Finite Group Databasebeta

← C2×C3×#6289 #13049641

Subgroups — C2×C3×#6289 #13049641

Order 1296.

1220 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 200 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
2C2×C32×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C33×Dic3C211yesmaximal, characteristic, complemented
4C3×#6289C214yesmaximal, direct product factor
5C2×C3×C4×C3⋊D3—31nomaximal
6C2×#6289C311yesmaximal, characteristic, direct product factor
7C2×C3×#624—34nomaximal
8C22×C34C2211yescharacteristic
9C2×C3×C3⋊(C3⋊D3)C414yescomplemented
10C2×C3×C3⋊(C3⋊D3)C2212yes—
11C32×C3⋊Dic3C2212yescomplemented
12C33×Dic3C2212yescomplemented
13C22×C3⋊(C3⋊D3)C2×C311yescharacteristic
14C22×C3×C3⋊D3—31no—
15C22×C3×C3⋊D3—34no—
16C22×C3×C3⋊D3—64no—
17C2×C33×C4—31no—
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3D311yescharacteristic, complemented
20C2×C3×C3⋊Dic3C2×C311yescharacteristic, complemented
21C3×C4×C3⋊D3—34no—
22C2×C32×Dic3—24no—
23C2×C32×Dic3—34no—
24C2×C32×Dic3D314yescomplemented
25C2×C32×Dic3C2×C311yescharacteristic, complemented
26#6289C2×C314yesdirect product factor
27C3×#624—316no—
28C2×C34C2×C412yescomplemented
29C2×C34C2311yescharacteristic
30C3×C3⋊(C3⋊D3)C2×C414yescomplemented
31C2×C4×C3⋊D3—31no—
32C2×C3×C4×D3—94no—
33C2×C3×C4×D3—91no—
34C2×#624—34no—
35C22×C33—24no—
36C22×C33—24no—
37C22×C33—21no—
38C22×C33—44no—
39C22×C33C2×D311yescharacteristic
40C22×C33C2×D314yes—
41C22×C33C22×C311yescharacteristic
42C2×C3⋊(C3⋊D3)C3×C414yescomplemented
43C2×C3⋊(C3⋊D3)C22×C312yes—
44C2×C3×C3⋊D3—34no—
45C2×C3×C3⋊D3—316no—
46C2×C3×C3⋊D3—32no—
47C2×C3×C3⋊D3—38no—
48C2×C3×C3⋊D3—616no—
49C2×C3×C3⋊D3—68no—
50C33×C4—32no—
51C3×C3⋊Dic3—22no—
52C3×C3⋊Dic3C2×D312yescomplemented
53C3×C3⋊Dic3C22×C312yescomplemented
54C32×Dic3—28no—
55C32×Dic3—38no—
56C32×Dic3C2×D318yescomplemented
57C32×Dic3C22×C312yescomplemented
58C34C22×C411yescharacteristic, complemented, Sylow
59C22×C3⋊D3—31no—
60C22×C3⋊D3—34no—
61C22×C3⋊D3—64no—
62C22×C3×D3—94no—
63C22×C3×D3—91no—
64C22×C3×D3—184no—
65C2×C32×C4—31no—
66C2×C32×C4—34no—
67C2×C32×C4—64no—
68C2×C32×C4—91no—
69C2×C3⋊Dic3C3×D311yescharacteristic, complemented
70C4×C3⋊D3—34no—
71C2×C3×Dic3—24no—
72C2×C3×Dic3—34no—
73C2×C3×Dic3—34no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3C3×D314yescomplemented
76C2×C3×Dic3C3⋊D311yescharacteristic, complemented
77C3×C4×D3—916no—
78C3×C4×D3—94no—
79#624—316no—
80C2×C33—28no—
81C2×C33—28no—
82C2×C33—22no—
83C2×C33—24no—
84C2×C33—24no—
85C2×C33—21no—
86C2×C33—48no—
87C2×C33—44no—
88C2×C33C4×D312yescomplemented
89C2×C33C4×D318yescomplemented
90C2×C33C2×C3×C412yescomplemented
91C2×C33C22×D311yescharacteristic
92C2×C33C22×D314yes—
93C2×C33C23×C311yescharacteristic
94C3⋊(C3⋊D3)C2×C3×C414yescomplemented
95C3×C3⋊D3—34no—
96C3×C3⋊D3—316no—
97C3×C3⋊D3—616no—
98C22×C3×C4—271no—
99C2×C4×D3—94no—
100C2×C4×D3—91no—
101C22×C32—24no—
102C22×C32—24no—
103C22×C32—24no—
104C22×C32—24no—
105C22×C32—24no—
106C22×C32—44no—
107C22×C32—412no—
108C22×C32—44no—
109C22×C32D3214yes—
110C22×C32C2×C3×D314yes—
111C22×C32C2×C3×D311yescharacteristic
112C22×C32C2×C3⋊D311yescharacteristic
113C2×C3⋊D3—34no—
114C2×C3⋊D3—32no—
115C2×C3⋊D3—316no—
116C2×C3⋊D3—38no—
117C2×C3⋊D3—616no—
118C2×C3⋊D3—68no—
119C2×C3×D3—916no—
120C2×C3×D3—94no—
121C2×C3×D3—98no—
122C2×C3×D3—92no—
123C2×C3×D3—1816no—
124C2×C3×D3—188no—
125C32×C4—32no—
126C32×C4—38no—
127C32×C4—68no—
128C32×C4—92no—
129C3⋊Dic3C2×C3×D312yescomplemented
130C3×Dic3—28no—
131C3×Dic3—38no—
132C3×Dic3—38no—
133C3×Dic3—68no—
134C3×Dic3C2×C3×D318yescomplemented
135C3×Dic3C2×C3⋊D312yescomplemented
136C33—24no—
137C33—24no—
138C33—21no—
139C33—44no—
140C33C2×C4×D311yescharacteristic, complemented
141C33C2×C4×D314yescomplemented
142C33C22×C3×C411yescharacteristic, complemented
143C23×C3—271no—
144C22×D3—94no—
145C22×D3—91no—
146C22×D3—184no—
147C2×C3×C4—34no—
148C2×C3×C4—31no—
149C2×C3×C4—64no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—181no—
153C2×C3×C4—274no—
154C2×Dic3—34no—
155C2×Dic3C3×C3⋊D311yescharacteristic, complemented
156C4×D3—916no—
157C4×D3—94no—
158C2×C32—28no—
159C2×C32—28no—
160C2×C32—24no—
161C2×C32—24no—
162C2×C32—28no—
163C2×C32—28no—
164C2×C32—28no—
165C2×C32—24no—
166C2×C32—24no—
167C2×C32—24no—
168C2×C32—48no—
169C2×C32—44no—
170C2×C32—412no—
171C2×C32—424no—
172C2×C32—48no—
173C2×C32—44no—
174C2×C32#62418yescomplemented
175C2×C32C3×C4×D318yescomplemented
176C2×C32C3×C4×D312yescomplemented
177C2×C32C4×C3⋊D312yescomplemented
178C2×C32C2×D3214yes—
179C2×C32C22×C3×D314yes—
180C2×C32C22×C3×D311yescharacteristic
181C2×C32C22×C3⋊D311yescharacteristic
182C3⋊D3—34no—
183C3⋊D3—316no—
184C3⋊D3—616no—
185C3×D3—916no—
186C3×D3—94no—
187C3×D3—1816no—
188C22×C4—271noSylow
189C22×C3—24no—
190C22×C3—21no—
191C22×C3—24no—
192C22×C3—44no—
193C22×C3—274no—
194C22×C3—272no—
195C22×C3C3×D3214yes—
196C22×C3D3×C3⋊D311yescharacteristic
197C22×C3C2×C3×C3⋊D311yescharacteristic
198C2×D3—916no—
199C2×D3—94no—
200C2×D3—98no—
201C2×D3—92no—
202C2×D3—188no—
203C2×D3—1816no—
204C3×C4—38no—
205C3×C4—32no—
206C3×C4—68no—
207C3×C4—92no—
208C3×C4—92no—
209C3×C4—182no—
210Dic3—38no—
211Dic3C2×C3×C3⋊D312yescomplemented
212C32—24no—
213C32—24no—
214C32—24no—
215C32—24no—
216C32—24no—
217C32—44no—
218C32—412no—
219C32—44no—
220C32C2×#62414yescomplemented
221C32C2×C3×C4×D314yescomplemented
222C32C2×C3×C4×D311yescharacteristic, complemented
223C32C2×C4×C3⋊D311yescharacteristic, complemented
224C23—271no—
225C2×C4—31no—
226C2×C4—91no—
227C2×C4—274no—
228C2×C3—24no—
229C2×C3—21no—
230C2×C3—24no—
231C2×C3—28no—
232C2×C3—28no—
233C2×C3—22no—
234C2×C3—44no—
235C2×C3—48no—
236C2×C3—274no—
237C2×C3C3×#62418yescomplemented
238C2×C3#628912yesdirect product factor
239C2×C3C3×C4×C3⋊D312yescomplemented
240C2×C3C2×C3×D3214yes—
241C2×C3C2×D3×C3⋊D311yescharacteristic
242C2×C3C22×C3×C3⋊D311yescharacteristic
243D3—916no—
244D3—94no—
245D3—1816no—
246C22—272no—
247C22—274no—
248C22C3×D3×C3⋊D311yescharacteristic
249C4—32no—
250C4—92no—
251C3—24no—
252C3—24no—
253C3—21no—
254C3—44no—
255C3C2×C3×#62414yescomplemented
256C3C2×#628911yescharacteristic, direct product factor
257C3C2×C3×C4×C3⋊D311yescharacteristic, complemented
258C2—274no—
259C2C3×#628912yesdirect product factor
260C2C2×C3×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×C3⋊D3
commutator, hypocenterC33C22×C3×C4
FrattiniC2C2×C3×D3×C3⋊D3
Fitting, socleC22×C34C22
prime radicalC22×C3×C3⋊(C3⋊D3)C2
cosocleC2×C33C23×C3
LCSG ▷ C33
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C32×C3⋊Dic3C2semidirect
3. C2×C33×Dic3C2semidirect
4. C3×#6289C2direct (N × H)
6. C2×#6289C3direct (N × H)
9. C2×C3×C3⋊(C3⋊D3)C4semidirect
11. C32×C3⋊Dic3C22semidirect
12. C33×Dic3C22semidirect
19. C2×C3×C3⋊Dic3D3semidirect
20. C2×C3×C3⋊Dic3C2×C3semidirect
24. C2×C32×Dic3D3semidirect
25. C2×C32×Dic3C2×C3semidirect
26. #6289C2×C3direct (N × H)
28. C2×C34C2×C4semidirect
30. C3×C3⋊(C3⋊D3)C2×C4semidirect
42. C2×C3⋊(C3⋊D3)C3×C4semidirect
52. C3×C3⋊Dic3C2×D3semidirect
53. C3×C3⋊Dic3C22×C3semidirect
56. C32×Dic3C2×D3semidirect
57. C32×Dic3C22×C3semidirect
58. C34C22×C4semidirect
69. C2×C3⋊Dic3C3×D3semidirect
75. C2×C3×Dic3C3×D3semidirect
76. C2×C3×Dic3C3⋊D3semidirect
88. C2×C33C4×D3semidirect
89. C2×C33C4×D3semidirect
90. C2×C33C2×C3×C4semidirect
94. C3⋊(C3⋊D3)C2×C3×C4semidirect
129. C3⋊Dic3C2×C3×D3semidirect
134. C3×Dic3C2×C3×D3semidirect
135. C3×Dic3C2×C3⋊D3semidirect
140. C33C2×C4×D3semidirect
141. C33C2×C4×D3semidirect
142. C33C22×C3×C4semidirect
155. C2×Dic3C3×C3⋊D3semidirect
174. C2×C32#624semidirect
175. C2×C32C3×C4×D3semidirect
176. C2×C32C3×C4×D3semidirect
177. C2×C32C4×C3⋊D3semidirect
211. Dic3C2×C3×C3⋊D3semidirect
220. C32C2×#624semidirect
221. C32C2×C3×C4×D3semidirect
222. C32C2×C3×C4×D3semidirect
223. C32C2×C4×C3⋊D3semidirect
237. C2×C3C3×#624semidirect
238. C2×C3#6289direct (N × H)
239. C2×C3C3×C4×C3⋊D3semidirect
255. C3C2×C3×#624semidirect
256. C3C2×#6289direct (N × H)
257. C3C2×C3×C4×C3⋊D3semidirect
259. C2C3×#6289direct (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
C2×C32×C3⋊Dic32
C2×C33×Dic32
C3×#62892
C2×C3×C4×C3⋊D33
C2×#62893
C2×C3×#6243