Finite Group Databasebeta

← C2×C3×#6290 #13049642

Subgroups — C2×C3×#6290 #13049642

Order 1296.

1268 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1268 conjugacy classes of subgroups — too many to draw. Only its 184 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×C32×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C33×D3C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C3×#6290C214yesmaximal, direct product factor
5C2×C3×#638—31nomaximal
6C2×#6290C311yesmaximal, characteristic, direct product factor
7C2×C3×#625—34nomaximal
8C22×C34C2211yescharacteristic
9C2×C32×C3⋊D3—22no—
10C2×C32×C3⋊D3C2212yescomplemented
11C2×C33×D3—22no—
12C2×C33×D3C2212yescomplemented
13C3×C3⋊(C3⋊Dic3)C2212yescomplemented
14C23×C33—31no—
15C22×C3×C3⋊D3—21no—
16C22×C3×C3⋊D3D311yescharacteristic, complemented
17C22×C3×C3⋊D3C2×C311yescharacteristic, complemented
18C22×C32×D3—24no—
19C22×C32×D3—34no—
20C22×C32×D3D314yescomplemented
21C22×C32×D3C2×C311yescharacteristic, complemented
22C2×C3⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
23C3×#638—34no—
24C2×C3×C3⋊Dic3—31no—
25C2×C3×C3⋊Dic3—34no—
26C2×C3×C3⋊Dic3—64no—
27#6290C2×C314yesdirect product factor
28C3×#625—316no—
29C2×C34D412yescomplemented
30C2×C34C2311yescharacteristic
31C32×C3⋊D3—22no—
32C33×D3—22no—
33C2×#638—31no—
34C2×C3×#67—94no—
35C2×C3×#67—91no—
36C2×#625—34no—
37C22×C33—24no—
38C22×C33—24no—
39C22×C33—21no—
40C22×C33—32no—
41C22×C33—44no—
42C22×C33—62no—
43C22×C33C2×D311yescharacteristic
44C22×C33C2×D314yes—
45C22×C33C22×C311yescharacteristic
46C2×C3×C3⋊D3—24no—
47C2×C3×C3⋊D3—22no—
48C2×C3×C3⋊D3—22no—
49C2×C3×C3⋊D3—22no—
50C2×C3×C3⋊D3C2×D312yescomplemented
51C2×C3×C3⋊D3C22×C312yescomplemented
52C2×C32×D3—216no—
53C2×C32×D3—28no—
54C2×C32×D3—28no—
55C2×C32×D3—22no—
56C2×C32×D3—38no—
57C2×C32×D3—68no—
58C2×C32×D3C2×D318yescomplemented
59C2×C32×D3C22×C312yescomplemented
60C3⋊(C3⋊Dic3)C22×C312yescomplemented
61C3×C3⋊Dic3—32no—
62C3×C3⋊Dic3—38no—
63C3×C3⋊Dic3—68no—
64C34C2×D411yescharacteristic, complemented, Sylow
65C23×C32—31no—
66C23×C32—34no—
67C23×C32—64no—
68C23×C32—91no—
69C22×C3⋊D3C3×D311yescharacteristic, complemented
70C22×C3×D3—24no—
71C22×C3×D3—34no—
72C22×C3×D3—34no—
73C22×C3×D3—64no—
74C22×C3×D3C3×D314yescomplemented
75C22×C3×D3C3⋊D311yescharacteristic, complemented
76#638—34no—
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—34no—
79C2×C3⋊Dic3—64no—
80C3×#67—916no—
81C3×#67—94no—
82C2×C3×Dic3—94no—
83C2×C3×Dic3—91no—
84C2×C3×Dic3—184no—
85#625—316no—
86C2×C33—28no—
87C2×C33—28no—
88C2×C33—22no—
89C2×C33—24no—
90C2×C33—24no—
91C2×C33—21no—
92C2×C33—48no—
93C2×C33—44no—
94C2×C33—62no—
95C2×C33#6712yescomplemented
96C2×C33#6718yescomplemented
97C2×C33C3×D412yescomplemented
98C2×C33C22×D311yescharacteristic
99C2×C33C22×D314yes—
100C2×C33C23×C311yescharacteristic
101C3×C3⋊D3—24no—
102C3×C3⋊D3—22no—
103C3×C3⋊D3—22no—
104C32×D3—216no—
105C32×D3—28no—
106C32×D3—22no—
107C32×D3—68no—
108C2×C3×D4—271no—
109C2×#67—94no—
110C2×#67—91no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32—24no—
116C22×C32—32no—
117C22×C32—38no—
118C22×C32—44no—
119C22×C32—412no—
120C22×C32—44no—
121C22×C32—68no—
122C22×C32—616no—
123C22×C32—68no—
124C22×C32—62no—
125C22×C32—92no—
126C22×C32—182no—
127C22×C32D3214yes—
128C22×C32C2×C3×D314yes—
129C22×C32C2×C3×D311yescharacteristic
130C22×C32C2×C3⋊D311yescharacteristic
131C2×C3⋊D3—22no—
132C2×C3⋊D3C2×C3×D312yescomplemented
133C2×C3×D3—216no—
134C2×C3×D3—28no—
135C2×C3×D3—22no—
136C2×C3×D3—28no—
137C2×C3×D3—38no—
138C2×C3×D3—38no—
139C2×C3×D3—68no—
140C2×C3×D3—616no—
141C2×C3×D3—68no—
142C2×C3×D3—68no—
143C2×C3×D3C2×C3×D318yescomplemented
144C2×C3×D3C2×C3⋊D312yescomplemented
145C3⋊Dic3—32no—
146C3⋊Dic3—38no—
147C3⋊Dic3—68no—
148C3×Dic3—98no—
149C3×Dic3—92no—
150C3×Dic3—188no—
151C33—24no—
152C33—24no—
153C33—21no—
154C33—44no—
155C33C2×#6711yescharacteristic, complemented
156C33C2×#6714yescomplemented
157C33C2×C3×D411yescharacteristic, complemented
158C23×C3—34no—
159C23×C3—31no—
160C23×C3—64no—
161C23×C3—91no—
162C23×C3—91no—
163C23×C3—181no—
164C22×D3—34no—
165C22×D3C3×C3⋊D311yescharacteristic, complemented
166C3×D4—274no—
167C2×C3×C4—271no—
168#67—916no—
169#67—94no—
170C2×Dic3—94no—
171C2×Dic3—91no—
172C2×Dic3—184no—
173C2×C32—28no—
174C2×C32—28no—
175C2×C32—24no—
176C2×C32—24no—
177C2×C32—28no—
178C2×C32—28no—
179C2×C32—28no—
180C2×C32—24no—
181C2×C32—24no—
182C2×C32—24no—
183C2×C32—48no—
184C2×C32—44no—
185C2×C32—412no—
186C2×C32—424no—
187C2×C32—48no—
188C2×C32—44no—
189C2×C32—616no—
190C2×C32—68no—
191C2×C32—62no—
192C2×C32—182no—
193C2×C32#62518yescomplemented
194C2×C32C3×#6718yescomplemented
195C2×C32C3×#6712yescomplemented
196C2×C32#63812yescomplemented
197C2×C32C2×D3214yes—
198C2×C32C22×C3×D314yes—
199C2×C32C22×C3×D311yescharacteristic
200C2×C32C22×C3⋊D311yescharacteristic
201C3⋊D3—22no—
202C3×D3—216no—
203C3×D3—28no—
204C3×D3—22no—
205C3×D3—616no—
206C3×D3—68no—
207C3×D3—68no—
208C2×D4—271noSylow
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—21no—
212C22×C3—38no—
213C22×C3—32no—
214C22×C3—44no—
215C22×C3—68no—
216C22×C3—616no—
217C22×C3—68no—
218C22×C3—62no—
219C22×C3—92no—
220C22×C3—92no—
221C22×C3—182no—
222C22×C3—184no—
223C22×C3—182no—
224C22×C3—182no—
225C22×C3C3×D3214yes—
226C22×C3D3×C3⋊D311yescharacteristic
227C22×C3C2×C3×C3⋊D311yescharacteristic
228C2×D3—22no—
229C2×D3—38no—
230C2×D3—68no—
231C2×D3C2×C3×C3⋊D312yescomplemented
232C3×C4—272no—
233Dic3—98no—
234Dic3—92no—
235Dic3—188no—
236C32—24no—
237C32—24no—
238C32—24no—
239C32—24no—
240C32—24no—
241C32—44no—
242C32—412no—
243C32—44no—
244C32C2×#62514yescomplemented
245C32C2×C3×#6714yescomplemented
246C32C2×C3×#6711yescharacteristic, complemented
247C32C2×#63811yescharacteristic, complemented
248C23—31no—
249C23—91no—
250D4—274no—
251C2×C4—271no—
252C2×C3—24no—
253C2×C3—24no—
254C2×C3—21no—
255C2×C3—28no—
256C2×C3—22no—
257C2×C3—28no—
258C2×C3—44no—
259C2×C3—48no—
260C2×C3—616no—
261C2×C3—68no—
262C2×C3—62no—
263C2×C3—184no—
264C2×C3—182no—
265C2×C3—182no—
266C2×C3C3×#62518yescomplemented
267C2×C3#629012yesdirect product factor
268C2×C3C3×#63812yescomplemented
269C2×C3C2×C3×D3214yes—
270C2×C3C2×D3×C3⋊D311yescharacteristic
271C2×C3C22×C3×C3⋊D311yescharacteristic
272D3—22no—
273D3—68no—
274C22—32no—
275C22—62no—
276C22—92no—
277C22—182no—
278C22C3×D3×C3⋊D311yescharacteristic
279C4—272no—
280C3—24no—
281C3—21no—
282C3—24no—
283C3—44no—
284C3C2×C3×#62514yescomplemented
285C3C2×#629011yescharacteristic, direct product factor
286C3C2×C3×#63811yescharacteristic, complemented
287C2—62no—
288C2—182no—
289C2C3×#629012yesdirect product factor
290C2C2×C3×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×C3⋊D3
commutator, cosocleC2×C33C23×C3
FrattiniC2C2×C3×D3×C3⋊D3
Fitting, socleC22×C34C22
hypocenterC33C2×C3×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×C3⋊D3C2semidirect
2. C22×C33×D3C2semidirect
3. C2×C3×C3⋊(C3⋊Dic3)C2semidirect
4. C3×#6290C2direct (N × H)
6. C2×#6290C3direct (N × H)
10. C2×C32×C3⋊D3C22semidirect
12. C2×C33×D3C22semidirect
13. C3×C3⋊(C3⋊Dic3)C22semidirect
16. C22×C3×C3⋊D3D3semidirect
17. C22×C3×C3⋊D3C2×C3semidirect
20. C22×C32×D3D3semidirect
21. C22×C32×D3C2×C3semidirect
22. C2×C3⋊(C3⋊Dic3)C2×C3semidirect
27. #6290C2×C3direct (N × H)
29. C2×C34D4semidirect
50. C2×C3×C3⋊D3C2×D3semidirect
51. C2×C3×C3⋊D3C22×C3semidirect
58. C2×C32×D3C2×D3semidirect
59. C2×C32×D3C22×C3semidirect
60. C3⋊(C3⋊Dic3)C22×C3semidirect
64. C34C2×D4semidirect
69. C22×C3⋊D3C3×D3semidirect
74. C22×C3×D3C3×D3semidirect
75. C22×C3×D3C3⋊D3semidirect
95. C2×C33#67semidirect
96. C2×C33#67semidirect
97. C2×C33C3×D4semidirect
132. C2×C3⋊D3C2×C3×D3semidirect
143. C2×C3×D3C2×C3×D3semidirect
144. C2×C3×D3C2×C3⋊D3semidirect
155. C33C2×#67semidirect
156. C33C2×#67semidirect
157. C33C2×C3×D4semidirect
165. C22×D3C3×C3⋊D3semidirect
193. C2×C32#625semidirect
194. C2×C32C3×#67semidirect
195. C2×C32C3×#67semidirect
196. C2×C32#638semidirect
231. C2×D3C2×C3×C3⋊D3semidirect
244. C32C2×#625semidirect
245. C32C2×C3×#67semidirect
246. C32C2×C3×#67semidirect
247. C32C2×#638semidirect
266. C2×C3C3×#625semidirect
267. C2×C3#6290direct (N × H)
268. C2×C3C3×#638semidirect
284. C3C2×C3×#625semidirect
285. C3C2×#6290direct (N × H)
286. C3C2×C3×#638semidirect
289. C2C3×#6290direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C3⋊D32
C22×C33×D32
C2×C3×C3⋊(C3⋊Dic3)2
C3×#62902
C2×C3×#6383
C2×#62903
C2×C3×#6253