Finite Group Databasebeta

← C23×#3532 #15671151

Subgroups — C23×#3532 #15671151

Order 2944.

1336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2C23×#560C213yesmaximal, complemented
3C23×#551C213yesmaximal, complemented
4C23×#546C216yesmaximal, complemented
5D42×C23C213yesmaximal, complemented
6C23×#538C216yesmaximal, complemented
7C23×#535C213yesmaximal, complemented
8C23×#534C213yesmaximal, complemented
9C2×C23×#120C213yesmaximal, complemented
10C25×C23C2211yescharacteristic, complemented
11C2×#41×C23C2213yescomplemented
12C22×D4×C23C2213yescomplemented
13C22×D4×C23C2216yescomplemented
14C22×D4×C23C2213yescomplemented
15C23×#127C2213yescomplemented
16C23×#127C2211yescharacteristic, complemented
17C23×#126C2216yescomplemented
18C23×#126C2212yescomplemented
19C23×#124C2213yescomplemented
20C23×#124C2216yescomplemented
21C23×#124C2213yescomplemented
22C23×#123C2213yescomplemented
23C23×#123C2216yescomplemented
24C23×#123C2213yescomplemented
25C23×#123C2216yescomplemented
26C23×C22⋊Q8C2216yescomplemented
27C23×#121C2216yescomplemented
28C23×#121C2216yescomplemented
29C23×#121C2216yescomplemented
30C23×#121C2216yescomplemented
31C23×#121C2216yescomplemented
32C23×#120C2213yes—
33C23×#120C2213yescomplemented
34C23×#120C2213yescomplemented
35C23×#120C2216yescomplemented
36C23×#120C2216yescomplemented
37C23×#120C2213yescomplemented
38C23×#120C2216yescomplemented
39C23×#120C2213yescomplemented
40C23×#120C2211yescharacteristic, complemented
41C4×D4×C23C2216yescomplemented
42C4×D4×C23C2216yescomplemented
43C23×#117C2213yescomplemented
44C2×C22⋊C4×C23C2216yescomplemented
45C2×C22⋊C4×C23C2213yescomplemented
46C2×C22⋊C4×C23C2213yescomplemented
47C24×C23—46no—
48C24×C23C2313yes—
49C24×C23C2313yes—
50C24×C23C2313yes—
51C24×C23C2311yescharacteristic
52C24×C23C2311yescharacteristic, complemented
53#41×C23—46no—
54C2×Q8×C23C2313yescomplemented
55C2×D4×C23—46no—
56C2×D4×C23—412no—
57C2×D4×C23—46no—
58C2×D4×C23C2316yes—
59C2×D4×C23C2316yes—
60C2×D4×C23C2313yes—
61C2×D4×C23C2313yes—
62C2×D4×C23C2316yes—
63C2×D4×C23C2313yes—
64C2×D4×C23C2316yes—
65C2×D4×C23C2313yes—
66C2×D4×C23C2313yescomplemented
67C2×D4×C23C2316yescomplemented
68C22×C4×C23C2313yes—
69C22×C4×C23C2316yes—
70C22×C4×C23C2316yescomplemented
71C22×C4×C23C2313yescomplemented
72C4⋊C4×C23C2316yescomplemented
73C4⋊C4×C23C2316yescomplemented
74C4⋊C4×C23C2316yescomplemented
75C22⋊C4×C23C2316yes—
76C22⋊C4×C23C2316yes—
77C22⋊C4×C23C2313yes—
78C22⋊C4×C23C2316yes—
79C22⋊C4×C23C2316yes—
80C22⋊C4×C23C2316yes—
81C22⋊C4×C23C2313yes—
82C22⋊C4×C23C2313yescomplemented
83C22⋊C4×C23C2313yescomplemented
84C22⋊C4×C23C2316yescomplemented
85C22⋊C4×C23C2316yescomplemented
86C42×C23C2311yescharacteristic, complemented
87C42×C23C2313yescomplemented
88C42×C23C2312yescomplemented
89C23×C23—212no—
90C23×C23—26no—
91C23×C23—26no—
92C23×C23—43no—
93C23×C23—46no—
94C23×C23—43no—
95C23×C23—46no—
96C23×C23—43no—
97C23×C23—416no—
98C23×C23C2413yes—
99C23×C23C2411yescharacteristic
100C23×C23C2413yes—
101C23×C23C2413yes—
102C23×C23C2413yes—
103Q8×C23—43no—
104D4×C23—212no—
105D4×C23—212no—
106D4×C23—43no—
107D4×C23—43no—
108D4×C23—43no—
109D4×C23—43no—
110D4×C23—46no—
111D4×C23—46no—
112D4×C23—46no—
113D4×C23—43no—
114C2×C4×C23—26no—
115C2×C4×C23—46no—
116C2×C4×C23—43no—
117C2×C4×C23—46no—
118C2×C4×C23C2413yes—
119C2×C4×C23C2416yes—
120C2×C4×C23C2416yes—
121C2×C4×C23C2413yes—
122#3532C2311yesmaximal, characteristic, direct product factor, Sylow
123C22×C23—26no—
124C22×C23—26no—
125C22×C23—23no—
126C22×C23—26no—
127C22×C23—23no—
128C22×C23—23no—
129C22×C23—26no—
130C22×C23—23no—
131C22×C23—412no—
132C22×C23—412no—
133C22×C23—44no—
134C22×C23—86no—
135C22×C23—82no—
136C22×C23H(5,𝔽2)16yes—
137C22×C23C2511yescharacteristic
138C4×C23—23no—
139C4×C23—23no—
140C4×C23—26no—
141C4×C23—26no—
142H(3,𝔽4)C2×C2311yescharacteristic, complemented
143#560C2×C2313yescomplemented
144#551C2×C2313yescomplemented
145#546C2×C2316yescomplemented
146D42C2×C2313yescomplemented
147#538C2×C2316yescomplemented
148#535C2×C2313yescomplemented
149#534C2×C2313yescomplemented
150C2×#120C2×C2313yescomplemented
151C2×C23—26no—
152C2×C23—43no—
153C2×C23—41no—
154C2×C23—43no—
155C2×C23—43no—
156C2×C23C2×H(5,𝔽2)13yes—
157C25C22×C2311yescharacteristic, complemented
158C2×#41C22×C2313yescomplemented
159C22×D4C22×C2313yescomplemented
160C22×D4C22×C2316yescomplemented
161C22×D4C22×C2313yescomplemented
162#127C22×C2311yescharacteristic, complemented
163#127C22×C2313yescomplemented
164#126C22×C2312yescomplemented
165#126C22×C2316yescomplemented
166#124C22×C2313yescomplemented
167#124C22×C2313yescomplemented
168#124C22×C2316yescomplemented
169#123C22×C2313yescomplemented
170#123C22×C2313yescomplemented
171#123C22×C2316yescomplemented
172#123C22×C2316yescomplemented
173C22⋊Q8C22×C2316yescomplemented
174#121C22×C2316yescomplemented
175#121C22×C2316yescomplemented
176#121C22×C2316yescomplemented
177#121C22×C2316yescomplemented
178#121C22×C2316yescomplemented
179#120C22×C2313yes—
180#120C22×C2316yescomplemented
181#120C22×C2316yescomplemented
182#120C22×C2313yescomplemented
183#120C22×C2313yescomplemented
184#120C22×C2313yescomplemented
185#120C22×C2316yescomplemented
186#120C22×C2311yescharacteristic, complemented
187#120C22×C2313yescomplemented
188C4×D4C22×C2316yescomplemented
189C4×D4C22×C2316yescomplemented
190#117C22×C2313yescomplemented
191C2×C22⋊C4C22×C2313yescomplemented
192C2×C22⋊C4C22×C2316yescomplemented
193C2×C22⋊C4C22×C2313yescomplemented
194C23#353211yescharacteristic, direct product factor, Sylow
195C24—46no—
196C24C23×C2313yes—
197C24C23×C2313yes—
198C24C23×C2313yes—
199C24C23×C2311yescharacteristic
200C24C23×C2311yescharacteristic, complemented
201#41—46no—
202C2×Q8C23×C2313yescomplemented
203C2×D4—46no—
204C2×D4—412no—
205C2×D4—46no—
206C2×D4C23×C2316yes—
207C2×D4C23×C2316yes—
208C2×D4C23×C2313yes—
209C2×D4C23×C2313yes—
210C2×D4C23×C2316yes—
211C2×D4C23×C2313yes—
212C2×D4C23×C2316yes—
213C2×D4C23×C2313yes—
214C2×D4C23×C2313yescomplemented
215C2×D4C23×C2316yescomplemented
216C22×C4C23×C2313yes—
217C22×C4C23×C2316yes—
218C22×C4C23×C2316yescomplemented
219C22×C4C23×C2313yescomplemented
220C4⋊C4C23×C2316yescomplemented
221C4⋊C4C23×C2316yescomplemented
222C4⋊C4C23×C2316yescomplemented
223C22⋊C4C23×C2316yes—
224C22⋊C4C23×C2316yes—
225C22⋊C4C23×C2313yes—
226C22⋊C4C23×C2316yes—
227C22⋊C4C23×C2316yes—
228C22⋊C4C23×C2316yes—
229C22⋊C4C23×C2313yes—
230C22⋊C4C23×C2313yescomplemented
231C22⋊C4C23×C2313yescomplemented
232C22⋊C4C23×C2316yescomplemented
233C22⋊C4C23×C2316yescomplemented
234C42C23×C2311yescharacteristic, complemented
235C42C23×C2313yescomplemented
236C42C23×C2312yescomplemented
237C23—212no—
238C23—26no—
239C23—26no—
240C23—43no—
241C23—46no—
242C23—43no—
243C23—46no—
244C23—43no—
245C23—416no—
246C23C24×C2313yes—
247C23C24×C2311yescharacteristic
248C23C24×C2313yes—
249C23C24×C2313yes—
250C23C24×C2313yes—
251Q8—43no—
252D4—212no—
253D4—212no—
254D4—43no—
255D4—43no—
256D4—43no—
257D4—43no—
258D4—46no—
259D4—46no—
260D4—46no—
261D4—43no—
262C2×C4—26no—
263C2×C4—46no—
264C2×C4—43no—
265C2×C4—46no—
266C2×C4C24×C2313yes—
267C2×C4C24×C2316yes—
268C2×C4C24×C2316yes—
269C2×C4C24×C2313yes—
270C22—26no—
271C22—26no—
272C22—23no—
273C22—26no—
274C22—23no—
275C22—23no—
276C22—26no—
277C22—23no—
278C22—412no—
279C22—412no—
280C22—44no—
281C22—86no—
282C22—82no—
283C22C23×H(5,𝔽2)16yes—
284C22C25×C2311yescharacteristic
285C4—23no—
286C4—23no—
287C4—26no—
288C4—26no—
289C2—26no—
290C2—43no—
291C2—41no—
292C2—43no—
293C2—43no—
294C2C2×C23×H(5,𝔽2)13yes—
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×H(3,𝔽4)C2semidirect
2. C23×#560C2semidirect
3. C23×#551C2semidirect
4. C23×#546C2semidirect
5. D42×C23C2semidirect
6. C23×#538C2semidirect
7. C23×#535C2semidirect
8. C23×#534C2semidirect
9. C2×C23×#120C2semidirect
10. C25×C23C22semidirect
11. C2×#41×C23C22semidirect
12. C22×D4×C23C22semidirect
13. C22×D4×C23C22semidirect
14. C22×D4×C23C22semidirect
15. C23×#127C22semidirect
16. C23×#127C22semidirect
17. C23×#126C22semidirect
18. C23×#126C22semidirect
19. C23×#124C22semidirect
20. C23×#124C22semidirect
21. C23×#124C22semidirect
22. C23×#123C22semidirect
23. C23×#123C22semidirect
24. C23×#123C22semidirect
25. C23×#123C22semidirect
26. C23×C22⋊Q8C22semidirect
27. C23×#121C22semidirect
28. C23×#121C22semidirect
29. C23×#121C22semidirect
30. C23×#121C22semidirect
31. C23×#121C22semidirect
33. C23×#120C22semidirect
34. C23×#120C22semidirect
35. C23×#120C22semidirect
36. C23×#120C22semidirect
37. C23×#120C22semidirect
38. C23×#120C22semidirect
39. C23×#120C22semidirect
40. C23×#120C22semidirect
41. C4×D4×C23C22semidirect
42. C4×D4×C23C22semidirect
43. C23×#117C22semidirect
44. C2×C22⋊C4×C23C22semidirect
45. C2×C22⋊C4×C23C22semidirect
46. C2×C22⋊C4×C23C22semidirect
52. C24×C23C23semidirect
54. C2×Q8×C23C23semidirect
66. C2×D4×C23C23semidirect
67. C2×D4×C23C23semidirect
70. C22×C4×C23C23semidirect
71. C22×C4×C23C23semidirect
72. C4⋊C4×C23C23semidirect
73. C4⋊C4×C23C23semidirect
74. C4⋊C4×C23C23semidirect
82. C22⋊C4×C23C23semidirect
83. C22⋊C4×C23C23semidirect
84. C22⋊C4×C23C23semidirect
85. C22⋊C4×C23C23semidirect
86. C42×C23C23semidirect
87. C42×C23C23semidirect
88. C42×C23C23semidirect
122. #3532C23direct (N × H)
142. H(3,𝔽4)C2×C23semidirect
143. #560C2×C23semidirect
144. #551C2×C23semidirect
145. #546C2×C23semidirect
146. D42C2×C23semidirect
147. #538C2×C23semidirect
148. #535C2×C23semidirect
149. #534C2×C23semidirect
150. C2×#120C2×C23semidirect
157. C25C22×C23semidirect
158. C2×#41C22×C23semidirect
159. C22×D4C22×C23semidirect
160. C22×D4C22×C23semidirect
161. C22×D4C22×C23semidirect
162. #127C22×C23semidirect
163. #127C22×C23semidirect
164. #126C22×C23semidirect
165. #126C22×C23semidirect
166. #124C22×C23semidirect
167. #124C22×C23semidirect
168. #124C22×C23semidirect
169. #123C22×C23semidirect
170. #123C22×C23semidirect
171. #123C22×C23semidirect
172. #123C22×C23semidirect
173. C22⋊Q8C22×C23semidirect
174. #121C22×C23semidirect
175. #121C22×C23semidirect
176. #121C22×C23semidirect
177. #121C22×C23semidirect
178. #121C22×C23semidirect
180. #120C22×C23semidirect
181. #120C22×C23semidirect
182. #120C22×C23semidirect
183. #120C22×C23semidirect
184. #120C22×C23semidirect
185. #120C22×C23semidirect
186. #120C22×C23semidirect
187. #120C22×C23semidirect
188. C4×D4C22×C23semidirect
189. C4×D4C22×C23semidirect
190. #117C22×C23semidirect
191. C2×C22⋊C4C22×C23semidirect
192. C2×C22⋊C4C22×C23semidirect
193. C2×C22⋊C4C22×C23semidirect
194. C23#3532direct (N × H)
200. C24C23×C23semidirect
202. C2×Q8C23×C23semidirect
214. C2×D4C23×C23semidirect
215. C2×D4C23×C23semidirect
218. C22×C4C23×C23semidirect
219. C22×C4C23×C23semidirect
220. C4⋊C4C23×C23semidirect
221. C4⋊C4C23×C23semidirect
222. C4⋊C4C23×C23semidirect
230. C22⋊C4C23×C23semidirect
231. C22⋊C4C23×C23semidirect
232. C22⋊C4C23×C23semidirect
233. C22⋊C4C23×C23semidirect
234. C42C23×C23semidirect
235. C42C23×C23semidirect
236. C42C23×C23semidirect

Maximal subgroups

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

subgroupindex
C23×H(3,𝔽4)2
C23×#5602
C23×#5512
C23×#5462
D42×C232
C23×#5382
C23×#5352
C23×#5342
C2×C23×#1202
#353223