Finite Group Databasebeta

← C23×#3310 #15670929

Subgroups — C23×#3310 #15670929

Order 2944.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#552C211yesmaximal, characteristic, complemented
2D42×C23C211yesmaximal, characteristic
3C23×#490C211yesmaximal, characteristic, complemented
4C23×#482C211yesmaximal, characteristic, complemented
5C23×#480C211yesmaximal, characteristic, complemented
6C23×#478C211yesmaximal, characteristic, complemented
7C23×#468C211yesmaximal, characteristic, complemented
8C23×#465C211yesmaximal, characteristic, complemented
9C23×#460C211yesmaximal, characteristic, complemented
10C23×#450C211yesmaximal, characteristic, complemented
11C23×#450C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C23×#435C211yesmaximal, characteristic, complemented
14C23×#418C211yesmaximal, characteristic, complemented
15C2×C23×#102C211yesmaximal, characteristic, complemented
16C22×D4×C23—21no—
17C22×D4×C23C2211yescharacteristic
18C22×D4×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×SD16×C23C2211yescharacteristic, complemented
21C2×SD16×C23C2211yescharacteristic, complemented
22C2×M16×C23C2211yescharacteristic, complemented
23C22×C8×C23C2211yescharacteristic, complemented
24C23×#127C2211yescharacteristic
25C23×#126—21no—
26C23×#125—21no—
27C23×#125C2211yescharacteristic, complemented
28C23×#123—21no—
29C23×C22⋊Q8C2211yescharacteristic, complemented
30C23×C22⋊Q8C2211yescharacteristic, complemented
31C23×#121—21no—
32C23×#121C2211yescharacteristic
33C23×#121C2211yescharacteristic
34C23×#120—21no—
35C23×#120—21no—
36C4×Q8×C23C2211yescharacteristic, complemented
37C4×D4×C23C2211yescharacteristic
38C4×D4×C23C2211yescharacteristic
39C23×#117—21no—
40C23×#117C2211yescharacteristic, complemented
41C2×C4⋊C4×C23C2211yescharacteristic, complemented
42C23×#107C2211yescharacteristic, complemented
43C23×#107C2211yescharacteristic, complemented
44C23×#106C2211yescharacteristic, complemented
45C23×C4⋊C8C2211yescharacteristic, complemented
46C23×#103C2211yescharacteristic, complemented
47C23×#102C2211yescharacteristic
48C23×#102C2211yescharacteristic
49C23×#102C2211yescharacteristic
50C23×#102C2211yescharacteristic, complemented
51C23×#102C2211yescharacteristic, complemented
52C23×#102C2211yescharacteristic, complemented
53C23×#102C2211yescharacteristic, complemented
54C23×#102C2211yescharacteristic, complemented
55C23×#102C2211yescharacteristic, complemented
56C23×C22⋊C8C2211yescharacteristic, complemented
57C23×C22⋊C8C2211yescharacteristic, complemented
58C23×#97C2211yescharacteristic, complemented
59C24×C23—21no—
60C24×C23—21no—
61C2×Q8×C23C2311yescharacteristic
62C2×D4×C23—22no—
63C2×D4×C23—22no—
64C2×D4×C23—21no—
65C2×D4×C23—21no—
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—44no—
73C2×D4×C23D411yescharacteristic
74C2×D4×C23C2311yescharacteristic
75C2×D4×C23C2311yescharacteristic
76C2×D4×C23C2311yescharacteristic
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23C2311yescharacteristic
80C22×C4×C23C2311yescharacteristic
81SD16×C23—22no—
82SD16×C23—41no—
83SD16×C23—41no—
84M16×C23—41no—
85C2×C8×C23—41no—
86C2×C8×C23C2311yescharacteristic
87C2×C8×C23C2311yescharacteristic
88C2×C8×C23C2311yescharacteristic
89C2×C8×C23C2311yescharacteristic
90C4⋊C4×C23—21no—
91C4⋊C4×C23—21no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23D411yescharacteristic
97C4⋊C4×C23C2311yescharacteristic
98C4⋊C4×C23C2311yescharacteristic
99C4⋊C4×C23C2311yescharacteristic
100C4⋊C4×C23C2311yescharacteristic
101C22⋊C4×C23—21no—
102C22⋊C4×C23—21no—
103C22⋊C4×C23—21no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23—21no—
107C22⋊C4×C23D411yescharacteristic
108C22⋊C4×C23D411yescharacteristic
109C42×C23—21no—
110C42×C23—21no—
111C42×C23C2311yescharacteristic
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—42no—
116C23×C23—42no—
117C23×C23—42no—
118C23×C23—42no—
119C23×C23—81no—
120C23×C23—81no—
121C23×C23C2×D411yescharacteristic
122C23×C23C2×D411yescharacteristic
123Q8×C23—21no—
124Q8×C23—21no—
125D4×C23—22no—
126D4×C23—21no—
127D4×C23—21no—
128D4×C23—21no—
129D4×C23—21no—
130D4×C23—21no—
131D4×C23—42no—
132D4×C23—42no—
133D4×C23—42no—
134D4×C23—42no—
135D4×C23—42no—
136D4×C23—81no—
137D4×C23#4112yes—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23C2×D411yescharacteristic
150C2×C4×C23C2×D411yescharacteristic
151C2×C4×C23C2×D411yescharacteristic
152C2×C4×C23C2×D411yescharacteristic
153C2×C4×C23C2411yescharacteristic
154C8×C23—21no—
155C8×C23—21no—
156C8×C23—21no—
157C8×C23—21no—
158#3310C2311yesmaximal, characteristic, direct product factor, Sylow
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—22no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—41no—
166C22×C23—41no—
167C22×C23—41no—
168C22×C23—41no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23—41no—
173C22×C23—84no—
174C22×C23—84no—
175C22×C23AGL(1,ℤ/8)12yes—
176C22×C23C22×D411yescharacteristic
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—41no—
184C4×C23—41no—
185C4×C23—41no—
186C4×C23C2×#4111yescharacteristic
187C4×C23H(5,𝔽2)11yescharacteristic
188#552C2×C2311yescharacteristic, complemented
189D42C2×C2311yescharacteristic
190#490C2×C2311yescharacteristic, complemented
191#482C2×C2311yescharacteristic, complemented
192#480C2×C2311yescharacteristic, complemented
193#478C2×C2311yescharacteristic, complemented
194#468C2×C2311yescharacteristic, complemented
195#465C2×C2311yescharacteristic, complemented
196#460C2×C2311yescharacteristic, complemented
197#450C2×C2311yescharacteristic, complemented
198#450C2×C2311yescharacteristic, complemented
199#440C2×C2311yescharacteristic, complemented
200#435C2×C2311yescharacteristic, complemented
201#418C2×C2311yescharacteristic, complemented
202C2×#102C2×C2311yescharacteristic, complemented
203C2×C23—22no—
204C2×C23—41no—
205C2×C23—42no—
206C2×C23—81no—
207C2×C23—81no—
208C2×C23#54611yescharacteristic
209C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
210C2×C23#57711yescharacteristic
211C22×D4—21no—
212C22×D4C22×C2311yescharacteristic
213C22×D4C22×C2311yescharacteristic
214C2×SD16C22×C2311yescharacteristic
215C2×SD16C22×C2311yescharacteristic, complemented
216C2×SD16C22×C2311yescharacteristic, complemented
217C2×M16C22×C2311yescharacteristic, complemented
218C22×C8C22×C2311yescharacteristic, complemented
219#127C22×C2311yescharacteristic
220#126—21no—
221#125—21no—
222#125C22×C2311yescharacteristic, complemented
223#123—21no—
224C22⋊Q8C22×C2311yescharacteristic, complemented
225C22⋊Q8C22×C2311yescharacteristic, complemented
226#121—21no—
227#121C22×C2311yescharacteristic
228#121C22×C2311yescharacteristic
229#120—21no—
230#120—21no—
231C4×Q8C22×C2311yescharacteristic, complemented
232C4×D4C22×C2311yescharacteristic
233C4×D4C22×C2311yescharacteristic
234#117—21no—
235#117C22×C2311yescharacteristic, complemented
236C2×C4⋊C4C22×C2311yescharacteristic, complemented
237#107C22×C2311yescharacteristic, complemented
238#107C22×C2311yescharacteristic, complemented
239#106C22×C2311yescharacteristic, complemented
240C4⋊C8C22×C2311yescharacteristic, complemented
241#103C22×C2311yescharacteristic, complemented
242#102C22×C2311yescharacteristic
243#102C22×C2311yescharacteristic
244#102C22×C2311yescharacteristic
245#102C22×C2311yescharacteristic, complemented
246#102C22×C2311yescharacteristic, complemented
247#102C22×C2311yescharacteristic, complemented
248#102C22×C2311yescharacteristic, complemented
249#102C22×C2311yescharacteristic, complemented
250#102C22×C2311yescharacteristic, complemented
251C22⋊C8C22×C2311yescharacteristic, complemented
252C22⋊C8C22×C2311yescharacteristic, complemented
253#97C22×C2311yescharacteristic, complemented
254C23#331011yescharacteristic, direct product factor, Sylow
255C24—21no—
256C24—21no—
257C2×Q8C23×C2311yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4D4×C2311yescharacteristic
270C2×D4C23×C2311yescharacteristic
271C2×D4C23×C2311yescharacteristic
272C2×D4C23×C2311yescharacteristic
273C22×C4—21no—
274C22×C4—21no—
275C22×C4C23×C2311yescharacteristic
276C22×C4C23×C2311yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—41no—
280M16—41no—
281C2×C8—41no—
282C2×C8C23×C2311yescharacteristic
283C2×C8C23×C2311yescharacteristic
284C2×C8C23×C2311yescharacteristic
285C2×C8C23×C2311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4D4×C2311yescharacteristic
293C4⋊C4C23×C2311yescharacteristic
294C4⋊C4C23×C2311yescharacteristic
295C4⋊C4C23×C2311yescharacteristic
296C4⋊C4C23×C2311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4D4×C2311yescharacteristic
304C22⋊C4D4×C2311yescharacteristic
305C42—21no—
306C42—21no—
307C42C23×C2311yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×D4×C2311yescharacteristic
318C23C2×D4×C2311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#41×C2312yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4C2×D4×C2311yescharacteristic
346C2×C4C2×D4×C2311yescharacteristic
347C2×C4C2×D4×C2311yescharacteristic
348C2×C4C2×D4×C2311yescharacteristic
349C2×C4C24×C2311yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22C23×AGL(1,ℤ/8)12yes—
371C22C22×D4×C2311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4C2×#41×C2311yescharacteristic
382C4C23×H(5,𝔽2)11yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2C23×#54611yescharacteristic
389C2C2×C23×AGL(1,ℤ/8)11yescharacteristic
390C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalD42×C23C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#552C2semidirect
3. C23×#490C2semidirect
4. C23×#482C2semidirect
5. C23×#480C2semidirect
6. C23×#478C2semidirect
7. C23×#468C2semidirect
8. C23×#465C2semidirect
9. C23×#460C2semidirect
10. C23×#450C2semidirect
11. C23×#450C2semidirect
12. C23×#440C2semidirect
13. C23×#435C2semidirect
14. C23×#418C2semidirect
15. C2×C23×#102C2semidirect
20. C2×SD16×C23C22semidirect
21. C2×SD16×C23C22semidirect
22. C2×M16×C23C22semidirect
23. C22×C8×C23C22semidirect
27. C23×#125C22semidirect
29. C23×C22⋊Q8C22semidirect
30. C23×C22⋊Q8C22semidirect
36. C4×Q8×C23C22semidirect
40. C23×#117C22semidirect
41. C2×C4⋊C4×C23C22semidirect
42. C23×#107C22semidirect
43. C23×#107C22semidirect
44. C23×#106C22semidirect
45. C23×C4⋊C8C22semidirect
46. C23×#103C22semidirect
50. C23×#102C22semidirect
51. C23×#102C22semidirect
52. C23×#102C22semidirect
53. C23×#102C22semidirect
54. C23×#102C22semidirect
55. C23×#102C22semidirect
56. C23×C22⋊C8C22semidirect
57. C23×C22⋊C8C22semidirect
58. C23×#97C22semidirect
158. #3310C23direct (N × H)
188. #552C2×C23semidirect
190. #490C2×C23semidirect
191. #482C2×C23semidirect
192. #480C2×C23semidirect
193. #478C2×C23semidirect
194. #468C2×C23semidirect
195. #465C2×C23semidirect
196. #460C2×C23semidirect
197. #450C2×C23semidirect
198. #450C2×C23semidirect
199. #440C2×C23semidirect
200. #435C2×C23semidirect
201. #418C2×C23semidirect
202. C2×#102C2×C23semidirect
215. C2×SD16C22×C23semidirect
216. C2×SD16C22×C23semidirect
217. C2×M16C22×C23semidirect
218. C22×C8C22×C23semidirect
222. #125C22×C23semidirect
224. C22⋊Q8C22×C23semidirect
225. C22⋊Q8C22×C23semidirect
231. C4×Q8C22×C23semidirect
235. #117C22×C23semidirect
236. C2×C4⋊C4C22×C23semidirect
237. #107C22×C23semidirect
238. #107C22×C23semidirect
239. #106C22×C23semidirect
240. C4⋊C8C22×C23semidirect
241. #103C22×C23semidirect
245. #102C22×C23semidirect
246. #102C22×C23semidirect
247. #102C22×C23semidirect
248. #102C22×C23semidirect
249. #102C22×C23semidirect
250. #102C22×C23semidirect
251. C22⋊C8C22×C23semidirect
252. C22⋊C8C22×C23semidirect
253. #97C22×C23semidirect
254. C23#3310direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5522
D42×C232
C23×#4902
C23×#4822
C23×#4802
C23×#4782
C23×#4682
C23×#4652
C23×#4602
C23×#4502
C23×#4402
C23×#4352
C23×#4182
C2×C23×#1022
#331023