D9C3×D311yescharacteristic
72C2×C3×D9C2×C3211yescharacteristic
73C4×3−1+2—33no—
74C4×3−1+2—63no—
75C3×C4×C9—31no—
76C3×C4×C9—61no—
77C3⋊Dic9C2×C3211yescharacteristic, complemented
783−1+2⋊C4—33no—
793−1+2⋊C4—63no—
80C9×Dic3—23no—
81C9×Dic3—21no—
82C9×Dic3C2×C3211yescharacteristic, complemented
83C3×Dic9—31no—
84C3×Dic9—61no—
85C34C4×D311yescharacteristic
86C3×3−1+2—23no—
87C3×3−1+2—21no—
88C3×3−1+2—23no—
89C3×3−1+2—21no—
90C3×3−1+2—41no—
91C3×3−1+2—43no—
92C3×3−1+2C2×Dic311yescharacteristic, complemented
93C3×3−1+2C2×C3×C413yescomplemented
94C32×C9—21no—
95C32×C9C2×C3×C411yescharacteristic, complemented
96C2×C32×C4—271no—
97C2×C3×Dic3—93no—
98C2×C3×Dic3—91no—
99C3×C4×D3—91no—
100C3×C4×D3—93no—
101D3×Dic3—31no—
102C4×D9—31no—
103C2×C33—23no—
104C2×C33—21no—
105C2×C33—31no—
106C2×C33—61no—
107C2×C33—61no—
108C2×C33—92no—
109C2×C33—121no—
110C2×C33D3211yescharacteristic
111C2×C33C2×C3×D311yescharacteristic
112C2×C33C2×C3×D313yes—
113C32×D3—32no—
114C32×D3—32no—
115C32×D3—36no—
116C32×D3—62no—
117C32×D3—66no—
118C2×3−1+2—23no—
119C2×3−1+2—23no—
120C2×3−1+2—23no—
121C2×3−1+2—43no—
122C2×3−1+2—46no—
123C2×3−1+2—43no—
124C2×3−1+2—43no—
125C2×3−1+2C2×C3×D313yes—
126C2×C3×C9—21no—
127C2×C3×C9—21no—
128C2×C3×C9—23no—
129C2×C3×C9—21no—
130C2×C3×C9—21no—
131C2×C3×C9—41no—
132C2×C3×C9—43no—
133C2×C3×C9—41no—
134C2×C3×C9C2×C3×D311yescharacteristic
135C2×C3×C9C22×C3211yescharacteristic
136AGL(1,ℤ/9)—26no—
137AGL(1,ℤ/9)C3×Dic316yesdirect product factor
138C3×D9—22no—
139C3×D9C3×Dic312yescomplemented
140C3×D9C32×C412yescomplemented
141C22×C32—91no—
142C22×C32—93no—
143C22×C32—91no—
144C22×C32—183no—
145C22×C32—181no—
146C2×C3×D3—31no—
147C2×C3×D3—33no—
148C2×C3×D3—31no—
149C2×C3×D3—61no—
150C2×C3×D3—63no—
151C32×C4—31no—
152C32×C4—33no—
153C32×C4—91no—
154C32×C4—181no—
155C32×C4—271no—
156C3⋊Dic3—31no—
157C3×Dic3—21no—
158C3×Dic3—33no—
159C3×Dic3—93no—
160C3×Dic3—91no—
161C3×Dic3—91no—
162C3×Dic3—93no—
163C3×Dic3—181no—
164C3×Dic3—183no—
165C3×Dic3AGL(1,ℤ/9)11yescharacteristic, direct product factor
166C3×Dic3C32×D311yescharacteristic
167C2×D9C32×D311yescharacteristic
168C4×C9—31no—
169C4×C9—61no—
170C4×C9—63no—
171Dic9—31no—
172Dic9—61no—
173C33—23no—
174C33—21no—
175C33—31no—
176C33—61no—
177C33—61no—
178C33—121no—
179C33D3×Dic311yescharacteristic
180C33C3×C4×D311yescharacteristic
181C33C3×C4×D313yes—
1823−1+2—23no—
1833−1+2—23no—
1843−1+2—23no—
1853−1+2—43no—
1863−1+2—46no—
1873−1+2—43no—
1883−1+2—43no—
1893−1+2C2×C3×Dic313yescomplemented
190C3×C9—21no—
191C3×C9—21no—
192C3×C9—23no—
193C3×C9—21no—
194C3×C9—21no—
195C3×C9—41no—
196C3×C9—43no—
197C3×C9—41no—
198C3×C9C2×C3×Dic311yescharacteristic, complemented
199C3×C9C2×C32×C411yescharacteristic, complemented
200C2×C3×C4—271no—
201C2×C3×C4—273no—
202C2×Dic3—91no—
203C4×D3—91no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—23no—
208C2×C32—31no—
209C2×C32—33no—
210C2×C32—41no—
211C2×C32—61no—
212C2×C32—63no—
213C2×C32—63no—
214C2×C32—61no—
215C2×C32—92no—
216C2×C32—96no—
217C2×C32—92no—
218C2×C32—121no—
219C2×C32—123no—
220C2×C32—186no—
221C2×C32—182no—
222C2×C32C2×AGL(1,ℤ/9)11yescharacteristic
223C2×C32C3×D3211yescharacteristic
224C2×C32C3×D3213yes—
225C2×C32C2×C32×D311yescharacteristic
226C3×D3—32no—
227C3×D3—36no—
228C3×D3—32no—
229C3×D3—62no—
230C3×D3—66no—
231C2×C9—23no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—41no—
235C2×C9—43no—
236C2×C9C2×C32×D311yescharacteristic
237D9C32×Dic312yescomplemented
238C22×C3—93no—
239C22×C3—91no—
240C22×C3—91no—
241C22×C3—181no—
242C22×C3—183no—
243C2×D3—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—61no—
247C3×C4—93no—
248C3×C4—273no—
249C3×C4—271no—
250Dic3—91no—
251Dic3—91no—
252Dic3—181no—
253Dic3C3×AGL(1,ℤ/9)11yescharacteristic, direct product factor
254C32—21no—
255C32—21no—
256C32—21no—
257C32—23no—
258C32—31no—
259C32—33no—
260C32—41no—
261C32—61no—
262C32—63no—
263C32—63no—
264C32—61no—
265C32—121no—
266C32—123no—
267C32C4×AGL(1,ℤ/9)11yescharacteristic, complemented
268C32C3×D3×Dic311yescharacteristic
269C32C3×D3×Dic313yes—
270C32C32×C4×D311yescharacteristic
271C9—23no—
272C9—21no—
273C9—21no—
274C9—41no—
275C9—43no—
276C9C2×C32×Dic311yescharacteristic, complemented
277C2×C4—271noSylow
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—33no—
282C2×C3—41no—
283C2×C3—63no—
284C2×C3—96no—
285C2×C3—92no—
286C2×C3—92no—
287C2×C3—182no—
288C2×C3—186no—
289C2×C3D3×AGL(1,ℤ/9)11yescharacteristic
290C2×C3C2×C3×AGL(1,ℤ/9)11yescharacteristic
291C2×C3C32×D3211yescharacteristic
292D3—32no—
293C22—91no—
294C4—31no—
295C4—271no—
296C3—21no—
297C3—21no—
298C3—21no—
299C3—33no—
300C3—41no—
301C3—63no—
302C3Dic3×AGL(1,ℤ/9)11yescharacteristic, direct product factor
303C3C3×C4×AGL(1,ℤ/9)11yescharacteristic, complemented
304C3C32×D3×Dic311yescharacteristic
305C2—92no—
306C2C3×D3×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3D3×AGL(1,ℤ/9)
commutator, hypocenterC3×C9C2×C32×C4
FrattiniC2×C3C32×D32
FittingC2×C32×3−1+2C22
socleC2×C33C2×C3×D3
prime radicalC2×C32×AGL(1,ℤ/9)C2
cosocleC2×C3×C9C22×C32
LCSG ▷ C3×C9
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊(3−1+2⋊C4)C2semidirect
3. C3×Dic3×3−1+2C2semidirect
6. Dic3×AGL(1,ℤ/9)C3direct (N × H)
7. C3×Dic3×D9C3semidirect
9. C32×AGL(1,ℤ/9)C4semidirect
19. C3⋊(3−1+2⋊C4)C2×C3semidirect
20. C3×C3⋊Dic9C2×C3semidirect
22. Dic3×3−1+2C2×C3semidirect
26. C3×C9×Dic3C2×C3semidirect
27. C32×3−1+2C2×C4semidirect
34. Dic3×D9C32semidirect
49. C3×AGL(1,ℤ/9)Dic3direct (N × H)
50. C3×AGL(1,ℤ/9)C3×C4semidirect
51. C32×D9C3×C4semidirect
77. C3⋊Dic9C2×C32semidirect
82. C9×Dic3C2×C32semidirect
92. C3×3−1+2C2×Dic3semidirect
93. C3×3−1+2C2×C3×C4semidirect
95. C32×C9C2×C3×C4semidirect
137. AGL(1,ℤ/9)C3×Dic3direct (N × H)
139. C3×D9C3×Dic3semidirect
140. C3×D9C32×C4semidirect
165. C3×Dic3AGL(1,ℤ/9)direct (N × H)
189. 3−1+2C2×C3×Dic3semidirect
198. C3×C9C2×C3×Dic3semidirect
199. C3×C9C2×C32×C4semidirect
237. D9C32×Dic3semidirect
253. Dic3C3×AGL(1,ℤ/9)direct (N × H)
267. C32C4×AGL(1,ℤ/9)semidirect
276. C9C2×C32×Dic3semidirect
302. C3Dic3×AGL(1,ℤ/9)direct (N × H)
303. C3C3×C4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/9)2
C3×C3⋊(3−1+2⋊C4)2
C3×Dic3×3−1+22
C32×D3×Dic33
C3×C4×AGL(1,ℤ/9)3
Dic3×AGL(1,ℤ/9)3
C3×Dic3×D93