Finite Group Databasebeta

← C3×#62972 #10681579

Subgroups — C3×#62972 #10681579

Order 768.

6822 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3568C217yesmaximal, complemented
2C3×#3560C2121yesmaximal, complemented
3C3×#3546C2121yesmaximal, complemented
4C3×#3532C2114yesmaximal, complemented
5#62972C311yesmaximal, characteristic, direct product factor, Sylow
6C3×#563C22121yescomplemented
7C3×H(3,𝔽4)C2217yescomplemented
8C3×#560C22114yescomplemented
9C3×#560C22142yescomplemented
10C3×#556C22121yescomplemented
11C3×#553C22142yescomplemented
12C3×#551C22142yescomplemented
13C3×#550C22121yescomplemented
14C3×#547C22121yescomplemented
15C3×#546C22142yescomplemented
16C3×#546C22142yescomplemented
17C3×D42C22121yescomplemented
18C3×#542C22142yescomplemented
19C3×#539C22121yescomplemented
20C3×#538C22121yescomplemented
21C3×#538C22142yescomplemented
22C3×#538C22142yescomplemented
23C3×#537C22121yescomplemented
24C3×#535C22142yescomplemented
25C3×#534C22121yescomplemented
26C3×#532C22121yescomplemented
27C3×#529C22121yescomplemented
28C2×C3×#126C2217yescomplemented
29C2×C3×#120C22114yescomplemented
30#3568C2×C317yescomplemented
31#3560C2×C3121yescomplemented
32#3546C2×C3121yescomplemented
33#3532C2×C3114yescomplemented
34C25×C3C2312yescomplemented
35C2×C3×#41C23142yescomplemented
36C22×C3×D4C23142yes—
37C3×C4⋊Q8C23121yescomplemented
38C3×#127C23121yes—
39C3×#127C23114yescomplemented
40C3×#126C23142yescomplemented
41C3×#126C23142yescomplemented
42C3×#126C23114yescomplemented
43C3×#126C23114yescomplemented
44C3×#126C23114yescomplemented
45C3×#125C23121yescomplemented
46C3×#124C23121yes—
47C3×#124C23142yes—
48C3×#124C23142yescomplemented
49C3×#123C23142yes—
50C3×#123C23142yes—
51C3×#123C23142yescomplemented
52C3×#123C23142yescomplemented
53C3×C22⋊Q8C23142yescomplemented
54C3×C22⋊Q8C23142yescomplemented
55C3×C22⋊Q8C23121yescomplemented
56C3×#121C23121yes—
57C3×#121C23142yes—
58C3×#121C23142yes—
59C3×#121C23142yes—
60C3×#121C23121yes—
61C3×#121C23142yescomplemented
62C3×#121C23121yescomplemented
63C3×#121C23142yescomplemented
64C3×#120C23142yes—
65C3×#120C23142yes—
66C3×#120C23114yes—
67C3×#120C23142yescomplemented
68C3×#120C23114yescomplemented
69C3×C4×Q8C23121yescomplemented
70C3×C4×D4C23121yes—
71C3×C4×D4C23121yes—
72C3×C4×D4C23142yescomplemented
73C3×C4×D4C23121yescomplemented
74C3×C4×D4C23142yescomplemented
75C3×#117C23142yescomplemented
76C2×C3×C4⋊C4C23121yescomplemented
77C2×C3×C22⋊C4C23121yes—
78C2×C3×C22⋊C4C23142yescomplemented
79C2×C3×C42C2317yescomplemented
80#563C22×C3121yescomplemented
81H(3,𝔽4)C22×C317yescomplemented
82#560C22×C3114yescomplemented
83#560C22×C3142yescomplemented
84#556C22×C3121yescomplemented
85#553C22×C3142yescomplemented
86#551C22×C3142yescomplemented
87#550C22×C3121yescomplemented
88#547C22×C3121yescomplemented
89#546C22×C3142yescomplemented
90#546C22×C3142yescomplemented
91D42C22×C3121yescomplemented
92#542C22×C3142yescomplemented
93#539C22×C3121yescomplemented
94#538C22×C3121yescomplemented
95#538C22×C3142yescomplemented
96#538C22×C3142yescomplemented
97#537C22×C3121yescomplemented
98#535C22×C3142yescomplemented
99#534C22×C3121yescomplemented
100#532C22×C3121yescomplemented
101#529C22×C3121yescomplemented
102C2×#126C22×C317yescomplemented
103C2×#120C22×C3114yescomplemented
104C24×C3—86no—
105C24×C3C24114yes—
106C24×C3C2417yes—
107C3×#41—842no—
108C2×C3×Q8C24121yes—
109C2×C3×D4—842no—
110C2×C3×D4C24121yes—
111C2×C3×D4C24121yes—
112C2×C3×D4C24121yes—
113C2×C3×D4C24142yes—
114C2×C3×D4C24142yes—
115C22×C3×C4C24121yes—
116C22×C3×C4C24121yes—
117C22×C3×C4C24142yes—
118C3×C4⋊C4C24121yes—
119C3×C4⋊C4C24142yes—
120C3×C4⋊C4C24121yes—
121C3×C4⋊C4C24121yes—
122C3×C4⋊C4C24121yes—
123C3×C22⋊C4C24142yes—
124C3×C22⋊C4C24142yes—
125C3×C22⋊C4C24142yes—
126C3×C22⋊C4C24142yes—
127C3×C22⋊C4C24142yes—
128C3×C42C24121yes—
129C3×C42C2417yes—
130C3×C42C24114yes—
131C25C23×C312yescomplemented
132C2×#41C23×C3142yescomplemented
133C22×D4C23×C3142yes—
134C4⋊Q8C23×C3121yescomplemented
135#127C23×C3121yes—
136#127C23×C3114yescomplemented
137#126C23×C3114yescomplemented
138#126C23×C3142yescomplemented
139#126C23×C3142yescomplemented
140#126C23×C3114yescomplemented
141#126C23×C3114yescomplemented
142#125C23×C3121yescomplemented
143#124C23×C3121yes—
144#124C23×C3142yes—
145#124C23×C3142yescomplemented
146#123C23×C3142yes—
147#123C23×C3142yes—
148#123C23×C3142yescomplemented
149#123C23×C3142yescomplemented
150C22⋊Q8C23×C3121yescomplemented
151C22⋊Q8C23×C3142yescomplemented
152C22⋊Q8C23×C3142yescomplemented
153#121C23×C3121yes—
154#121C23×C3121yes—
155#121C23×C3142yes—
156#121C23×C3142yes—
157#121C23×C3142yes—
158#121C23×C3142yescomplemented
159#121C23×C3121yescomplemented
160#121C23×C3142yescomplemented
161#120C23×C3114yes—
162#120C23×C3142yes—
163#120C23×C3142yes—
164#120C23×C3114yescomplemented
165#120C23×C3142yescomplemented
166C4×Q8C23×C3121yescomplemented
167C4×D4C23×C3121yes—
168C4×D4C23×C3121yes—
169C4×D4C23×C3142yescomplemented
170C4×D4C23×C3121yescomplemented
171C4×D4C23×C3142yescomplemented
172#117C23×C3142yescomplemented
173C2×C4⋊C4C23×C3121yescomplemented
174C2×C22⋊C4C23×C3121yes—
175C2×C22⋊C4C23×C3142yescomplemented
176C2×C42C23×C317yescomplemented
177C23×C3—442no—
178C23×C3—421no—
179C23×C3—816no—
180C23×C3C25114yes—
181C23×C3C2517yes—
182C3×Q8—421no—
183C3×D4—421no—
184C3×D4—421no—
185C3×D4—442no—
186C3×D4—442no—
187C3×D4—421no—
188C2×C3×C4—442no—
189C2×C3×C4—421no—
190C2×C3×C4—421no—
191C2×C3×C4C25121yes—
192C2×C3×C4C25121yes—
193C24—86no—
194C24C24×C317yes—
195C24C24×C3114yes—
196#41—842no—
197C2×Q8C24×C3121yes—
198C2×D4—842no—
199C2×D4C24×C3121yes—
200C2×D4C24×C3121yes—
201C2×D4C24×C3121yes—
202C2×D4C24×C3142yes—
203C2×D4C24×C3142yes—
204C22×C4C24×C3121yes—
205C22×C4C24×C3121yes—
206C22×C4C24×C3142yes—
207C4⋊C4C24×C3121yes—
208C4⋊C4C24×C3142yes—
209C4⋊C4C24×C3121yes—
210C4⋊C4C24×C3121yes—
211C4⋊C4C24×C3121yes—
212C22⋊C4C24×C3142yes—
213C22⋊C4C24×C3142yes—
214C22⋊C4C24×C3142yes—
215C22⋊C4C24×C3142yes—
216C22⋊C4C24×C3142yes—
217C42C24×C3121yes—
218C42C24×C317yes—
219C42C24×C3114yes—
220C22×C3—221no—
221C22×C3—242no—
222C22×C3—828no—
223C22×C3—167no—
224C22×C3C2611yescharacteristic
225C3×C4—221no—
226C3×C4—221no—
227C23—442no—
228C23—421no—
229C23—816no—
230C23C25×C3114yes—
231C23C25×C317yes—
232Q8—421no—
233D4—421no—
234D4—421no—
235D4—442no—
236D4—442no—
237D4—421no—
238C2×C4—442no—
239C2×C4—421no—
240C2×C4—421no—
241C2×C4C25×C3121yes—
242C2×C4C25×C3121yes—
243C2×C3—414no—
244C2×C3—47no—
245C2×C3H(7,𝔽2)13yes—
246C22—221no—
247C22—242no—
248C22—828no—
249C22—167no—
250C22C26×C311yescharacteristic
251C4—221no—
252C4—221no—
253C3#6297211yescharacteristic, direct product factor, Sylow
254C2—414no—
255C2—47no—
256C2C3×H(7,𝔽2)13yes—
subgroupG/H
center, socleC22×C3C26
commutator, Frattini, cosocleC22C26×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3568C2semidirect
2. C3×#3560C2semidirect
3. C3×#3546C2semidirect
4. C3×#3532C2semidirect
5. #62972C3direct (N × H)
6. C3×#563C22semidirect
7. C3×H(3,𝔽4)C22semidirect
8. C3×#560C22semidirect
9. C3×#560C22semidirect
10. C3×#556C22semidirect
11. C3×#553C22semidirect
12. C3×#551C22semidirect
13. C3×#550C22semidirect
14. C3×#547C22semidirect
15. C3×#546C22semidirect
16. C3×#546C22semidirect
17. C3×D42C22semidirect
18. C3×#542C22semidirect
19. C3×#539C22semidirect
20. C3×#538C22semidirect
21. C3×#538C22semidirect
22. C3×#538C22semidirect
23. C3×#537C22semidirect
24. C3×#535C22semidirect
25. C3×#534C22semidirect
26. C3×#532C22semidirect
27. C3×#529C22semidirect
28. C2×C3×#126C22semidirect
29. C2×C3×#120C22semidirect
30. #3568C2×C3semidirect
31. #3560C2×C3semidirect
32. #3546C2×C3semidirect
33. #3532C2×C3semidirect
34. C25×C3C23semidirect
35. C2×C3×#41C23semidirect
37. C3×C4⋊Q8C23semidirect
39. C3×#127C23semidirect
40. C3×#126C23semidirect
41. C3×#126C23semidirect
42. C3×#126C23semidirect
43. C3×#126C23semidirect
44. C3×#126C23semidirect
45. C3×#125C23semidirect
48. C3×#124C23semidirect
51. C3×#123C23semidirect
52. C3×#123C23semidirect
53. C3×C22⋊Q8C23semidirect
54. C3×C22⋊Q8C23semidirect
55. C3×C22⋊Q8C23semidirect
61. C3×#121C23semidirect
62. C3×#121C23semidirect
63. C3×#121C23semidirect
67. C3×#120C23semidirect
68. C3×#120C23semidirect
69. C3×C4×Q8C23semidirect
72. C3×C4×D4C23semidirect
73. C3×C4×D4C23semidirect
74. C3×C4×D4C23semidirect
75. C3×#117C23semidirect
76. C2×C3×C4⋊C4C23semidirect
78. C2×C3×C22⋊C4C23semidirect
79. C2×C3×C42C23semidirect
80. #563C22×C3semidirect
81. H(3,𝔽4)C22×C3semidirect
82. #560C22×C3semidirect
83. #560C22×C3semidirect
84. #556C22×C3semidirect
85. #553C22×C3semidirect
86. #551C22×C3semidirect
87. #550C22×C3semidirect
88. #547C22×C3semidirect
89. #546C22×C3semidirect
90. #546C22×C3semidirect
91. D42C22×C3semidirect
92. #542C22×C3semidirect
93. #539C22×C3semidirect
94. #538C22×C3semidirect
95. #538C22×C3semidirect
96. #538C22×C3semidirect
97. #537C22×C3semidirect
98. #535C22×C3semidirect
99. #534C22×C3semidirect
100. #532C22×C3semidirect
101. #529C22×C3semidirect
102. C2×#126C22×C3semidirect
103. C2×#120C22×C3semidirect
131. C25C23×C3semidirect
132. C2×#41C23×C3semidirect
134. C4⋊Q8C23×C3semidirect
136. #127C23×C3semidirect
137. #126C23×C3semidirect
138. #126C23×C3semidirect
139. #126C23×C3semidirect
140. #126C23×C3semidirect
141. #126C23×C3semidirect
142. #125C23×C3semidirect
145. #124C23×C3semidirect
148. #123C23×C3semidirect
149. #123C23×C3semidirect
150. C22⋊Q8C23×C3semidirect
151. C22⋊Q8C23×C3semidirect
152. C22⋊Q8C23×C3semidirect
158. #121C23×C3semidirect
159. #121C23×C3semidirect
160. #121C23×C3semidirect
164. #120C23×C3semidirect
165. #120C23×C3semidirect
166. C4×Q8C23×C3semidirect
169. C4×D4C23×C3semidirect
170. C4×D4C23×C3semidirect
171. C4×D4C23×C3semidirect
172. #117C23×C3semidirect
173. C2×C4⋊C4C23×C3semidirect
175. C2×C22⋊C4C23×C3semidirect
176. C2×C42C23×C3semidirect
253. C3#62972direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35682
C3×#35602
C3×#35462
C3×#35322
#629723