Finite Group Databasebeta

← C23×#3490 #15671109

Subgroups — C23×#3490 #15671109

Order 2944.

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
1C23×#551C218yesmaximal, complemented
2C23×#546C218yesmaximal, complemented
3C23×#543C211yesmaximal, characteristic, complemented
4C23×#538C214yesmaximal, complemented
5C23×#534C211yesmaximal, characteristic, complemented
6C23×#525C214yesmaximal, complemented
7C2×C23×#120C212yesmaximal, complemented
8C23×#518C212yesmaximal, complemented
9C22×C22⋊C4×C23C211yesmaximal, characteristic, complemented
10C25×C23C2211yescharacteristic
11C2×#41×C23C2214yescomplemented
12C22×D4×C23C2212yes—
13C22×D4×C23C2214yescomplemented
14C23×C4×C23C2212yescomplemented
15C23×#126C2218yescomplemented
16C23×#124C2218yescomplemented
17C23×#123C2218yescomplemented
18C23×#123C2214yescomplemented
19C23×#123C2218yescomplemented
20C23×C22⋊Q8C2214yescomplemented
21C23×C22⋊Q8C2218yescomplemented
22C23×C22⋊Q8C2214yescomplemented
23C23×#121C2218yescomplemented
24C23×#121C2214yescomplemented
25C23×#121C2214yescomplemented
26C23×#120C2218yes—
27C23×#120C2214yescomplemented
28C23×#120C2214yescomplemented
29C23×#120C2214yescomplemented
30C4×D4×C23C2218yescomplemented
31C4×D4×C23C2218yescomplemented
32C4×D4×C23C2218yescomplemented
33C23×#117C2214yescomplemented
34C23×#117C2214yescomplemented
35C2×C22⋊C4×C23C2211yescharacteristic
36C2×C22⋊C4×C23C2211yescharacteristic
37C2×C22⋊C4×C23C2212yes—
38C2×C22⋊C4×C23C2214yescomplemented
39C2×C22⋊C4×C23C2218yescomplemented
40C2×C22⋊C4×C23C2212yescomplemented
41C2×C22⋊C4×C23C2214yescomplemented
42C24×C23—24no—
43C24×C23—44no—
44C24×C23C2311yescharacteristic
45C24×C23C2314yes—
46C24×C23C2311yescharacteristic
47C24×C23C2312yes—
48C24×C23C2311yescharacteristic
49#41×C23—48no—
50C2×Q8×C23C2314yescomplemented
51C2×D4×C23—28no—
52C2×D4×C23—48no—
53C2×D4×C23C2314yes—
54C2×D4×C23C2314yes—
55C2×D4×C23C2318yes—
56C2×D4×C23C2318yes—
57C2×D4×C23C2314yescomplemented
58C22×C4×C23—44no—
59C22×C4×C23C2312yes—
60C22×C4×C23C2314yes—
61C22×C4×C23C2312yes—
62C22×C4×C23C2314yes—
63C22×C4×C23C2312yes—
64C22×C4×C23C2314yes—
65C22×C4×C23C2318yescomplemented
66C4⋊C4×C23C2318yescomplemented
67C4⋊C4×C23C2314yescomplemented
68C4⋊C4×C23C2314yescomplemented
69C4⋊C4×C23C2318yescomplemented
70C22⋊C4×C23C2314yes—
71C22⋊C4×C23C2314yes—
72C22⋊C4×C23C2314yes—
73C22⋊C4×C23C2314yes—
74C22⋊C4×C23C2318yes—
75C22⋊C4×C23C2318yes—
76C22⋊C4×C23C2318yes—
77C22⋊C4×C23C2318yescomplemented
78C22⋊C4×C23C2318yescomplemented
79C42×C23C2318yescomplemented
80C23×C23—22no—
81C23×C23—24no—
82C23×C23—28no—
83C23×C23—22no—
84C23×C23—28no—
85C23×C23—24no—
86C23×C23—42no—
87C23×C23—44no—
88C23×C23—42no—
89C23×C23—44no—
90C23×C23—416no—
91C23×C23#4114yescomplemented
92C23×C23C2412yes—
93C23×C23C2414yes—
94C23×C23C2412yes—
95C23×C23C2411yescharacteristic
96C23×C23C2412yes—
97Q8×C23—44no—
98D4×C23—28no—
99D4×C23—28no—
100D4×C23—216no—
101D4×C23—48no—
102D4×C23—44no—
103C2×C4×C23—24no—
104C2×C4×C23—28no—
105C2×C4×C23—42no—
106C2×C4×C23—44no—
107C2×C4×C23—42no—
108C2×C4×C23—44no—
109C2×C4×C23—48no—
110C2×C4×C23C2414yes—
111C2×C4×C23C2414yes—
112C2×C4×C23C2418yes—
113C2×C4×C23C2414yes—
114#3490C2311yesmaximal, characteristic, direct product factor, Sylow
115C22×C23—28no—
116C22×C23—22no—
117C22×C23—22no—
118C22×C23—22no—
119C22×C23—24no—
120C22×C23—24no—
121C22×C23—22no—
122C22×C23—28no—
123C22×C23—24no—
124C22×C23—44no—
125C22×C23—416no—
126C22×C23—44no—
127C22×C23—84no—
128C22×C23C2×#4114yes—
129C22×C23C2×#4112yes—
130C22×C23H(5,𝔽2)14yes—
131C22×C23C2511yescharacteristic
132C4×C23—24no—
133C4×C23—24no—
134C4×C23—24no—
135C4×C23—28no—
136#551C2×C2318yescomplemented
137#546C2×C2318yescomplemented
138#543C2×C2311yescharacteristic, complemented
139#538C2×C2314yescomplemented
140#534C2×C2311yescharacteristic, complemented
141#525C2×C2314yescomplemented
142C2×#120C2×C2312yescomplemented
143#518C2×C2312yescomplemented
144C22×C22⋊C4C2×C2311yescharacteristic, complemented
145C2×C23—22no—
146C2×C23—24no—
147C2×C23—24no—
148C2×C23—42no—
149C2×C23—44no—
150C2×C23C22×#4111yescharacteristic
151C2×C23C2×H(5,𝔽2)12yes—
152C25C22×C2311yescharacteristic
153C2×#41C22×C2314yescomplemented
154C22×D4C22×C2312yes—
155C22×D4C22×C2314yescomplemented
156C23×C4C22×C2312yescomplemented
157#126C22×C2318yescomplemented
158#124C22×C2318yescomplemented
159#123C22×C2318yescomplemented
160#123C22×C2318yescomplemented
161#123C22×C2314yescomplemented
162C22⋊Q8C22×C2314yescomplemented
163C22⋊Q8C22×C2314yescomplemented
164C22⋊Q8C22×C2318yescomplemented
165#121C22×C2318yescomplemented
166#121C22×C2314yescomplemented
167#121C22×C2314yescomplemented
168#120C22×C2318yes—
169#120C22×C2314yescomplemented
170#120C22×C2314yescomplemented
171#120C22×C2314yescomplemented
172C4×D4C22×C2318yescomplemented
173C4×D4C22×C2318yescomplemented
174C4×D4C22×C2318yescomplemented
175#117C22×C2314yescomplemented
176#117C22×C2314yescomplemented
177C2×C22⋊C4C22×C2311yescharacteristic
178C2×C22⋊C4C22×C2311yescharacteristic
179C2×C22⋊C4C22×C2312yes—
180C2×C22⋊C4C22×C2312yescomplemented
181C2×C22⋊C4C22×C2318yescomplemented
182C2×C22⋊C4C22×C2314yescomplemented
183C2×C22⋊C4C22×C2314yescomplemented
184C23#349011yescharacteristic, direct product factor, Sylow
185C24—24no—
186C24—44no—
187C24C23×C2311yescharacteristic
188C24C23×C2314yes—
189C24C23×C2311yescharacteristic
190C24C23×C2312yes—
191C24C23×C2311yescharacteristic
192#41—48no—
193C2×Q8C23×C2314yescomplemented
194C2×D4—28no—
195C2×D4—48no—
196C2×D4C23×C2314yes—
197C2×D4C23×C2314yes—
198C2×D4C23×C2318yes—
199C2×D4C23×C2318yes—
200C2×D4C23×C2314yescomplemented
201C22×C4—44no—
202C22×C4C23×C2312yes—
203C22×C4C23×C2314yes—
204C22×C4C23×C2312yes—
205C22×C4C23×C2314yes—
206C22×C4C23×C2312yes—
207C22×C4C23×C2314yes—
208C22×C4C23×C2318yescomplemented
209C4⋊C4C23×C2318yescomplemented
210C4⋊C4C23×C2314yescomplemented
211C4⋊C4C23×C2314yescomplemented
212C4⋊C4C23×C2318yescomplemented
213C22⋊C4C23×C2314yes—
214C22⋊C4C23×C2314yes—
215C22⋊C4C23×C2314yes—
216C22⋊C4C23×C2314yes—
217C22⋊C4C23×C2318yes—
218C22⋊C4C23×C2318yes—
219C22⋊C4C23×C2318yes—
220C22⋊C4C23×C2318yescomplemented
221C22⋊C4C23×C2318yescomplemented
222C42C23×C2318yescomplemented
223C23—22no—
224C23—24no—
225C23—28no—
226C23—22no—
227C23—28no—
228C23—24no—
229C23—42no—
230C23—44no—
231C23—42no—
232C23—44no—
233C23—416no—
234C23#41×C2314yescomplemented
235C23C24×C2312yes—
236C23C24×C2314yes—
237C23C24×C2312yes—
238C23C24×C2311yescharacteristic
239C23C24×C2312yes—
240Q8—44no—
241D4—28no—
242D4—28no—
243D4—216no—
244D4—48no—
245D4—44no—
246C2×C4—24no—
247C2×C4—28no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4C24×C2314yes—
254C2×C4C24×C2314yes—
255C2×C4C24×C2318yes—
256C2×C4C24×C2314yes—
257C22—28no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—24no—
263C22—22no—
264C22—28no—
265C22—24no—
266C22—44no—
267C22—416no—
268C22—44no—
269C22—84no—
270C22C2×#41×C2314yes—
271C22C2×#41×C2312yes—
272C22C23×H(5,𝔽2)14yes—
273C22C25×C2311yescharacteristic
274C4—24no—
275C4—24no—
276C4—24no—
277C4—28no—
278C2—22no—
279C2—24no—
280C2—24no—
281C2—42no—
282C2—44no—
283C2C22×#41×C2311yescharacteristic
284C2C2×C23×H(5,𝔽2)12yes—
subgroupG/H
center, socleC22×C23C25
commutator, Frattini, cosocleC22C25×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#551C2semidirect
2. C23×#546C2semidirect
3. C23×#543C2semidirect
4. C23×#538C2semidirect
5. C23×#534C2semidirect
6. C23×#525C2semidirect
7. C2×C23×#120C2semidirect
8. C23×#518C2semidirect
9. C22×C22⋊C4×C23C2semidirect
11. C2×#41×C23C22semidirect
13. C22×D4×C23C22semidirect
14. C23×C4×C23C22semidirect
15. C23×#126C22semidirect
16. C23×#124C22semidirect
17. C23×#123C22semidirect
18. C23×#123C22semidirect
19. C23×#123C22semidirect
20. C23×C22⋊Q8C22semidirect
21. C23×C22⋊Q8C22semidirect
22. C23×C22⋊Q8C22semidirect
23. C23×#121C22semidirect
24. C23×#121C22semidirect
25. C23×#121C22semidirect
27. C23×#120C22semidirect
28. C23×#120C22semidirect
29. C23×#120C22semidirect
30. C4×D4×C23C22semidirect
31. C4×D4×C23C22semidirect
32. C4×D4×C23C22semidirect
33. C23×#117C22semidirect
34. C23×#117C22semidirect
38. C2×C22⋊C4×C23C22semidirect
39. C2×C22⋊C4×C23C22semidirect
40. C2×C22⋊C4×C23C22semidirect
41. C2×C22⋊C4×C23C22semidirect
50. C2×Q8×C23C23semidirect
57. C2×D4×C23C23semidirect
65. C22×C4×C23C23semidirect
66. C4⋊C4×C23C23semidirect
67. C4⋊C4×C23C23semidirect
68. C4⋊C4×C23C23semidirect
69. C4⋊C4×C23C23semidirect
77. C22⋊C4×C23C23semidirect
78. C22⋊C4×C23C23semidirect
79. C42×C23C23semidirect
91. C23×C23#41semidirect
114. #3490C23direct (N × H)
136. #551C2×C23semidirect
137. #546C2×C23semidirect
138. #543C2×C23semidirect
139. #538C2×C23semidirect
140. #534C2×C23semidirect
141. #525C2×C23semidirect
142. C2×#120C2×C23semidirect
143. #518C2×C23semidirect
144. C22×C22⋊C4C2×C23semidirect
153. C2×#41C22×C23semidirect
155. C22×D4C22×C23semidirect
156. C23×C4C22×C23semidirect
157. #126C22×C23semidirect
158. #124C22×C23semidirect
159. #123C22×C23semidirect
160. #123C22×C23semidirect
161. #123C22×C23semidirect
162. C22⋊Q8C22×C23semidirect
163. C22⋊Q8C22×C23semidirect
164. C22⋊Q8C22×C23semidirect
165. #121C22×C23semidirect
166. #121C22×C23semidirect
167. #121C22×C23semidirect
169. #120C22×C23semidirect
170. #120C22×C23semidirect
171. #120C22×C23semidirect
172. C4×D4C22×C23semidirect
173. C4×D4C22×C23semidirect
174. C4×D4C22×C23semidirect
175. #117C22×C23semidirect
176. #117C22×C23semidirect
180. C2×C22⋊C4C22×C23semidirect
181. C2×C22⋊C4C22×C23semidirect
182. C2×C22⋊C4C22×C23semidirect
183. C2×C22⋊C4C22×C23semidirect
184. C23#3490direct (N × H)
193. C2×Q8C23×C23semidirect
200. C2×D4C23×C23semidirect
208. C22×C4C23×C23semidirect
209. C4⋊C4C23×C23semidirect
210. C4⋊C4C23×C23semidirect
211. C4⋊C4C23×C23semidirect
212. C4⋊C4C23×C23semidirect
220. C22⋊C4C23×C23semidirect
221. C22⋊C4C23×C23semidirect
222. C42C23×C23semidirect
234. C23#41×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5512
C23×#5462
C23×#5432
C23×#5382
C23×#5342
C23×#5252
C2×C23×#1202
C23×#5182
C22×C22⋊C4×C232
#349023