Finite Group Databasebeta

← C2×C3×(C3⋊D9)⋊C4 #13048576

Subgroups — C2×C3×(C3⋊D9)⋊C4 #13048576

Order 1296.

464 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 122 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×C3⋊D9C211yesmaximal, characteristic
2C2×C3×C9×Dic3C211yesmaximal, characteristic, complemented
3C2×C32×Dic9C211yesmaximal, characteristic, complemented
4C3×(C3⋊D9)⋊C4C214yesmaximal, direct product factor
5C2×C3×#624—31nomaximal
6C2×C3×C4×D9—31nomaximal
7C2×(C3⋊D9)⋊C4C311yesmaximal, characteristic, direct product factor
8C22×C32×C9C2211yescharacteristic
9C2×C3×C3⋊D9C414yescomplemented
10C2×C3×C3⋊D9C2212yes—
11C3×C9×Dic3C2212yescomplemented
12C32×Dic9C2212yescomplemented
13C22×C3×C3⋊D3—31no—
14C2×C32×Dic3—31no—
15C2×C32×Dic3D311yescharacteristic
16C3×#624—34no—
17C22×C3⋊D9C2×C311yescharacteristic
18C22×C3×D9—31no—
19C22×C3×D9—61no—
20C2×C3×C4×C9—31no—
21C2×C9×Dic3—21no—
22C2×C9×Dic3C2×C311yescharacteristic, complemented
23C2×C3×Dic9—21no—
24C2×C3×Dic9D311yescharacteristic, complemented
25C2×C3×Dic9C2×C311yescharacteristic, complemented
26C3×C4×D9—34no—
27(C3⋊D9)⋊C4C2×C314yesdirect product factor
28C2×C32×C9C2×C412yescomplemented
29C2×C32×C9C2311yescharacteristic
30C3×C3⋊D9C2×C414yescomplemented
31C2×C3×C4×D3—91no—
32C2×C3×C4×D3—91no—
33C2×#624—31no—
34C2×C4×D9—31no—
35C22×C33C2×D311yescharacteristic
36C2×C3×C3⋊D3—34no—
37C2×C3×C3⋊D3—32no—
38C32×Dic3—32no—
39C32×Dic3C2×D312yes—
40C22×C3×C9—21no—
41C22×C3×C9—21no—
42C22×C3×C9—21no—
43C22×C3×C9—41no—
44C22×C3×C9C2×D311yescharacteristic
45C22×C3×C9C22×C311yescharacteristic
46C2×C3⋊D9C3×C414yescomplemented
47C2×C3⋊D9C22×C312yes—
48C2×C3×D9—34no—
49C2×C3×D9—32no—
50C2×C3×D9—64no—
51C2×C3×D9—62no—
52C3×C4×C9—32no—
53C9×Dic3—22no—
54C9×Dic3C22×C312yescomplemented
55C3×Dic9—22no—
56C3×Dic9C2×D312yescomplemented
57C3×Dic9C22×C312yescomplemented
58C32×C9C22×C411yescharacteristic, complemented, Sylow
59C22×C3⋊D3—31no—
60C22×C3×D3—91no—
61C22×C3×D3—91no—
62C22×C3×D3—181no—
63C2×C32×C4—31no—
64C2×C32×C4—91no—
65C2×C3×Dic3—21no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—31no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3D911yescharacteristic, complemented
70C2×C3×Dic3C3×D311yescharacteristic
71C3×C4×D3—94no—
72C3×C4×D3—94no—
73#624—34no—
74C22×D9—31no—
75C22×D9—61no—
76C2×C4×C9—31no—
77C2×C4×C9—61no—
78C2×Dic9C3×D311yescharacteristic, complemented
79C4×D9—34no—
80C2×C33C4×D312yes—
81C2×C33C22×D311yescharacteristic
82C3×C3⋊D3—34no—
83C2×C3×C9—22no—
84C2×C3×C9—22no—
85C2×C3×C9—21no—
86C2×C3×C9—21no—
87C2×C3×C9—22no—
88C2×C3×C9—21no—
89C2×C3×C9—42no—
90C2×C3×C9—41no—
91C2×C3×C9C4×D312yescomplemented
92C2×C3×C9C2×C3×C412yescomplemented
93C2×C3×C9C22×D311yescharacteristic
94C2×C3×C9C23×C311yescharacteristic
95C3⋊D9C2×C3×C414yescomplemented
96C3×D9—34no—
97C3×D9—64no—
98C22×C3×C4—271no—
99C2×C4×D3—91no—
100C2×C4×D3—91no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—41no—
105C22×C32C2×D911yescharacteristic
106C22×C32D3211yescharacteristic
107C22×C32C2×C3×D311yescharacteristic
108C2×C3⋊D3—34no—
109C2×C3⋊D3—32no—
110C2×C3×D3—94no—
111C2×C3×D3—92no—
112C2×C3×D3—94no—
113C2×C3×D3—92no—
114C2×C3×D3—184no—
115C2×C3×D3—182no—
116C32×C4—32no—
117C32×C4—92no—
118C3×Dic3—22no—
119C3×Dic3—32no—
120C3×Dic3—32no—
121C3×Dic3—62no—
122C3×Dic3C2×D912yescomplemented
123C3×Dic3C2×C3×D312yes—
124C22×C9—21no—
125C22×C9—21no—
126C22×C9—41no—
127C22×C9C2×C3×D311yescharacteristic
128C2×D9—34no—
129C2×D9—32no—
130C2×D9—64no—
131C2×D9—62no—
132C4×C9—32no—
133C4×C9—62no—
134Dic9C2×C3×D312yescomplemented
135C33C2×C4×D311yescharacteristic
136C3×C9—21no—
137C3×C9—21no—
138C3×C9—21no—
139C3×C9—41no—
140C3×C9C2×C4×D311yescharacteristic, complemented
141C3×C9C22×C3×C411yescharacteristic, complemented
142C23×C3—271no—
143C22×D3—91no—
144C22×D3—91no—
145C22×D3—181no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—61no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—181no—
152C2×C3×C4—274no—
153C2×Dic3—31no—
154C2×Dic3C3×D911yescharacteristic, complemented
155C4×D3—94no—
156C4×D3—94no—
157C2×C32—22no—
158C2×C32—22no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—22no—
162C2×C32—21no—
163C2×C32—42no—
164C2×C32—41no—
165C2×C32C4×D912yescomplemented
166C2×C32C22×D911yescharacteristic
167C2×C32#62412yes—
168C2×C32C3×C4×D312yes—
169C2×C32C2×D3211yescharacteristic
170C2×C32C22×C3×D311yescharacteristic
171C3⋊D3—34no—
172C3×D3—94no—
173C3×D3—94no—
174C3×D3—184no—
175C2×C9—22no—
176C2×C9—22no—
177C2×C9—21no—
178C2×C9—21no—
179C2×C9—42no—
180C2×C9—41no—
181C2×C9C3×C4×D312yescomplemented
182C2×C9C22×C3×D311yescharacteristic
183D9—34no—
184D9—64no—
185C22×C4—271noSylow
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—41no—
190C22×C3—274no—
191C22×C3—272no—
192C22×C3D3×D911yescharacteristic
193C22×C3C2×C3×D911yescharacteristic
194C22×C3C3×D3211yescharacteristic
195C2×D3—94no—
196C2×D3—92no—
197C2×D3—94no—
198C2×D3—92no—
199C2×D3—182no—
200C2×D3—184no—
201C3×C4—32no—
202C3×C4—32no—
203C3×C4—62no—
204C3×C4—92no—
205C3×C4—92no—
206C3×C4—182no—
207Dic3—32no—
208Dic3C2×C3×D912yescomplemented
209C32—21no—
210C32—21no—
211C32—21no—
212C32—41no—
213C32C2×C4×D911yescharacteristic, complemented
214C32C2×#62411yescharacteristic
215C32C2×C3×C4×D311yescharacteristic
216C9—21no—
217C9—21no—
218C9—41no—
219C9C2×C3×C4×D311yescharacteristic, complemented
220C23—271no—
221C2×C4—31no—
222C2×C4—91no—
223C2×C4—274no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—41no—
231C2×C3—42no—
232C2×C3—274no—
233C2×C3(C3⋊D9)⋊C412yesdirect product factor
234C2×C3C3×C4×D912yescomplemented
235C2×C3C2×D3×D911yescharacteristic
236C2×C3C22×C3×D911yescharacteristic
237C2×C3C3×#62412yes—
238C2×C3C2×C3×D3211yescharacteristic
239D3—94no—
240D3—94no—
241D3—184no—
242C22—272no—
243C22—274no—
244C22C3×D3×D911yescharacteristic
245C4—32no—
246C4—92no—
247C3—21no—
248C3—21no—
249C3—21no—
250C3—41no—
251C3C2×(C3⋊D9)⋊C411yescharacteristic, direct product factor
252C3C2×C3×C4×D911yescharacteristic, complemented
253C3C2×C3×#62411yescharacteristic
254C2—274no—
255C2C3×(C3⋊D9)⋊C412yesdirect product factor
256C2C2×C3×D3×D911yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×D9
commutator, hypocenterC3×C9C22×C3×C4
FrattiniC2×C3C2×C3×D32
FittingC22×C32×C9C22
socleC22×C33C2×D3
prime radicalC22×C3×C3⋊D9C2
cosocleC2×C3×C9C23×C3
LCSG ▷ C3×C9
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×Dic3C2semidirect
3. C2×C32×Dic9C2semidirect
4. C3×(C3⋊D9)⋊C4C2direct (N × H)
7. C2×(C3⋊D9)⋊C4C3direct (N × H)
9. C2×C3×C3⋊D9C4semidirect
11. C3×C9×Dic3C22semidirect
12. C32×Dic9C22semidirect
22. C2×C9×Dic3C2×C3semidirect
24. C2×C3×Dic9D3semidirect
25. C2×C3×Dic9C2×C3semidirect
27. (C3⋊D9)⋊C4C2×C3direct (N × H)
28. C2×C32×C9C2×C4semidirect
30. C3×C3⋊D9C2×C4semidirect
46. C2×C3⋊D9C3×C4semidirect
54. C9×Dic3C22×C3semidirect
56. C3×Dic9C2×D3semidirect
57. C3×Dic9C22×C3semidirect
58. C32×C9C22×C4semidirect
69. C2×C3×Dic3D9semidirect
78. C2×Dic9C3×D3semidirect
91. C2×C3×C9C4×D3semidirect
92. C2×C3×C9C2×C3×C4semidirect
95. C3⋊D9C2×C3×C4semidirect
122. C3×Dic3C2×D9semidirect
134. Dic9C2×C3×D3semidirect
140. C3×C9C2×C4×D3semidirect
141. C3×C9C22×C3×C4semidirect
154. C2×Dic3C3×D9semidirect
165. C2×C32C4×D9semidirect
181. C2×C9C3×C4×D3semidirect
208. Dic3C2×C3×D9semidirect
213. C32C2×C4×D9semidirect
219. C9C2×C3×C4×D3semidirect
233. C2×C3(C3⋊D9)⋊C4direct (N × H)
234. C2×C3C3×C4×D9semidirect
251. C3C2×(C3⋊D9)⋊C4direct (N × H)
252. C3C2×C3×C4×D9semidirect
255. C2C3×(C3⋊D9)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊D92
C2×C3×C9×Dic32
C2×C32×Dic92
C3×(C3⋊D9)⋊C42
C2×C3×#6243
C2×C3×C4×D93
C2×(C3⋊D9)⋊C43