Finite Group Databasebeta

← C2×C3×#6292 #13049643

Subgroups — C2×C3×#6292 #13049643

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×C3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C22×C32×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C33×Dic3C211yesmaximal, characteristic, complemented
4C3×#6292C214yesmaximal, direct product factor
5C2×C3×C4⋊(C3⋊D3)—31nomaximal
6C2×#6292C311yesmaximal, characteristic, direct product factor
7C2×C3×#626—34nomaximal
8C22×C34C2211yescharacteristic
9C2×C3×C3⋊(C3⋊D3)—22no—
10C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
11C2×C32×C3⋊D3—22no—
12C2×C32×C3⋊D3C2212yescomplemented
13C33×Dic3C2212yescomplemented
14C22×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
15C22×C3×C3⋊D3—21no—
16C22×C3×C3⋊D3—31no—
17C22×C3×C3⋊D3—34no—
18C22×C3×C3⋊D3—64no—
19C22×C3×C3⋊D3D311yescharacteristic, complemented
20C22×C3×C3⋊D3C2×C311yescharacteristic, complemented
21C22×C32×D3—34no—
22C2×C33×C4—31no—
23C3×C4⋊(C3⋊D3)—34no—
24C2×C32×Dic3—24no—
25C2×C32×Dic3D314yescomplemented
26C2×C32×Dic3C2×C311yescharacteristic, complemented
27#6292C2×C314yesdirect product factor
28C3×#626—316no—
29C2×C34D412yescomplemented
30C2×C34C2311yescharacteristic
31C3×C3⋊(C3⋊D3)—22no—
32C32×C3⋊D3—22no—
33C2×C4⋊(C3⋊D3)—31no—
34C2×C3×#67—91no—
35C2×C3×D12—94no—
36C2×#626—34no—
37C22×C33—24no—
38C22×C33—24no—
39C22×C33—21no—
40C22×C33—44no—
41C22×C33C2×D314yes—
42C22×C33C2×D311yescharacteristic
43C22×C33C22×C311yescharacteristic
44C2×C3⋊(C3⋊D3)—22no—
45C2×C3⋊(C3⋊D3)C22×C312yescomplemented
46C2×C3×C3⋊D3—24no—
47C2×C3×C3⋊D3—22no—
48C2×C3×C3⋊D3—22no—
49C2×C3×C3⋊D3—22no—
50C2×C3×C3⋊D3—32no—
51C2×C3×C3⋊D3—38no—
52C2×C3×C3⋊D3—68no—
53C2×C3×C3⋊D3—616no—
54C2×C3×C3⋊D3—68no—
55C2×C3×C3⋊D3—62no—
56C2×C3×C3⋊D3C2×D312yescomplemented
57C2×C3×C3⋊D3C22×C312yescomplemented
58C2×C32×D3—38no—
59C2×C32×D3—68no—
60C33×C4—32no—
61C32×Dic3—28no—
62C32×Dic3C2×D318yescomplemented
63C32×Dic3C22×C312yescomplemented
64C34C2×D411yescharacteristic, complemented, Sylow
65C23×C32—91no—
66C22×C3⋊D3—31no—
67C22×C3⋊D3—34no—
68C22×C3⋊D3—64no—
69C22×C3⋊D3C3×D311yescharacteristic, complemented
70C22×C3×D3—34no—
71C22×C3×D3—34no—
72C22×C3×D3—64no—
73C22×C3×D3—94no—
74C22×C3×D3—91no—
75C22×C3×D3—184no—
76C2×C32×C4—31no—
77C2×C32×C4—34no—
78C2×C32×C4—64no—
79C4⋊(C3⋊D3)—34no—
80C3×#67—94no—
81C2×C3×Dic3—24no—
82C2×C3×Dic3C3×D314yescomplemented
83C2×C3×Dic3C3⋊D311yescharacteristic, complemented
84C3×D12—916no—
85#626—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×C33D1218yescomplemented
95C2×C33#6712yescomplemented
96C2×C33C3×D412yescomplemented
97C2×C33C22×D311yescharacteristic
98C2×C33C22×D314yes—
99C2×C33C23×C311yescharacteristic
100C3⋊(C3⋊D3)—22no—
101C3×C3⋊D3—24no—
102C3×C3⋊D3—22no—
103C3×C3⋊D3—22no—
104C3×C3⋊D3—616no—
105C3×C3⋊D3—68no—
106C3×C3⋊D3—62no—
107C32×D3—68no—
108C2×C3×D4—271no—
109C2×#67—91no—
110C2×D12—94no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32—24no—
116C22×C32—44no—
117C22×C32—412no—
118C22×C32—44no—
119C22×C32—92no—
120C22×C32—182no—
121C22×C32D3214yes—
122C22×C32C2×C3×D314yes—
123C22×C32C2×C3×D311yescharacteristic
124C22×C32C2×C3⋊D311yescharacteristic
125C2×C3⋊D3—22no—
126C2×C3⋊D3—32no—
127C2×C3⋊D3—38no—
128C2×C3⋊D3—68no—
129C2×C3⋊D3—616no—
130C2×C3⋊D3—68no—
131C2×C3⋊D3—62no—
132C2×C3⋊D3C2×C3×D312yescomplemented
133C2×C3×D3—38no—
134C2×C3×D3—38no—
135C2×C3×D3—68no—
136C2×C3×D3—616no—
137C2×C3×D3—68no—
138C2×C3×D3—68no—
139C2×C3×D3—98no—
140C2×C3×D3—92no—
141C2×C3×D3—188no—
142C2×C3×D3—1816no—
143C2×C3×D3—188no—
144C2×C3×D3—182no—
145C32×C4—32no—
146C32×C4—38no—
147C32×C4—68no—
148C3×Dic3—28no—
149C3×Dic3C2×C3×D318yescomplemented
150C3×Dic3C2×C3⋊D312yescomplemented
151C33—24no—
152C33—24no—
153C33—21no—
154C33—44no—
155C33C2×D1214yescomplemented
156C33C2×#6711yescharacteristic, complemented
157C33C2×C3×D411yescharacteristic, complemented
158C23×C3—91no—
159C23×C3—91no—
160C23×C3—181no—
161C23×C3—271no—
162C22×D3—34no—
163C22×D3—94no—
164C22×D3—91no—
165C22×D3—184no—
166C3×D4—274no—
167C2×C3×C4—34no—
168C2×C3×C4—31no—
169C2×C3×C4—64no—
170#67—94no—
171C2×Dic3C3×C3⋊D311yescharacteristic, complemented
172D12—916no—
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—182no—
190C2×C32#62618yescomplemented
191C2×C32C3×D1218yescomplemented
192C2×C32C3×#6712yescomplemented
193C2×C32C4⋊(C3⋊D3)12yescomplemented
194C2×C32C2×D3214yes—
195C2×C32C22×C3×D314yes—
196C2×C32C22×C3×D311yescharacteristic
197C2×C32C22×C3⋊D311yescharacteristic
198C3⋊D3—22no—
199C3⋊D3—616no—
200C3⋊D3—68no—
201C3⋊D3—62no—
202C3×D3—616no—
203C3×D3—68no—
204C3×D3—68no—
205C3×D3—1816no—
206C3×D3—188no—
207C3×D3—182no—
208C2×D4—271noSylow
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—21no—
212C22×C3—44no—
213C22×C3—92no—
214C22×C3—92no—
215C22×C3—182no—
216C22×C3—184no—
217C22×C3—182no—
218C22×C3—182no—
219C22×C3—272no—
220C22×C3—542no—
221C22×C3C3×D3214yes—
222C22×C3D3×C3⋊D311yescharacteristic
223C22×C3C2×C3×C3⋊D311yescharacteristic
224C2×D3—38no—
225C2×D3—68no—
226C2×D3—98no—
227C2×D3—92no—
228C2×D3—188no—
229C2×D3—1816no—
230C2×D3—188no—
231C2×D3—182no—
232C3×C4—38no—
233C3×C4—32no—
234C3×C4—68no—
235Dic3C2×C3×C3⋊D312yescomplemented
236C32—24no—
237C32—24no—
238C32—24no—
239C32—24no—
240C32—24no—
241C32—44no—
242C32—412no—
243C32—44no—
244C32C2×#62614yescomplemented
245C32C2×C3×D1214yescomplemented
246C32C2×C3×#6711yescharacteristic, complemented
247C32C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
248C23—91no—
249C23—271no—
250D4—274no—
251C2×C4—31no—
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—184no—
261C2×C3—182no—
262C2×C3—182no—
263C2×C3—542no—
264C2×C3C3×#62618yescomplemented
265C2×C3#629212yesdirect product factor
266C2×C3C3×C4⋊(C3⋊D3)12yescomplemented
267C2×C3C2×C3×D3214yes—
268C2×C3C2×D3×C3⋊D311yescharacteristic
269C2×C3C22×C3×C3⋊D311yescharacteristic
270D3—68no—
271D3—1816no—
272D3—188no—
273D3—182no—
274C22—92no—
275C22—182no—
276C22—272no—
277C22—542no—
278C22C3×D3×C3⋊D311yescharacteristic
279C4—32no—
280C3—24no—
281C3—21no—
282C3—24no—
283C3—44no—
284C3C2×C3×#62614yescomplemented
285C3C2×#629211yescharacteristic, direct product factor
286C3C2×C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
287C2—182no—
288C2—542no—
289C2C3×#629212yesdirect 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×C3×C3⋊(C3⋊D3)C2semidirect
2. C22×C32×C3⋊D3C2semidirect
3. C2×C33×Dic3C2semidirect
4. C3×#6292C2direct (N × H)
6. C2×#6292C3direct (N × H)
10. C2×C3×C3⋊(C3⋊D3)C22semidirect
12. C2×C32×C3⋊D3C22semidirect
13. C33×Dic3C22semidirect
14. C22×C3⋊(C3⋊D3)C2×C3semidirect
19. C22×C3×C3⋊D3D3semidirect
20. C22×C3×C3⋊D3C2×C3semidirect
25. C2×C32×Dic3D3semidirect
26. C2×C32×Dic3C2×C3semidirect
27. #6292C2×C3direct (N × H)
29. C2×C34D4semidirect
45. C2×C3⋊(C3⋊D3)C22×C3semidirect
56. C2×C3×C3⋊D3C2×D3semidirect
57. C2×C3×C3⋊D3C22×C3semidirect
62. C32×Dic3C2×D3semidirect
63. C32×Dic3C22×C3semidirect
64. C34C2×D4semidirect
69. C22×C3⋊D3C3×D3semidirect
82. C2×C3×Dic3C3×D3semidirect
83. C2×C3×Dic3C3⋊D3semidirect
94. C2×C33D12semidirect
95. C2×C33#67semidirect
96. C2×C33C3×D4semidirect
132. C2×C3⋊D3C2×C3×D3semidirect
149. C3×Dic3C2×C3×D3semidirect
150. C3×Dic3C2×C3⋊D3semidirect
155. C33C2×D12semidirect
156. C33C2×#67semidirect
157. C33C2×C3×D4semidirect
171. C2×Dic3C3×C3⋊D3semidirect
190. C2×C32#626semidirect
191. C2×C32C3×D12semidirect
192. C2×C32C3×#67semidirect
193. C2×C32C4⋊(C3⋊D3)semidirect
235. Dic3C2×C3×C3⋊D3semidirect
244. C32C2×#626semidirect
245. C32C2×C3×D12semidirect
246. C32C2×C3×#67semidirect
247. C32C2×C4⋊(C3⋊D3)semidirect
264. C2×C3C3×#626semidirect
265. C2×C3#6292direct (N × H)
266. C2×C3C3×C4⋊(C3⋊D3)semidirect
284. C3C2×C3×#626semidirect
285. C3C2×#6292direct (N × H)
286. C3C2×C3×C4⋊(C3⋊D3)semidirect
289. C2C3×#6292direct (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⋊(C3⋊D3)2
C22×C32×C3⋊D32
C2×C33×Dic32
C3×#62922
C2×C3×C4⋊(C3⋊D3)3
C2×#62923
C2×C3×#6263