Finite Group Databasebeta

← #3490×C211 #41011110

Subgroups — #3490×C211 #41011110

Order 27008.

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
1#551×C211C218yesmaximal, complemented
2#546×C211C218yesmaximal, complemented
3#543×C211C211yesmaximal, characteristic, complemented
4#538×C211C214yesmaximal, complemented
5#534×C211C211yesmaximal, characteristic, complemented
6#525×C211C214yesmaximal, complemented
7C2×#120×C211C212yesmaximal, complemented
8#518×C211C212yesmaximal, complemented
9C22×C22⋊C4×C211C211yesmaximal, characteristic, complemented
10C25×C211C2211yescharacteristic
11C2×#41×C211C2214yescomplemented
12C22×D4×C211C2212yes—
13C22×D4×C211C2214yescomplemented
14C23×C4×C211C2212yescomplemented
15#126×C211C2218yescomplemented
16#124×C211C2218yescomplemented
17#123×C211C2218yescomplemented
18#123×C211C2218yescomplemented
19#123×C211C2214yescomplemented
20C22⋊Q8×C211C2214yescomplemented
21C22⋊Q8×C211C2214yescomplemented
22C22⋊Q8×C211C2218yescomplemented
23#121×C211C2218yescomplemented
24#121×C211C2214yescomplemented
25#121×C211C2214yescomplemented
26#120×C211C2218yes—
27#120×C211C2214yescomplemented
28#120×C211C2214yescomplemented
29#120×C211C2214yescomplemented
30C4×D4×C211C2218yescomplemented
31C4×D4×C211C2218yescomplemented
32C4×D4×C211C2218yescomplemented
33#117×C211C2214yescomplemented
34#117×C211C2214yescomplemented
35C2×C22⋊C4×C211C2211yescharacteristic
36C2×C22⋊C4×C211C2212yes—
37C2×C22⋊C4×C211C2211yescharacteristic
38C2×C22⋊C4×C211C2212yescomplemented
39C2×C22⋊C4×C211C2218yescomplemented
40C2×C22⋊C4×C211C2214yescomplemented
41C2×C22⋊C4×C211C2214yescomplemented
42C24×C211—24no—
43C24×C211—44no—
44C24×C211C2311yescharacteristic
45C24×C211C2314yes—
46C24×C211C2311yescharacteristic
47C24×C211C2312yes—
48C24×C211C2311yescharacteristic
49#41×C211—48no—
50C2×Q8×C211C2314yescomplemented
51C2×D4×C211—28no—
52C2×D4×C211—48no—
53C2×D4×C211C2314yes—
54C2×D4×C211C2314yes—
55C2×D4×C211C2318yes—
56C2×D4×C211C2318yes—
57C2×D4×C211C2314yescomplemented
58C22×C4×C211—44no—
59C22×C4×C211C2312yes—
60C22×C4×C211C2314yes—
61C22×C4×C211C2312yes—
62C22×C4×C211C2314yes—
63C22×C4×C211C2312yes—
64C22×C4×C211C2314yes—
65C22×C4×C211C2318yescomplemented
66C4⋊C4×C211C2318yescomplemented
67C4⋊C4×C211C2314yescomplemented
68C4⋊C4×C211C2314yescomplemented
69C4⋊C4×C211C2318yescomplemented
70C22⋊C4×C211C2314yes—
71C22⋊C4×C211C2314yes—
72C22⋊C4×C211C2314yes—
73C22⋊C4×C211C2314yes—
74C22⋊C4×C211C2318yes—
75C22⋊C4×C211C2318yes—
76C22⋊C4×C211C2318yes—
77C22⋊C4×C211C2318yescomplemented
78C22⋊C4×C211C2318yescomplemented
79C42×C211C2318yescomplemented
80C23×C211—22no—
81C23×C211—24no—
82C23×C211—28no—
83C23×C211—22no—
84C23×C211—28no—
85C23×C211—24no—
86C23×C211—42no—
87C23×C211—44no—
88C23×C211—42no—
89C23×C211—44no—
90C23×C211—416no—
91C23×C211#4114yescomplemented
92C23×C211C2412yes—
93C23×C211C2414yes—
94C23×C211C2412yes—
95C23×C211C2411yescharacteristic
96C23×C211C2412yes—
97Q8×C211—44no—
98D4×C211—28no—
99D4×C211—28no—
100D4×C211—216no—
101D4×C211—48no—
102D4×C211—44no—
103C2×C4×C211—24no—
104C2×C4×C211—28no—
105C2×C4×C211—42no—
106C2×C4×C211—44no—
107C2×C4×C211—42no—
108C2×C4×C211—44no—
109C2×C4×C211—48no—
110C2×C4×C211C2414yes—
111C2×C4×C211C2414yes—
112C2×C4×C211C2418yes—
113C2×C4×C211C2414yes—
114C22×C211—28no—
115C22×C211—22no—
116C22×C211—22no—
117C22×C211—22no—
118C22×C211—24no—
119C22×C211—24no—
120C22×C211—22no—
121C22×C211—28no—
122C22×C211—24no—
123C22×C211—44no—
124C22×C211—416no—
125C22×C211—44no—
126C22×C211—84no—
127C22×C211C2×#4114yes—
128C22×C211C2×#4112yes—
129C22×C211H(5,𝔽2)14yes—
130C22×C211C2511yescharacteristic
131C4×C211—24no—
132C4×C211—24no—
133C4×C211—24no—
134C4×C211—28no—
135C2×C211—22no—
136C2×C211—24no—
137C2×C211—24no—
138C2×C211—42no—
139C2×C211—44no—
140C2×C211C22×#4111yescharacteristic
141C2×C211C2×H(5,𝔽2)12yes—
142C211#349011yescharacteristic, direct product factor, Sylow
143#3490C21111yesmaximal, characteristic, direct product factor, Sylow
144#551C2×C21118yescomplemented
145#546C2×C21118yescomplemented
146#543C2×C21111yescharacteristic, complemented
147#538C2×C21114yescomplemented
148#534C2×C21111yescharacteristic, complemented
149#525C2×C21114yescomplemented
150C2×#120C2×C21112yescomplemented
151#518C2×C21112yescomplemented
152C22×C22⋊C4C2×C21111yescharacteristic, complemented
153C25C22×C21111yescharacteristic
154C2×#41C22×C21114yescomplemented
155C22×D4C22×C21112yes—
156C22×D4C22×C21114yescomplemented
157C23×C4C22×C21112yescomplemented
158#126C22×C21118yescomplemented
159#124C22×C21118yescomplemented
160#123C22×C21118yescomplemented
161#123C22×C21118yescomplemented
162#123C22×C21114yescomplemented
163C22⋊Q8C22×C21114yescomplemented
164C22⋊Q8C22×C21114yescomplemented
165C22⋊Q8C22×C21118yescomplemented
166#121C22×C21118yescomplemented
167#121C22×C21114yescomplemented
168#121C22×C21114yescomplemented
169#120C22×C21118yes—
170#120C22×C21114yescomplemented
171#120C22×C21114yescomplemented
172#120C22×C21114yescomplemented
173C4×D4C22×C21118yescomplemented
174C4×D4C22×C21118yescomplemented
175C4×D4C22×C21118yescomplemented
176#117C22×C21114yescomplemented
177#117C22×C21114yescomplemented
178C2×C22⋊C4C22×C21111yescharacteristic
179C2×C22⋊C4C22×C21111yescharacteristic
180C2×C22⋊C4C22×C21112yes—
181C2×C22⋊C4C22×C21112yescomplemented
182C2×C22⋊C4C22×C21118yescomplemented
183C2×C22⋊C4C22×C21114yescomplemented
184C2×C22⋊C4C22×C21114yescomplemented
185C24—24no—
186C24—44no—
187C24C23×C21111yescharacteristic
188C24C23×C21114yes—
189C24C23×C21111yescharacteristic
190C24C23×C21112yes—
191C24C23×C21111yescharacteristic
192#41—48no—
193C2×Q8C23×C21114yescomplemented
194C2×D4—28no—
195C2×D4—48no—
196C2×D4C23×C21114yes—
197C2×D4C23×C21114yes—
198C2×D4C23×C21118yes—
199C2×D4C23×C21118yes—
200C2×D4C23×C21114yescomplemented
201C22×C4—44no—
202C22×C4C23×C21112yes—
203C22×C4C23×C21114yes—
204C22×C4C23×C21112yes—
205C22×C4C23×C21114yes—
206C22×C4C23×C21112yes—
207C22×C4C23×C21114yes—
208C22×C4C23×C21118yescomplemented
209C4⋊C4C23×C21118yescomplemented
210C4⋊C4C23×C21114yescomplemented
211C4⋊C4C23×C21114yescomplemented
212C4⋊C4C23×C21118yescomplemented
213C22⋊C4C23×C21114yes—
214C22⋊C4C23×C21114yes—
215C22⋊C4C23×C21114yes—
216C22⋊C4C23×C21114yes—
217C22⋊C4C23×C21118yes—
218C22⋊C4C23×C21118yes—
219C22⋊C4C23×C21118yes—
220C22⋊C4C23×C21118yescomplemented
221C22⋊C4C23×C21118yescomplemented
222C42C23×C21118yescomplemented
223C23—22no—
224C23—24no—
225C23—28no—
226C23—22no—
227C23—28no—
228C23—24no—
229C23—42no—
230C23—44no—
231C23—42no—
232C23—44no—
233C23—416no—
234C23#41×C21114yescomplemented
235C23C24×C21112yes—
236C23C24×C21114yes—
237C23C24×C21112yes—
238C23C24×C21111yescharacteristic
239C23C24×C21112yes—
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×C21114yes—
254C2×C4C24×C21114yes—
255C2×C4C24×C21118yes—
256C2×C4C24×C21114yes—
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×C21114yes—
271C22C2×#41×C21112yes—
272C22H(5,𝔽2)×C21114yes—
273C22C25×C21111yescharacteristic
274C4—24no—
275C4—24no—
276C4—24no—
277C4—28no—
278C2—22no—
279C2—24no—
280C2—24no—
281C2—42no—
282C2—44no—
283C2C22×#41×C21111yescharacteristic
284C2C2×H(5,𝔽2)×C21112yes—
subgroupG/H
center, socleC22×C211C25
commutator, Frattini, cosocleC22C25×C211
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. #551×C211C2semidirect
2. #546×C211C2semidirect
3. #543×C211C2semidirect
4. #538×C211C2semidirect
5. #534×C211C2semidirect
6. #525×C211C2semidirect
7. C2×#120×C211C2semidirect
8. #518×C211C2semidirect
9. C22×C22⋊C4×C211C2semidirect
11. C2×#41×C211C22semidirect
13. C22×D4×C211C22semidirect
14. C23×C4×C211C22semidirect
15. #126×C211C22semidirect
16. #124×C211C22semidirect
17. #123×C211C22semidirect
18. #123×C211C22semidirect
19. #123×C211C22semidirect
20. C22⋊Q8×C211C22semidirect
21. C22⋊Q8×C211C22semidirect
22. C22⋊Q8×C211C22semidirect
23. #121×C211C22semidirect
24. #121×C211C22semidirect
25. #121×C211C22semidirect
27. #120×C211C22semidirect
28. #120×C211C22semidirect
29. #120×C211C22semidirect
30. C4×D4×C211C22semidirect
31. C4×D4×C211C22semidirect
32. C4×D4×C211C22semidirect
33. #117×C211C22semidirect
34. #117×C211C22semidirect
38. C2×C22⋊C4×C211C22semidirect
39. C2×C22⋊C4×C211C22semidirect
40. C2×C22⋊C4×C211C22semidirect
41. C2×C22⋊C4×C211C22semidirect
50. C2×Q8×C211C23semidirect
57. C2×D4×C211C23semidirect
65. C22×C4×C211C23semidirect
66. C4⋊C4×C211C23semidirect
67. C4⋊C4×C211C23semidirect
68. C4⋊C4×C211C23semidirect
69. C4⋊C4×C211C23semidirect
77. C22⋊C4×C211C23semidirect
78. C22⋊C4×C211C23semidirect
79. C42×C211C23semidirect
91. C23×C211#41semidirect
142. C211#3490direct (N × H)
143. #3490C211direct (N × H)
144. #551C2×C211semidirect
145. #546C2×C211semidirect
146. #543C2×C211semidirect
147. #538C2×C211semidirect
148. #534C2×C211semidirect
149. #525C2×C211semidirect
150. C2×#120C2×C211semidirect
151. #518C2×C211semidirect
152. C22×C22⋊C4C2×C211semidirect
154. C2×#41C22×C211semidirect
156. C22×D4C22×C211semidirect
157. C23×C4C22×C211semidirect
158. #126C22×C211semidirect
159. #124C22×C211semidirect
160. #123C22×C211semidirect
161. #123C22×C211semidirect
162. #123C22×C211semidirect
163. C22⋊Q8C22×C211semidirect
164. C22⋊Q8C22×C211semidirect
165. C22⋊Q8C22×C211semidirect
166. #121C22×C211semidirect
167. #121C22×C211semidirect
168. #121C22×C211semidirect
170. #120C22×C211semidirect
171. #120C22×C211semidirect
172. #120C22×C211semidirect
173. C4×D4C22×C211semidirect
174. C4×D4C22×C211semidirect
175. C4×D4C22×C211semidirect
176. #117C22×C211semidirect
177. #117C22×C211semidirect
181. C2×C22⋊C4C22×C211semidirect
182. C2×C22⋊C4C22×C211semidirect
183. C2×C22⋊C4C22×C211semidirect
184. C2×C22⋊C4C22×C211semidirect
193. C2×Q8C23×C211semidirect
200. C2×D4C23×C211semidirect
208. C22×C4C23×C211semidirect
209. C4⋊C4C23×C211semidirect
210. C4⋊C4C23×C211semidirect
211. C4⋊C4C23×C211semidirect
212. C4⋊C4C23×C211semidirect
220. C22⋊C4C23×C211semidirect
221. C22⋊C4C23×C211semidirect
222. C42C23×C211semidirect
234. C23#41×C211semidirect

Maximal subgroups

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

subgroupindex
#551×C2112
#546×C2112
#543×C2112
#538×C2112
#534×C2112
#525×C2112
C2×#120×C2112
#518×C2112
C22×C22⋊C4×C2112
#3490211