Finite Group Databasebeta

← C3×#3490 #87374

Subgroups — C3×#3490 #87374

Order 384.

1328 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#551C218yesmaximal, complemented
2C3×#546C218yesmaximal, complemented
3C3×#543C211yesmaximal, characteristic, complemented
4C3×#538C214yesmaximal, complemented
5C3×#534C211yesmaximal, characteristic, complemented
6C3×#525C214yesmaximal, complemented
7C2×C3×#120C212yesmaximal, complemented
8C3×#518C212yesmaximal, complemented
9C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
10#3490C311yesmaximal, characteristic, direct product factor, Sylow
11C25×C3C2211yescharacteristic
12C2×C3×#41C2214yescomplemented
13C22×C3×D4C2212yes—
14C22×C3×D4C2214yescomplemented
15C23×C3×C4C2212yescomplemented
16C3×#126C2218yescomplemented
17C3×#124C2218yescomplemented
18C3×#123C2218yescomplemented
19C3×#123C2218yescomplemented
20C3×#123C2214yescomplemented
21C3×C22⋊Q8C2214yescomplemented
22C3×C22⋊Q8C2214yescomplemented
23C3×C22⋊Q8C2218yescomplemented
24C3×#121C2218yescomplemented
25C3×#121C2214yescomplemented
26C3×#121C2214yescomplemented
27C3×#120C2218yes—
28C3×#120C2214yescomplemented
29C3×#120C2214yescomplemented
30C3×#120C2214yescomplemented
31C3×C4×D4C2218yescomplemented
32C3×C4×D4C2218yescomplemented
33C3×C4×D4C2218yescomplemented
34C3×#117C2214yescomplemented
35C3×#117C2214yescomplemented
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×C3×C22⋊C4C2211yescharacteristic
38C2×C3×C22⋊C4C2212yes—
39C2×C3×C22⋊C4C2212yescomplemented
40C2×C3×C22⋊C4C2218yescomplemented
41C2×C3×C22⋊C4C2214yescomplemented
42C2×C3×C22⋊C4C2214yescomplemented
43#551C2×C318yescomplemented
44#546C2×C318yescomplemented
45#543C2×C311yescharacteristic, complemented
46#538C2×C314yescomplemented
47#534C2×C311yescharacteristic, complemented
48#525C2×C314yescomplemented
49C2×#120C2×C312yescomplemented
50#518C2×C312yescomplemented
51C22×C22⋊C4C2×C311yescharacteristic, complemented
52C24×C3—24no—
53C24×C3—44no—
54C24×C3C2311yescharacteristic
55C24×C3C2314yes—
56C24×C3C2311yescharacteristic
57C24×C3C2312yes—
58C24×C3C2311yescharacteristic
59C3×#41—48no—
60C2×C3×Q8C2314yescomplemented
61C2×C3×D4—28no—
62C2×C3×D4—48no—
63C2×C3×D4C2314yes—
64C2×C3×D4C2314yes—
65C2×C3×D4C2318yes—
66C2×C3×D4C2318yes—
67C2×C3×D4C2314yescomplemented
68C22×C3×C4—44no—
69C22×C3×C4C2312yes—
70C22×C3×C4C2314yes—
71C22×C3×C4C2312yes—
72C22×C3×C4C2314yes—
73C22×C3×C4C2312yes—
74C22×C3×C4C2314yes—
75C22×C3×C4C2318yescomplemented
76C3×C4⋊C4C2318yescomplemented
77C3×C4⋊C4C2314yescomplemented
78C3×C4⋊C4C2314yescomplemented
79C3×C4⋊C4C2318yescomplemented
80C3×C22⋊C4C2314yes—
81C3×C22⋊C4C2314yes—
82C3×C22⋊C4C2314yes—
83C3×C22⋊C4C2314yes—
84C3×C22⋊C4C2318yes—
85C3×C22⋊C4C2318yes—
86C3×C22⋊C4C2318yes—
87C3×C22⋊C4C2318yescomplemented
88C3×C22⋊C4C2318yescomplemented
89C3×C42C2318yescomplemented
90C25C22×C311yescharacteristic
91C2×#41C22×C314yescomplemented
92C22×D4C22×C312yes—
93C22×D4C22×C314yescomplemented
94C23×C4C22×C312yescomplemented
95#126C22×C318yescomplemented
96#124C22×C318yescomplemented
97#123C22×C318yescomplemented
98#123C22×C318yescomplemented
99#123C22×C314yescomplemented
100C22⋊Q8C22×C314yescomplemented
101C22⋊Q8C22×C314yescomplemented
102C22⋊Q8C22×C318yescomplemented
103#121C22×C318yescomplemented
104#121C22×C314yescomplemented
105#121C22×C314yescomplemented
106#120C22×C318yes—
107#120C22×C314yescomplemented
108#120C22×C314yescomplemented
109#120C22×C314yescomplemented
110C4×D4C22×C318yescomplemented
111C4×D4C22×C318yescomplemented
112C4×D4C22×C318yescomplemented
113#117C22×C314yescomplemented
114#117C22×C314yescomplemented
115C2×C22⋊C4C22×C311yescharacteristic
116C2×C22⋊C4C22×C311yescharacteristic
117C2×C22⋊C4C22×C312yes—
118C2×C22⋊C4C22×C312yescomplemented
119C2×C22⋊C4C22×C318yescomplemented
120C2×C22⋊C4C22×C314yescomplemented
121C2×C22⋊C4C22×C314yescomplemented
122C23×C3—22no—
123C23×C3—24no—
124C23×C3—28no—
125C23×C3—22no—
126C23×C3—28no—
127C23×C3—24no—
128C23×C3—42no—
129C23×C3—44no—
130C23×C3—42no—
131C23×C3—44no—
132C23×C3—416no—
133C23×C3#4114yescomplemented
134C23×C3C2412yes—
135C23×C3C2414yes—
136C23×C3C2412yes—
137C23×C3C2411yescharacteristic
138C23×C3C2412yes—
139C3×Q8—44no—
140C3×D4—28no—
141C3×D4—28no—
142C3×D4—216no—
143C3×D4—48no—
144C3×D4—44no—
145C2×C3×C4—24no—
146C2×C3×C4—28no—
147C2×C3×C4—42no—
148C2×C3×C4—44no—
149C2×C3×C4—42no—
150C2×C3×C4—44no—
151C2×C3×C4—48no—
152C2×C3×C4C2414yes—
153C2×C3×C4C2414yes—
154C2×C3×C4C2418yes—
155C2×C3×C4C2414yes—
156C24—24no—
157C24—44no—
158C24C23×C311yescharacteristic
159C24C23×C314yes—
160C24C23×C311yescharacteristic
161C24C23×C312yes—
162C24C23×C311yescharacteristic
163#41—48no—
164C2×Q8C23×C314yescomplemented
165C2×D4—28no—
166C2×D4—48no—
167C2×D4C23×C314yes—
168C2×D4C23×C314yes—
169C2×D4C23×C318yes—
170C2×D4C23×C318yes—
171C2×D4C23×C314yescomplemented
172C22×C4—44no—
173C22×C4C23×C312yes—
174C22×C4C23×C314yes—
175C22×C4C23×C312yes—
176C22×C4C23×C314yes—
177C22×C4C23×C312yes—
178C22×C4C23×C314yes—
179C22×C4C23×C318yescomplemented
180C4⋊C4C23×C318yescomplemented
181C4⋊C4C23×C314yescomplemented
182C4⋊C4C23×C314yescomplemented
183C4⋊C4C23×C318yescomplemented
184C22⋊C4C23×C314yes—
185C22⋊C4C23×C314yes—
186C22⋊C4C23×C314yes—
187C22⋊C4C23×C314yes—
188C22⋊C4C23×C318yes—
189C22⋊C4C23×C318yes—
190C22⋊C4C23×C318yes—
191C22⋊C4C23×C318yescomplemented
192C22⋊C4C23×C318yescomplemented
193C42C23×C318yescomplemented
194C22×C3—28no—
195C22×C3—22no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—24no—
199C22×C3—24no—
200C22×C3—22no—
201C22×C3—28no—
202C22×C3—24no—
203C22×C3—44no—
204C22×C3—416no—
205C22×C3—44no—
206C22×C3—84no—
207C22×C3C2×#4114yes—
208C22×C3C2×#4112yes—
209C22×C3H(5,𝔽2)14yes—
210C22×C3C2511yescharacteristic
211C3×C4—24no—
212C3×C4—24no—
213C3×C4—24no—
214C3×C4—28no—
215C23—22no—
216C23—24no—
217C23—28no—
218C23—22no—
219C23—28no—
220C23—24no—
221C23—42no—
222C23—44no—
223C23—42no—
224C23—44no—
225C23—416no—
226C23C3×#4114yescomplemented
227C23C24×C312yes—
228C23C24×C314yes—
229C23C24×C312yes—
230C23C24×C311yescharacteristic
231C23C24×C312yes—
232Q8—44no—
233D4—28no—
234D4—28no—
235D4—216no—
236D4—48no—
237D4—44no—
238C2×C4—24no—
239C2×C4—28no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4C24×C314yes—
246C2×C4C24×C314yes—
247C2×C4C24×C318yes—
248C2×C4C24×C314yes—
249C2×C3—22no—
250C2×C3—24no—
251C2×C3—24no—
252C2×C3—42no—
253C2×C3—44no—
254C2×C3C22×#4111yescharacteristic
255C2×C3C2×H(5,𝔽2)12yes—
256C22—28no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—28no—
264C22—24no—
265C22—44no—
266C22—416no—
267C22—44no—
268C22—84no—
269C22C2×C3×#4114yes—
270C22C2×C3×#4112yes—
271C22C3×H(5,𝔽2)14yes—
272C22C25×C311yescharacteristic
273C4—24no—
274C4—24no—
275C4—24no—
276C4—28no—
277C3#349011yescharacteristic, direct product factor, Sylow
278C2—22no—
279C2—24no—
280C2—24no—
281C2—42no—
282C2—44no—
283C2C22×C3×#4111yescharacteristic
284C2C2×C3×H(5,𝔽2)12yes—
subgroupG/H
center, socleC22×C3C25
commutator, Frattini, cosocleC22C25×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#551C2semidirect
2. C3×#546C2semidirect
3. C3×#543C2semidirect
4. C3×#538C2semidirect
5. C3×#534C2semidirect
6. C3×#525C2semidirect
7. C2×C3×#120C2semidirect
8. C3×#518C2semidirect
9. C22×C3×C22⋊C4C2semidirect
10. #3490C3direct (N × H)
12. C2×C3×#41C22semidirect
14. C22×C3×D4C22semidirect
15. C23×C3×C4C22semidirect
16. C3×#126C22semidirect
17. C3×#124C22semidirect
18. C3×#123C22semidirect
19. C3×#123C22semidirect
20. C3×#123C22semidirect
21. C3×C22⋊Q8C22semidirect
22. C3×C22⋊Q8C22semidirect
23. C3×C22⋊Q8C22semidirect
24. C3×#121C22semidirect
25. C3×#121C22semidirect
26. C3×#121C22semidirect
28. C3×#120C22semidirect
29. C3×#120C22semidirect
30. C3×#120C22semidirect
31. C3×C4×D4C22semidirect
32. C3×C4×D4C22semidirect
33. C3×C4×D4C22semidirect
34. C3×#117C22semidirect
35. C3×#117C22semidirect
39. C2×C3×C22⋊C4C22semidirect
40. C2×C3×C22⋊C4C22semidirect
41. C2×C3×C22⋊C4C22semidirect
42. C2×C3×C22⋊C4C22semidirect
43. #551C2×C3semidirect
44. #546C2×C3semidirect
45. #543C2×C3semidirect
46. #538C2×C3semidirect
47. #534C2×C3semidirect
48. #525C2×C3semidirect
49. C2×#120C2×C3semidirect
50. #518C2×C3semidirect
51. C22×C22⋊C4C2×C3semidirect
60. C2×C3×Q8C23semidirect
67. C2×C3×D4C23semidirect
75. C22×C3×C4C23semidirect
76. C3×C4⋊C4C23semidirect
77. C3×C4⋊C4C23semidirect
78. C3×C4⋊C4C23semidirect
79. C3×C4⋊C4C23semidirect
87. C3×C22⋊C4C23semidirect
88. C3×C22⋊C4C23semidirect
89. C3×C42C23semidirect
91. C2×#41C22×C3semidirect
93. C22×D4C22×C3semidirect
94. C23×C4C22×C3semidirect
95. #126C22×C3semidirect
96. #124C22×C3semidirect
97. #123C22×C3semidirect
98. #123C22×C3semidirect
99. #123C22×C3semidirect
100. C22⋊Q8C22×C3semidirect
101. C22⋊Q8C22×C3semidirect
102. C22⋊Q8C22×C3semidirect
103. #121C22×C3semidirect
104. #121C22×C3semidirect
105. #121C22×C3semidirect
107. #120C22×C3semidirect
108. #120C22×C3semidirect
109. #120C22×C3semidirect
110. C4×D4C22×C3semidirect
111. C4×D4C22×C3semidirect
112. C4×D4C22×C3semidirect
113. #117C22×C3semidirect
114. #117C22×C3semidirect
118. C2×C22⋊C4C22×C3semidirect
119. C2×C22⋊C4C22×C3semidirect
120. C2×C22⋊C4C22×C3semidirect
121. C2×C22⋊C4C22×C3semidirect
133. C23×C3#41semidirect
164. C2×Q8C23×C3semidirect
171. C2×D4C23×C3semidirect
179. C22×C4C23×C3semidirect
180. C4⋊C4C23×C3semidirect
181. C4⋊C4C23×C3semidirect
182. C4⋊C4C23×C3semidirect
183. C4⋊C4C23×C3semidirect
191. C22⋊C4C23×C3semidirect
192. C22⋊C4C23×C3semidirect
193. C42C23×C3semidirect
226. C23C3×#41semidirect
277. C3#3490direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5512
C3×#5462
C3×#5432
C3×#5382
C3×#5342
C3×#5252
C2×C3×#1202
C3×#5182
C22×C3×C22⋊C42
#34903