Finite Group Databasebeta

← C2×C3×#6192 #13048577

Subgroups — C2×C3×#6192 #13048577

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×C3⋊D9C211yesmaximal, characteristic, complemented
2C22×C32×D9C211yesmaximal, characteristic, complemented
3C2×C3×C9×Dic3C211yesmaximal, characteristic, complemented
4C3×#6192C214yesmaximal, direct product factor
5C2×C3×#626—31nomaximal
6C2×C3×D36—31nomaximal
7C2×#6192C311yesmaximal, characteristic, direct product factor
8C22×C32×C9C2211yescharacteristic
9C2×C3×C3⋊D9—22no—
10C2×C3×C3⋊D9C2212yescomplemented
11C2×C32×D9—22no—
12C2×C32×D9C2212yescomplemented
13C3×C9×Dic3C2212yescomplemented
14C22×C3×C3⋊D3—31no—
15C22×C32×D3—31no—
16C2×C32×Dic3D311yescharacteristic
17C3×#626—34no—
18C22×C3⋊D9C2×C311yescharacteristic, complemented
19C22×C3×D9—21no—
20C22×C3×D9—31no—
21C22×C3×D9—61no—
22C22×C3×D9D311yescharacteristic, complemented
23C22×C3×D9C2×C311yescharacteristic, complemented
24C2×C3×C4×C9—31no—
25C2×C9×Dic3—21no—
26C2×C9×Dic3C2×C311yescharacteristic, complemented
27C3×D36—34no—
28#6192C2×C314yesdirect product factor
29C2×C32×C9D412yescomplemented
30C2×C32×C9C2311yescharacteristic
31C3×C3⋊D9—22no—
32C32×D9—22no—
33C2×C3×#67—91no—
34C2×C3×D12—91no—
35C2×#626—31no—
36C2×D36—31no—
37C22×C33C2×D311yescharacteristic
38C2×C3×C3⋊D3—32no—
39C2×C3×C3⋊D3—62no—
40C2×C32×D3—32no—
41C2×C32×D3—62no—
42C32×Dic3C2×D312yes—
43C22×C3×C9—21no—
44C22×C3×C9—21no—
45C22×C3×C9—21no—
46C22×C3×C9—41no—
47C22×C3×C9C2×D311yescharacteristic
48C22×C3×C9C22×C311yescharacteristic
49C2×C3⋊D9—22no—
50C2×C3⋊D9C22×C312yescomplemented
51C2×C3×D9—24no—
52C2×C3×D9—22no—
53C2×C3×D9—22no—
54C2×C3×D9—22no—
55C2×C3×D9—32no—
56C2×C3×D9—62no—
57C2×C3×D9—64no—
58C2×C3×D9—62no—
59C2×C3×D9C2×D312yescomplemented
60C2×C3×D9C22×C312yescomplemented
61C3×C4×C9—32no—
62C9×Dic3—22no—
63C9×Dic3C22×C312yescomplemented
64C32×C9C2×D411yescharacteristic, complemented, Sylow
65C23×C32—91no—
66C22×C3⋊D3—31no—
67C22×C3×D3—31no—
68C22×C3×D3—31no—
69C22×C3×D3—61no—
70C22×C3×D3—91no—
71C22×C3×D3—91no—
72C22×C3×D3—181no—
73C2×C32×C4—31no—
74C3×#67—94no—
75C2×C3×Dic3—21no—
76C2×C3×Dic3D911yescharacteristic, complemented
77C2×C3×Dic3C3×D311yescharacteristic
78C3×D12—94no—
79#626—34no—
80C22×D9—31no—
81C22×D9—61no—
82C22×D9C3×D311yescharacteristic, complemented
83C2×C4×C9—31no—
84C2×C4×C9—61no—
85D36—34no—
86C2×C33D1212yes—
87C2×C33C22×D311yescharacteristic
88C3×C3⋊D3—62no—
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#6712yescomplemented
99C2×C3×C9C3×D412yescomplemented
100C2×C3×C9C22×D311yescharacteristic
101C2×C3×C9C23×C311yescharacteristic
102C3⋊D9—22no—
103C3×D9—24no—
104C3×D9—22no—
105C3×D9—22no—
106C3×D9—64no—
107C3×D9—62no—
108C2×C3×D4—271no—
109C2×#67—91no—
110C2×D12—91no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—41no—
115C22×C32—92no—
116C22×C32—182no—
117C22×C32C2×D911yescharacteristic
118C22×C32D3211yescharacteristic
119C22×C32C2×C3×D311yescharacteristic
120C2×C3⋊D3—32no—
121C2×C3⋊D3—62no—
122C2×C3×D3—32no—
123C2×C3×D3—32no—
124C2×C3×D3—62no—
125C2×C3×D3—64no—
126C2×C3×D3—62no—
127C2×C3×D3—62no—
128C2×C3×D3—92no—
129C2×C3×D3—92no—
130C2×C3×D3—182no—
131C2×C3×D3—184no—
132C2×C3×D3—182no—
133C2×C3×D3—182no—
134C32×C4—32no—
135C3×Dic3—22no—
136C3×Dic3C2×D912yescomplemented
137C3×Dic3C2×C3×D312yes—
138C22×C9—21no—
139C22×C9—21no—
140C22×C9—41no—
141C22×C9C2×C3×D311yescharacteristic
142C2×D9—22no—
143C2×D9—32no—
144C2×D9—62no—
145C2×D9—64no—
146C2×D9—62no—
147C2×D9C2×C3×D312yescomplemented
148C4×C9—32no—
149C4×C9—62no—
150C33C2×D1211yescharacteristic
151C3×C9—21no—
152C3×C9—21no—
153C3×C9—21no—
154C3×C9—41no—
155C3×C9C2×#6711yescharacteristic, complemented
156C3×C9C2×C3×D411yescharacteristic, complemented
157C23×C3—91no—
158C23×C3—91no—
159C23×C3—181no—
160C23×C3—271no—
161C22×D3—31no—
162C22×D3—91no—
163C22×D3—91no—
164C22×D3—181no—
165C3×D4—274no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—61no—
169#67—94no—
170C2×Dic3C3×D911yescharacteristic, complemented
171D12—94no—
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—182no—
181C2×C32D3612yescomplemented
182C2×C32C22×D911yescharacteristic
183C2×C32#62612yes—
184C2×C32C3×D1212yes—
185C2×C32C2×D3211yescharacteristic
186C2×C32C22×C3×D311yescharacteristic
187C3⋊D3—62no—
188C3×D3—64no—
189C3×D3—62no—
190C3×D3—62no—
191C3×D3—184no—
192C3×D3—182no—
193C3×D3—182no—
194C2×C9—22no—
195C2×C9—22no—
196C2×C9—21no—
197C2×C9—21no—
198C2×C9—42no—
199C2×C9—41no—
200C2×C9C3×#6712yescomplemented
201C2×C9C22×C3×D311yescharacteristic
202D9—22no—
203D9—64no—
204D9—62no—
205C2×D4—271noSylow
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—41no—
210C22×C3—92no—
211C22×C3—92no—
212C22×C3—182no—
213C22×C3—184no—
214C22×C3—182no—
215C22×C3—182no—
216C22×C3—272no—
217C22×C3—542no—
218C22×C3D3×D911yescharacteristic
219C22×C3C2×C3×D911yescharacteristic
220C22×C3C3×D3211yescharacteristic
221C2×D3—32no—
222C2×D3—62no—
223C2×D3—92no—
224C2×D3—92no—
225C2×D3—182no—
226C2×D3—184no—
227C2×D3—182no—
228C2×D3—182no—
229C3×C4—32no—
230C3×C4—32no—
231C3×C4—62no—
232Dic3C2×C3×D912yescomplemented
233C32—21no—
234C32—21no—
235C32—21no—
236C32—41no—
237C32C2×D3611yescharacteristic, complemented
238C32C2×#62611yescharacteristic
239C32C2×C3×D1211yescharacteristic
240C9—21no—
241C9—21no—
242C9—41no—
243C9C2×C3×#6711yescharacteristic, complemented
244C23—91no—
245C23—271no—
246D4—274no—
247C2×C4—31no—
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—184no—
257C2×C3—182no—
258C2×C3—182no—
259C2×C3—542no—
260C2×C3#619212yesdirect product factor
261C2×C3C3×D3612yescomplemented
262C2×C3C2×D3×D911yescharacteristic
263C2×C3C22×C3×D911yescharacteristic
264C2×C3C3×#62612yes—
265C2×C3C2×C3×D3211yescharacteristic
266D3—62no—
267D3—184no—
268D3—182no—
269D3—182no—
270C22—92no—
271C22—182no—
272C22—272no—
273C22—542no—
274C22C3×D3×D911yescharacteristic
275C4—32no—
276C3—21no—
277C3—21no—
278C3—21no—
279C3—41no—
280C3C2×#619211yescharacteristic, direct product factor
281C3C2×C3×D3611yescharacteristic, complemented
282C3C2×C3×#62611yescharacteristic
283C2—182no—
284C2—542no—
285C2C3×#619212yesdirect 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×C3⋊D9C2semidirect
2. C22×C32×D9C2semidirect
3. C2×C3×C9×Dic3C2semidirect
4. C3×#6192C2direct (N × H)
7. C2×#6192C3direct (N × H)
10. C2×C3×C3⋊D9C22semidirect
12. C2×C32×D9C22semidirect
13. C3×C9×Dic3C22semidirect
18. C22×C3⋊D9C2×C3semidirect
22. C22×C3×D9D3semidirect
23. C22×C3×D9C2×C3semidirect
26. C2×C9×Dic3C2×C3semidirect
28. #6192C2×C3direct (N × H)
29. C2×C32×C9D4semidirect
50. C2×C3⋊D9C22×C3semidirect
59. C2×C3×D9C2×D3semidirect
60. C2×C3×D9C22×C3semidirect
63. C9×Dic3C22×C3semidirect
64. C32×C9C2×D4semidirect
76. C2×C3×Dic3D9semidirect
82. C22×D9C3×D3semidirect
98. C2×C3×C9#67semidirect
99. C2×C3×C9C3×D4semidirect
136. C3×Dic3C2×D9semidirect
147. C2×D9C2×C3×D3semidirect
155. C3×C9C2×#67semidirect
156. C3×C9C2×C3×D4semidirect
170. C2×Dic3C3×D9semidirect
181. C2×C32D36semidirect
200. C2×C9C3×#67semidirect
232. Dic3C2×C3×D9semidirect
237. C32C2×D36semidirect
243. C9C2×C3×#67semidirect
260. C2×C3#6192direct (N × H)
261. C2×C3C3×D36semidirect
280. C3C2×#6192direct (N × H)
281. C3C2×C3×D36semidirect
285. C2C3×#6192direct (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
C22×C32×D92
C2×C3×C9×Dic32
C3×#61922
C2×C3×#6263
C2×C3×D363
C2×#61923