Finite Group Databasebeta

← C2×C3×#6291 #13049644

Subgroups — C2×C3×#6291 #13049644

Order 1296.

1348 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1348 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×C33×D3C211yesmaximal, characteristic, complemented
3C2×C32×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#6291C214yesmaximal, direct product factor
5C2×C3×#638—31nomaximal
6C2×#6291C311yesmaximal, characteristic, direct product factor
7C2×C3×#626—34nomaximal
8C22×C34C2211yescharacteristic
9C2×C3×C3⋊(C3⋊D3)—22no—
10C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
11C2×C33×D3—22no—
12C2×C33×D3C2212yescomplemented
13C32×C3⋊Dic3C2212yescomplemented
14C23×C33—31no—
15C22×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
16C22×C3×C3⋊D3—31no—
17C22×C3×C3⋊D3—34no—
18C22×C3×C3⋊D3—64no—
19C22×C32×D3—24no—
20C22×C32×D3D314yescomplemented
21C22×C32×D3C2×C311yescharacteristic, complemented
22C3×#638—34no—
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3D311yescharacteristic, complemented
25C2×C3×C3⋊Dic3C2×C311yescharacteristic, complemented
26C2×C32×Dic3—34no—
27#6291C2×C314yesdirect product factor
28C3×#626—316no—
29C2×C34D412yescomplemented
30C2×C34C2311yescharacteristic
31C3×C3⋊(C3⋊D3)—22no—
32C33×D3—22no—
33C2×#638—31no—
34C2×C3×#67—94no—
35C2×C3×D12—91no—
36C2×#626—34no—
37C22×C33—24no—
38C22×C33—21no—
39C22×C33—24no—
40C22×C33—32no—
41C22×C33—44no—
42C22×C33—62no—
43C22×C33C2×D311yescharacteristic
44C22×C33C2×D314yes—
45C22×C33C22×C311yescharacteristic
46C2×C3⋊(C3⋊D3)—22no—
47C2×C3⋊(C3⋊D3)C22×C312yescomplemented
48C2×C3×C3⋊D3—32no—
49C2×C3×C3⋊D3—38no—
50C2×C3×C3⋊D3—68no—
51C2×C3×C3⋊D3—616no—
52C2×C3×C3⋊D3—62no—
53C2×C3×C3⋊D3—68no—
54C2×C32×D3—216no—
55C2×C32×D3—28no—
56C2×C32×D3—28no—
57C2×C32×D3—22no—
58C2×C32×D3C2×D318yescomplemented
59C2×C32×D3C22×C312yescomplemented
60C3×C3⋊Dic3—22no—
61C3×C3⋊Dic3C2×D312yescomplemented
62C3×C3⋊Dic3C22×C312yescomplemented
63C32×Dic3—38no—
64C34C2×D411yescharacteristic, complemented, Sylow
65C23×C32—31no—
66C23×C32—34no—
67C23×C32—64no—
68C22×C3⋊D3—31no—
69C22×C3⋊D3—34no—
70C22×C3⋊D3—64no—
71C22×C3×D3—24no—
72C22×C3×D3—94no—
73C22×C3×D3—91no—
74C22×C3×D3—184no—
75C22×C3×D3C3×D314yescomplemented
76C22×C3×D3C3⋊D311yescharacteristic, complemented
77C2×C32×C4—91no—
78#638—34no—
79C2×C3⋊Dic3C3×D311yescharacteristic, complemented
80C3×#67—916no—
81C2×C3×Dic3—34no—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—64no—
84C3×D12—94no—
85#626—316no—
86C2×C33—28no—
87C2×C33—22no—
88C2×C33—28no—
89C2×C33—24no—
90C2×C33—21no—
91C2×C33—24no—
92C2×C33—48no—
93C2×C33—44no—
94C2×C33—62no—
95C2×C33D1212yescomplemented
96C2×C33#6718yescomplemented
97C2×C33C3×D412yescomplemented
98C2×C33C22×D311yescharacteristic
99C2×C33C22×D314yes—
100C2×C33C23×C311yescharacteristic
101C3⋊(C3⋊D3)—22no—
102C3×C3⋊D3—616no—
103C3×C3⋊D3—62no—
104C3×C3⋊D3—68no—
105C32×D3—216no—
106C32×D3—28no—
107C32×D3—22no—
108C2×C3×D4—271no—
109C2×#67—94no—
110C2×D12—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×C32D3214yes—
126C22×C32C2×C3×D314yes—
127C22×C32C2×C3×D311yescharacteristic
128C22×C32C2×C3⋊D311yescharacteristic
129C2×C3⋊D3—32no—
130C2×C3⋊D3—38no—
131C2×C3⋊D3—68no—
132C2×C3⋊D3—616no—
133C2×C3⋊D3—62no—
134C2×C3⋊D3—68no—
135C2×C3×D3—216no—
136C2×C3×D3—28no—
137C2×C3×D3—22no—
138C2×C3×D3—28no—
139C2×C3×D3—98no—
140C2×C3×D3—92no—
141C2×C3×D3—188no—
142C2×C3×D3—1816no—
143C2×C3×D3—188no—
144C2×C3×D3—182no—
145C2×C3×D3C2×C3×D318yescomplemented
146C2×C3×D3C2×C3⋊D312yescomplemented
147C32×C4—92no—
148C3⋊Dic3C2×C3×D312yescomplemented
149C3×Dic3—38no—
150C3×Dic3—38no—
151C3×Dic3—68no—
152C33—24no—
153C33—21no—
154C33—24no—
155C33—44no—
156C33C2×D1211yescharacteristic, complemented
157C33C2×#6714yescomplemented
158C33C2×C3×D411yescharacteristic, complemented
159C23×C3—34no—
160C23×C3—31no—
161C23×C3—64no—
162C23×C3—271no—
163C22×D3—94no—
164C22×D3—91no—
165C22×D3—184no—
166C22×D3C3×C3⋊D311yescharacteristic, complemented
167C3×D4—274no—
168C2×C3×C4—91no—
169C2×C3×C4—91no—
170C2×C3×C4—181no—
171#67—916no—
172C2×Dic3—34no—
173D12—94no—
174C2×C32—28no—
175C2×C32—28no—
176C2×C32—24no—
177C2×C32—24no—
178C2×C32—28no—
179C2×C32—28no—
180C2×C32—28no—
181C2×C32—24no—
182C2×C32—24no—
183C2×C32—24no—
184C2×C32—48no—
185C2×C32—44no—
186C2×C32—412no—
187C2×C32—424no—
188C2×C32—48no—
189C2×C32—44no—
190C2×C32—616no—
191C2×C32—68no—
192C2×C32—62no—
193C2×C32#62618yescomplemented
194C2×C32C3×D1212yescomplemented
195C2×C32C3×#6718yescomplemented
196C2×C32#63812yescomplemented
197C2×C32C2×D3214yes—
198C2×C32C22×C3×D314yes—
199C2×C32C22×C3×D311yescharacteristic
200C2×C32C22×C3⋊D311yescharacteristic
201C3⋊D3—616no—
202C3⋊D3—62no—
203C3⋊D3—68no—
204C3×D3—216no—
205C3×D3—28no—
206C3×D3—22no—
207C3×D3—1816no—
208C3×D3—188no—
209C3×D3—182no—
210C2×D4—271noSylow
211C22×C3—21no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—38no—
215C22×C3—32no—
216C22×C3—44no—
217C22×C3—68no—
218C22×C3—616no—
219C22×C3—68no—
220C22×C3—62no—
221C22×C3—272no—
222C22×C3—542no—
223C22×C3C3×D3214yes—
224C22×C3D3×C3⋊D311yescharacteristic
225C22×C3C2×C3×C3⋊D311yescharacteristic
226C2×D3—22no—
227C2×D3—98no—
228C2×D3—92no—
229C2×D3—188no—
230C2×D3—1816no—
231C2×D3—188no—
232C2×D3—182no—
233C2×D3C2×C3×C3⋊D312yescomplemented
234C3×C4—92no—
235C3×C4—92no—
236C3×C4—182no—
237Dic3—38no—
238C32—24no—
239C32—24no—
240C32—24no—
241C32—24no—
242C32—24no—
243C32—44no—
244C32—412no—
245C32—44no—
246C32C2×#62614yescomplemented
247C32C2×C3×D1211yescharacteristic, complemented
248C32C2×C3×#6714yescomplemented
249C32C2×#63811yescharacteristic, complemented
250C23—31no—
251C23—271no—
252D4—274no—
253C2×C4—91no—
254C2×C3—21no—
255C2×C3—24no—
256C2×C3—24no—
257C2×C3—28no—
258C2×C3—28no—
259C2×C3—22no—
260C2×C3—44no—
261C2×C3—48no—
262C2×C3—616no—
263C2×C3—68no—
264C2×C3—62no—
265C2×C3—542no—
266C2×C3C3×#62618yescomplemented
267C2×C3#629112yesdirect product factor
268C2×C3C3×#63812yescomplemented
269C2×C3C2×C3×D3214yes—
270C2×C3C2×D3×C3⋊D311yescharacteristic
271C2×C3C22×C3×C3⋊D311yescharacteristic
272D3—22no—
273D3—1816no—
274D3—188no—
275D3—182no—
276C22—32no—
277C22—62no—
278C22—272no—
279C22—542no—
280C22C3×D3×C3⋊D311yescharacteristic
281C4—92no—
282C3—24no—
283C3—24no—
284C3—21no—
285C3—44no—
286C3C2×C3×#62614yescomplemented
287C3C2×#629111yescharacteristic, direct product factor
288C3C2×C3×#63811yescharacteristic, complemented
289C2—62no—
290C2—542no—
291C2C3×#629112yesdirect product factor
292C2C2×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×C33×D3C2semidirect
3. C2×C32×C3⋊Dic3C2semidirect
4. C3×#6291C2direct (N × H)
6. C2×#6291C3direct (N × H)
10. C2×C3×C3⋊(C3⋊D3)C22semidirect
12. C2×C33×D3C22semidirect
13. C32×C3⋊Dic3C22semidirect
15. C22×C3⋊(C3⋊D3)C2×C3semidirect
20. C22×C32×D3D3semidirect
21. C22×C32×D3C2×C3semidirect
24. C2×C3×C3⋊Dic3D3semidirect
25. C2×C3×C3⋊Dic3C2×C3semidirect
27. #6291C2×C3direct (N × H)
29. C2×C34D4semidirect
47. C2×C3⋊(C3⋊D3)C22×C3semidirect
58. C2×C32×D3C2×D3semidirect
59. C2×C32×D3C22×C3semidirect
61. C3×C3⋊Dic3C2×D3semidirect
62. C3×C3⋊Dic3C22×C3semidirect
64. C34C2×D4semidirect
75. C22×C3×D3C3×D3semidirect
76. C22×C3×D3C3⋊D3semidirect
79. C2×C3⋊Dic3C3×D3semidirect
95. C2×C33D12semidirect
96. C2×C33#67semidirect
97. C2×C33C3×D4semidirect
145. C2×C3×D3C2×C3×D3semidirect
146. C2×C3×D3C2×C3⋊D3semidirect
148. C3⋊Dic3C2×C3×D3semidirect
156. C33C2×D12semidirect
157. C33C2×#67semidirect
158. C33C2×C3×D4semidirect
166. C22×D3C3×C3⋊D3semidirect
193. C2×C32#626semidirect
194. C2×C32C3×D12semidirect
195. C2×C32C3×#67semidirect
196. C2×C32#638semidirect
233. C2×D3C2×C3×C3⋊D3semidirect
246. C32C2×#626semidirect
247. C32C2×C3×D12semidirect
248. C32C2×C3×#67semidirect
249. C32C2×#638semidirect
266. C2×C3C3×#626semidirect
267. C2×C3#6291direct (N × H)
268. C2×C3C3×#638semidirect
286. C3C2×C3×#626semidirect
287. C3C2×#6291direct (N × H)
288. C3C2×C3×#638semidirect
291. C2C3×#6291direct (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×C33×D32
C2×C32×C3⋊Dic32
C3×#62912
C2×C3×#6383
C2×#62913
C2×C3×#6263