Finite Group Databasebeta

← C2×C3×D3×Dic9 #13048578

Subgroups — C2×C3×D3×Dic9 #13048578

Order 1296.

464 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 154 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×D3×C9C211yesmaximal, characteristic
2C2×C3×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C2×C32×Dic9C211yesmaximal, characteristic, complemented
4C3×D3×Dic9C214yesmaximal, direct product factor
5C2×C3×D3×Dic3—31nomaximal
6C22×C3×Dic9—31nomaximal
7C2×D3×Dic9C311yesmaximal, characteristic, direct product factor
8C22×C32×C9C2211yescharacteristic
9C2×C3×D3×C9C414yescomplemented
10C2×C3×D3×C9C2212yes—
11C3×C3⋊Dic9C2212yescomplemented
12C32×Dic9C2212yescomplemented
13C22×C32×D3D311yescharacteristic
14C2×C3×C3⋊Dic3—31no—
15C2×C32×Dic3—31no—
16C3×D3×Dic3—34no—
17C23×C3×C9—31no—
18C22×D3×C9—21no—
19C22×D3×C9C2×C311yescharacteristic
20C2×C3⋊Dic9C2×C311yescharacteristic, complemented
21C2×C3×Dic9—21no—
22C2×C3×Dic9—31no—
23C2×C3×Dic9—34no—
24C2×C3×Dic9—61no—
25C2×C3×Dic9D311yescharacteristic, direct product factor
26C2×C3×Dic9C2×C311yescharacteristic, complemented
27D3×Dic9C2×C314yesdirect product factor
28C2×C32×C9C2×C412yescomplemented
29C2×C32×C9C2311yescharacteristic
30C3×D3×C9C2×C414yescomplemented
31C22×C3×Dic3—91no—
32C2×C3×C4×D3—91no—
33C2×D3×Dic3—31no—
34C22×Dic9—31no—
35C22×C33C2×D311yescharacteristic
36C2×C32×D3Dic314yes—
37C2×C32×D3C2×D312yes—
38C3×C3⋊Dic3—32no—
39C32×Dic3—32no—
40C22×C3×C9—21no—
41C22×C3×C9—21no—
42C22×C3×C9—21no—
43C22×C3×C9—34no—
44C22×C3×C9—32no—
45C22×C3×C9—41no—
46C22×C3×C9C2×D311yescharacteristic
47C22×C3×C9C22×C311yescharacteristic
48C2×D3×C9—24no—
49C2×D3×C9—22no—
50C2×D3×C9C3×C414yescomplemented
51C2×D3×C9C22×C312yes—
52C3⋊Dic9C22×C312yescomplemented
53C3×Dic9—22no—
54C3×Dic9—32no—
55C3×Dic9—62no—
56C3×Dic9C2×D312yesdirect product factor
57C3×Dic9C22×C312yescomplemented
58C32×C9C22×C411yescharacteristic, complemented, Sylow
59C23×C32—31no—
60C22×C3×D3—21no—
61C22×C3×D3D911yescharacteristic
62C22×C3×D3C3×D311yescharacteristic
63C2×C32×C4—91no—
64C2×C3⋊Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—94no—
69C2×C3×Dic3—91no—
70C2×C3×Dic3—91no—
71C2×C3×Dic3—181no—
72C3×C4×D3—94no—
73D3×Dic3—34no—
74C23×C9—31no—
75C23×C9—61no—
76C2×Dic9—31no—
77C2×Dic9—34no—
78C2×Dic9—61no—
79C2×Dic9C3×D311yescharacteristic, direct product factor
80C2×C33C2×Dic312yes—
81C2×C33C22×D311yescharacteristic
82C32×D3C2×Dic314yes—
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—34no—
90C2×C3×C9—42no—
91C2×C3×C9—41no—
92C2×C3×C9C4×D312yescomplemented
93C2×C3×C9C2×C3×C412yescomplemented
94C2×C3×C9C22×D311yescharacteristic
95C2×C3×C9C23×C311yescharacteristic
96D3×C9—24no—
97D3×C9C2×C3×C414yescomplemented
98C22×C3×C4—271no—
99C22×Dic3—91no—
100C2×C4×D3—91no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—34no—
105C22×C32—32no—
106C22×C32—41no—
107C22×C32C2×D911yescharacteristic
108C22×C32D3211yescharacteristic
109C22×C32C2×C3×D311yescharacteristic
110C2×C3×D3—24no—
111C2×C3×D3—22no—
112C2×C3×D3Dic914yesdirect product factor
113C2×C3×D3C2×D912yes—
114C2×C3×D3C3×Dic314yes—
115C2×C3×D3C2×C3×D312yes—
116C32×C4—92no—
117C3⋊Dic3—32no—
118C3×Dic3—32no—
119C3×Dic3—32no—
120C3×Dic3—62no—
121C3×Dic3—92no—
122C3×Dic3—92no—
123C3×Dic3—182no—
124C22×C9—21no—
125C22×C9—21no—
126C22×C9—34no—
127C22×C9—32no—
128C22×C9—41no—
129C22×C9—64no—
130C22×C9—62no—
131C22×C9C2×C3×D311yescharacteristic
132Dic9—32no—
133Dic9—62no—
134Dic9C2×C3×D312yesdirect product factor
135C33C22×Dic311yescharacteristic
136C3×C9—21no—
137C3×C9—21no—
138C3×C9—21no—
139C3×C9—41no—
140C3×C9C2×C4×D311yescharacteristic, complemented
141C3×C9C22×C3×C411yescharacteristic, complemented
142C23×C3—31no—
143C23×C3—31no—
144C23×C3—61no—
145C22×D3C3×D911yescharacteristic
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—181no—
149C2×C3×C4—274no—
150C2×C3×C4—271no—
151C2×Dic3—31no—
152C2×Dic3—94no—
153C2×Dic3—91no—
154C2×Dic3—91no—
155C2×Dic3—181no—
156C4×D3—94no—
157C2×C32—22no—
158C2×C32—22no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—22no—
162C2×C32—21no—
163C2×C32—34no—
164C2×C32—42no—
165C2×C32—41no—
166C2×C32C2×Dic912yescomplemented
167C2×C32C22×D911yescharacteristic
168C2×C32D3×Dic312yes—
169C2×C32C2×C3×Dic312yes—
170C2×C32C2×D3211yescharacteristic
171C2×C32C22×C3×D311yescharacteristic
172C3×D3—24no—
173C3×D3C2×Dic914yesdirect product factor
174C3×D3C2×C3×Dic314yes—
175C2×C9—22no—
176C2×C9—22no—
177C2×C9—21no—
178C2×C9—21no—
179C2×C9—34no—
180C2×C9—42no—
181C2×C9—41no—
182C2×C9—64no—
183C2×C9C3×C4×D312yescomplemented
184C2×C9C22×C3×D311yescharacteristic
185C22×C4—271noSylow
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—34no—
190C22×C3—32no—
191C22×C3—34no—
192C22×C3—32no—
193C22×C3—41no—
194C22×C3—62no—
195C22×C3—64no—
196C22×C3D3×D911yescharacteristic
197C22×C3C2×C3×D911yescharacteristic
198C22×C3C3×D3211yescharacteristic
199C2×D3C3×Dic914yesdirect product factor
200C2×D3C2×C3×D912yes—
201C3×C4—92no—
202C3×C4—92no—
203C3×C4—182no—
204C3×C4—272no—
205Dic3—32no—
206Dic3—92no—
207Dic3—92no—
208Dic3—182no—
209C32—21no—
210C32—21no—
211C32—21no—
212C32—41no—
213C32C22×Dic911yescharacteristic, complemented
214C32C2×D3×Dic311yescharacteristic
215C32C22×C3×Dic311yescharacteristic
216C9—21no—
217C9—21no—
218C9—41no—
219C9C2×C3×C4×D311yescharacteristic, complemented
220C23—31no—
221C2×C4—91no—
222C2×C4—274no—
223C2×C4—271no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—34no—
231C2×C3—34no—
232C2×C3—41no—
233C2×C3—42no—
234C2×C3—64no—
235C2×C3D3×Dic912yesdirect product factor
236C2×C3C2×C3×Dic912yescomplemented
237C2×C3C2×D3×D911yescharacteristic
238C2×C3C22×C3×D911yescharacteristic
239C2×C3C3×D3×Dic312yes—
240C2×C3C2×C3×D3211yescharacteristic
241D3C2×C3×Dic914yesdirect product factor
242C22—32no—
243C22—34no—
244C22C3×D3×D911yescharacteristic
245C4—92no—
246C4—272no—
247C3—21no—
248C3—21no—
249C3—21no—
250C3—41no—
251C3C2×D3×Dic911yescharacteristic, direct product factor
252C3C22×C3×Dic911yescharacteristic, complemented
253C3C2×C3×D3×Dic311yescharacteristic
254C2—34no—
255C2C3×D3×Dic912yesdirect 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×C32×D3D3
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×C3⋊Dic9C2semidirect
3. C2×C32×Dic9C2semidirect
4. C3×D3×Dic9C2direct (N × H)
7. C2×D3×Dic9C3direct (N × H)
9. C2×C3×D3×C9C4semidirect
11. C3×C3⋊Dic9C22semidirect
12. C32×Dic9C22semidirect
20. C2×C3⋊Dic9C2×C3semidirect
25. C2×C3×Dic9D3direct (N × H)
26. C2×C3×Dic9C2×C3semidirect
27. D3×Dic9C2×C3direct (N × H)
28. C2×C32×C9C2×C4semidirect
30. C3×D3×C9C2×C4semidirect
50. C2×D3×C9C3×C4semidirect
52. C3⋊Dic9C22×C3semidirect
56. C3×Dic9C2×D3direct (N × H)
57. C3×Dic9C22×C3semidirect
58. C32×C9C22×C4semidirect
79. C2×Dic9C3×D3direct (N × H)
92. C2×C3×C9C4×D3semidirect
93. C2×C3×C9C2×C3×C4semidirect
97. D3×C9C2×C3×C4semidirect
112. C2×C3×D3Dic9direct (N × H)
134. Dic9C2×C3×D3direct (N × H)
140. C3×C9C2×C4×D3semidirect
141. C3×C9C22×C3×C4semidirect
166. C2×C32C2×Dic9semidirect
173. C3×D3C2×Dic9direct (N × H)
183. C2×C9C3×C4×D3semidirect
199. C2×D3C3×Dic9direct (N × H)
213. C32C22×Dic9semidirect
219. C9C2×C3×C4×D3semidirect
235. C2×C3D3×Dic9direct (N × H)
236. C2×C3C2×C3×Dic9semidirect
241. D3C2×C3×Dic9direct (N × H)
251. C3C2×D3×Dic9direct (N × H)
252. C3C22×C3×Dic9semidirect
255. C2C3×D3×Dic9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D3×C92
C2×C3×C3⋊Dic92
C2×C32×Dic92
C3×D3×Dic92
C2×C3×D3×Dic33
C22×C3×Dic93
C2×D3×Dic93