Finite Group Databasebeta

← C2×C3×#6194 #13048580

Subgroups — C2×C3×#6194 #13048580

Order 1296.

488 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 106 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, complemented
2C22×C32×D9C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊Dic9C211yesmaximal, characteristic, complemented
4C3×#6194C214yesmaximal, direct product factor
5C2×C3×#625—31nomaximal
6C2×C3×#611—31nomaximal
7C2×#6194C311yesmaximal, characteristic, direct product factor
8C22×C32×C9C2211yescharacteristic
9C2×C3×D3×C9—22no—
10C2×C3×D3×C9C2212yescomplemented
11C2×C32×D9—22no—
12C2×C32×D9C2212yescomplemented
13C3×C3⋊Dic9C2212yescomplemented
14C22×C32×D3—31no—
15C22×C32×D3D311yescharacteristic
16C2×C3×C3⋊Dic3—31no—
17C3×#625—34no—
18C23×C3×C9—31no—
19C22×D3×C9—21no—
20C22×D3×C9C2×C311yescharacteristic, complemented
21C22×C3×D9—21no—
22C22×C3×D9D311yescharacteristic, complemented
23C22×C3×D9C2×C311yescharacteristic, complemented
24C2×C3⋊Dic9C2×C311yescharacteristic, complemented
25C3×#611—34no—
26C2×C3×Dic9—31no—
27C2×C3×Dic9—61no—
28#6194C2×C314yesdirect product factor
29C2×C32×C9D412yescomplemented
30C2×C32×C9C2311yescharacteristic
31C3×D3×C9—22no—
32C32×D9—22no—
33C2×C3×#67—91no—
34C2×C3×#67—91no—
35C2×#625—31no—
36C2×#611—31no—
37C22×C33C2×D311yescharacteristic
38C2×C32×D3—22no—
39C2×C32×D3—32no—
40C2×C32×D3—62no—
41C2×C32×D3C2×D312yes—
42C3×C3⋊Dic3—32no—
43C22×C3×C9—21no—
44C22×C3×C9—21no—
45C22×C3×C9—21no—
46C22×C3×C9—32no—
47C22×C3×C9—41no—
48C22×C3×C9—62no—
49C22×C3×C9C2×D311yescharacteristic
50C22×C3×C9C22×C311yescharacteristic
51C2×D3×C9—24no—
52C2×D3×C9—22no—
53C2×D3×C9—22no—
54C2×D3×C9C22×C312yescomplemented
55C2×C3×D9—24no—
56C2×C3×D9—22no—
57C2×C3×D9—22no—
58C2×C3×D9—22no—
59C2×C3×D9C2×D312yescomplemented
60C2×C3×D9C22×C312yescomplemented
61C3⋊Dic9C22×C312yescomplemented
62C3×Dic9—32no—
63C3×Dic9—62no—
64C32×C9C2×D411yescharacteristic, complemented, Sylow
65C23×C32—31no—
66C23×C32—91no—
67C22×C3×D3—21no—
68C22×C3×D3—31no—
69C22×C3×D3—31no—
70C22×C3×D3—61no—
71C22×C3×D3D911yescharacteristic, complemented
72C22×C3×D3C3×D311yescharacteristic
73C2×C3⋊Dic3—31no—
74C3×#67—94no—
75C3×#67—94no—
76C2×C3×Dic3—91no—
77C2×C3×Dic3—91no—
78C2×C3×Dic3—181no—
79#625—34no—
80C23×C9—31no—
81C23×C9—61no—
82C22×D9C3×D311yescharacteristic, complemented
83#611—34no—
84C2×Dic9—31no—
85C2×Dic9—61no—
86C2×C33#6712yes—
87C2×C33C22×D311yescharacteristic
88C32×D3—22no—
89C32×D3—62no—
90C2×C3×C9—22no—
91C2×C3×C9—22no—
92C2×C3×C9—21no—
93C2×C3×C9—21no—
94C2×C3×C9—22no—
95C2×C3×C9—21no—
96C2×C3×C9—42no—
97C2×C3×C9—41no—
98C2×C3×C9—62no—
99C2×C3×C9#6712yescomplemented
100C2×C3×C9C3×D412yescomplemented
101C2×C3×C9C22×D311yescharacteristic
102C2×C3×C9C23×C311yescharacteristic
103D3×C9—24no—
104D3×C9—22no—
105C3×D9—24no—
106C3×D9—22no—
107C3×D9—22no—
108C2×C3×D4—271no—
109C2×#67—91no—
110C2×#67—91no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—32no—
115C22×C32—41no—
116C22×C32—62no—
117C22×C32—92no—
118C22×C32—182no—
119C22×C32C2×D911yescharacteristic
120C22×C32D3211yescharacteristic
121C22×C32C2×C3×D311yescharacteristic
122C2×C3×D3—24no—
123C2×C3×D3—22no—
124C2×C3×D3—22no—
125C2×C3×D3—22no—
126C2×C3×D3—32no—
127C2×C3×D3—32no—
128C2×C3×D3—62no—
129C2×C3×D3—64no—
130C2×C3×D3—62no—
131C2×C3×D3—62no—
132C2×C3×D3C2×D912yescomplemented
133C2×C3×D3C2×C3×D312yes—
134C3⋊Dic3—32no—
135C3×Dic3—92no—
136C3×Dic3—92no—
137C3×Dic3—182no—
138C22×C9—21no—
139C22×C9—21no—
140C22×C9—32no—
141C22×C9—41no—
142C22×C9—62no—
143C22×C9—64no—
144C22×C9—62no—
145C22×C9C2×C3×D311yescharacteristic
146C2×D9—22no—
147C2×D9C2×C3×D312yescomplemented
148Dic9—32no—
149Dic9—62no—
150C33C2×#6711yescharacteristic
151C3×C9—21no—
152C3×C9—21no—
153C3×C9—21no—
154C3×C9—41no—
155C3×C9C2×#6711yescharacteristic, complemented
156C3×C9C2×C3×D411yescharacteristic, complemented
157C23×C3—31no—
158C23×C3—31no—
159C23×C3—61no—
160C23×C3—91no—
161C23×C3—91no—
162C23×C3—181no—
163C22×D3—31no—
164C22×D3C3×D911yescharacteristic, complemented
165C3×D4—274no—
166C2×C3×C4—271no—
167#67—94no—
168#67—94no—
169C2×Dic3—91no—
170C2×Dic3—91no—
171C2×Dic3—181no—
172C2×C32—22no—
173C2×C32—22no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—22no—
177C2×C32—21no—
178C2×C32—42no—
179C2×C32—41no—
180C2×C32—62no—
181C2×C32—182no—
182C2×C32#61112yescomplemented
183C2×C32C22×D911yescharacteristic
184C2×C32#62512yes—
185C2×C32C3×#6712yes—
186C2×C32C2×D3211yescharacteristic
187C2×C32C22×C3×D311yescharacteristic
188C3×D3—24no—
189C3×D3—22no—
190C3×D3—22no—
191C3×D3—64no—
192C3×D3—62no—
193C3×D3—62no—
194C2×C9—22no—
195C2×C9—22no—
196C2×C9—21no—
197C2×C9—21no—
198C2×C9—42no—
199C2×C9—41no—
200C2×C9—64no—
201C2×C9—62no—
202C2×C9C3×#6712yescomplemented
203C2×C9C22×C3×D311yescharacteristic
204D9—22no—
205C2×D4—271noSylow
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—32no—
210C22×C3—32no—
211C22×C3—41no—
212C22×C3—62no—
213C22×C3—64no—
214C22×C3—62no—
215C22×C3—62no—
216C22×C3—92no—
217C22×C3—92no—
218C22×C3—182no—
219C22×C3—184no—
220C22×C3—182no—
221C22×C3—182no—
222C22×C3D3×D911yescharacteristic
223C22×C3C2×C3×D911yescharacteristic
224C22×C3C3×D3211yescharacteristic
225C2×D3—22no—
226C2×D3—32no—
227C2×D3—62no—
228C2×D3C2×C3×D912yescomplemented
229C3×C4—272no—
230Dic3—92no—
231Dic3—92no—
232Dic3—182no—
233C32—21no—
234C32—21no—
235C32—21no—
236C32—41no—
237C32C2×#61111yescharacteristic, complemented
238C32C2×#62511yescharacteristic
239C32C2×C3×#6711yescharacteristic
240C9—21no—
241C9—21no—
242C9—41no—
243C9C2×C3×#6711yescharacteristic, complemented
244C23—31no—
245C23—91no—
246D4—274no—
247C2×C4—271no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—41no—
255C2×C3—42no—
256C2×C3—64no—
257C2×C3—62no—
258C2×C3—62no—
259C2×C3—184no—
260C2×C3—182no—
261C2×C3—182no—
262C2×C3#619412yesdirect product factor
263C2×C3C3×#61112yescomplemented
264C2×C3C2×D3×D911yescharacteristic
265C2×C3C22×C3×D911yescharacteristic
266C2×C3C3×#62512yes—
267C2×C3C2×C3×D3211yescharacteristic
268D3—22no—
269D3—62no—
270C22—32no—
271C22—62no—
272C22—92no—
273C22—182no—
274C22C3×D3×D911yescharacteristic
275C4—272no—
276C3—21no—
277C3—21no—
278C3—21no—
279C3—41no—
280C3C2×#619411yescharacteristic, direct product factor
281C3C2×C3×#61111yescharacteristic, complemented
282C3C2×C3×#62511yescharacteristic
283C2—62no—
284C2—182no—
285C2C3×#619412yesdirect product factor
286C2C2×C3×D3×D911yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×D9
commutator, cosocleC2×C3×C9C23×C3
FrattiniC2×C3C2×C3×D32
FittingC22×C32×C9C22
socleC22×C33C2×D3
hypocenterC3×C9C2×C3×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×D3×C9C2semidirect
2. C22×C32×D9C2semidirect
3. C2×C3×C3⋊Dic9C2semidirect
4. C3×#6194C2direct (N × H)
7. C2×#6194C3direct (N × H)
10. C2×C3×D3×C9C22semidirect
12. C2×C32×D9C22semidirect
13. C3×C3⋊Dic9C22semidirect
20. C22×D3×C9C2×C3semidirect
22. C22×C3×D9D3semidirect
23. C22×C3×D9C2×C3semidirect
24. C2×C3⋊Dic9C2×C3semidirect
28. #6194C2×C3direct (N × H)
29. C2×C32×C9D4semidirect
54. C2×D3×C9C22×C3semidirect
59. C2×C3×D9C2×D3semidirect
60. C2×C3×D9C22×C3semidirect
61. C3⋊Dic9C22×C3semidirect
64. C32×C9C2×D4semidirect
71. C22×C3×D3D9semidirect
82. C22×D9C3×D3semidirect
99. C2×C3×C9#67semidirect
100. C2×C3×C9C3×D4semidirect
132. C2×C3×D3C2×D9semidirect
147. C2×D9C2×C3×D3semidirect
155. C3×C9C2×#67semidirect
156. C3×C9C2×C3×D4semidirect
164. C22×D3C3×D9semidirect
182. C2×C32#611semidirect
202. C2×C9C3×#67semidirect
228. C2×D3C2×C3×D9semidirect
237. C32C2×#611semidirect
243. C9C2×C3×#67semidirect
262. C2×C3#6194direct (N × H)
263. C2×C3C3×#611semidirect
280. C3C2×#6194direct (N × H)
281. C3C2×C3×#611semidirect
285. C2C3×#6194direct (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
C22×C32×D92
C2×C3×C3⋊Dic92
C3×#61942
C2×C3×#6253
C2×C3×#6113
C2×#61943