Finite Group Databasebeta

← C23×#3296 #15670915

Subgroups — C23×#3296 #15670915

Order 2944.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#552C211yesmaximal, characteristic, complemented
2C23×#546C211yesmaximal, characteristic
3C23×#488C211yesmaximal, characteristic, complemented
4C23×#483C211yesmaximal, characteristic, complemented
5C23×#482C211yesmaximal, characteristic, complemented
6C23×#478C211yesmaximal, characteristic, complemented
7C23×#467C211yesmaximal, characteristic, complemented
8C23×#465C211yesmaximal, characteristic, complemented
9C23×#464C211yesmaximal, characteristic, complemented
10C23×#452C211yesmaximal, characteristic, complemented
11C23×#450C211yesmaximal, characteristic, complemented
12C4×SD16×C23C211yesmaximal, characteristic, complemented
13C8×D4×C23C211yesmaximal, characteristic, complemented
14C23×#416C211yesmaximal, characteristic, complemented
15C2×C23×#102C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic
18C2×Q16×C23C2211yescharacteristic, complemented
19C2×SD16×C23C2211yescharacteristic
20C2×SD16×C23C2211yescharacteristic
21C22×C8×C23C2211yescharacteristic
22C22×C8×C23C2211yescharacteristic, complemented
23C23×#126—21no—
24C23×#125—21no—
25C23×#125C2211yescharacteristic, complemented
26C23×#124C2211yescharacteristic
27C23×#123—21no—
28C23×#123—21no—
29C23×C22⋊Q8C2211yescharacteristic
30C23×C22⋊Q8C2211yescharacteristic
31C23×C22⋊Q8C2211yescharacteristic, complemented
32C23×#121—21no—
33C23×#121C2211yescharacteristic
34C23×#120—21no—
35C4×Q8×C23C2211yescharacteristic, complemented
36C4×D4×C23C2211yescharacteristic
37C4×D4×C23C2211yescharacteristic
38C23×#117—21no—
39C23×#117C2211yescharacteristic, complemented
40C2×C4⋊C4×C23C2211yescharacteristic
41C2×C22⋊C4×C23—21no—
42C23×#107C2211yescharacteristic, complemented
43C23×#106C2211yescharacteristic, complemented
44C23×#106C2211yescharacteristic, complemented
45C23×C4⋊C8C2211yescharacteristic, complemented
46C23×#103C2211yescharacteristic, complemented
47C23×#103C2211yescharacteristic, complemented
48C23×#103C2211yescharacteristic, complemented
49C23×#103C2211yescharacteristic, complemented
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic, complemented
58C4×C8×C23C2211yescharacteristic, complemented
59C24×C23—21no—
60#41×C23—22no—
61#41×C23—41no—
62C2×Q8×C23C2311yescharacteristic
63C2×Q8×C23C2311yescharacteristic
64C2×D4×C23—22no—
65C2×D4×C23—21no—
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—41no—
69C2×D4×C23D411yescharacteristic
70C2×D4×C23C2311yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23C2311yescharacteristic
76C22×C4×C23C2311yescharacteristic
77Q16×C23—41no—
78SD16×C23—22no—
79SD16×C23—41no—
80C2×C8×C23—41no—
81C2×C8×C23—41no—
82C2×C8×C23C2311yescharacteristic
83C2×C8×C23C2311yescharacteristic
84C2×C8×C23C2311yescharacteristic
85C2×C8×C23C2311yescharacteristic
86C4⋊C4×C23—21no—
87C4⋊C4×C23—21no—
88C4⋊C4×C23—21no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23—21no—
91C4⋊C4×C23—21no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23D411yescharacteristic
94C4⋊C4×C23C2311yescharacteristic
95C4⋊C4×C23C2311yescharacteristic
96C4⋊C4×C23C2311yescharacteristic
97C4⋊C4×C23C2311yescharacteristic
98C22⋊C4×C23—21no—
99C22⋊C4×C23—21no—
100C22⋊C4×C23—21no—
101C22⋊C4×C23—21no—
102C22⋊C4×C23—21no—
103C22⋊C4×C23—21no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23D411yescharacteristic
107C22⋊C4×C23D411yescharacteristic
108C42×C23—21no—
109C42×C23—21no—
110C42×C23C2311yescharacteristic
111C23×C23—21no—
112C23×C23—21no—
113C23×C23—42no—
114C23×C23—42no—
115C23×C23—81no—
116C23×C23C2×D411yescharacteristic
117C23×C23C2×D411yescharacteristic
118Q8×C23—21no—
119Q8×C23—21no—
120Q8×C23—21no—
121Q8×C23—21no—
122D4×C23—22no—
123D4×C23—21no—
124D4×C23—21no—
125D4×C23—21no—
126D4×C23—42no—
127D4×C23—42no—
128D4×C23—81no—
129D4×C23#4112yes—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—42no—
141C2×C4×C23—42no—
142C2×C4×C23—81no—
143C2×C4×C23C2×D411yescharacteristic
144C2×C4×C23C2×D411yescharacteristic
145C2×C4×C23C2×D411yescharacteristic
146C2×C4×C23C2×D411yescharacteristic
147C2×C4×C23C2411yescharacteristic
148C8×C23—21no—
149C8×C23—21no—
150C8×C23—21no—
151C8×C23—21no—
152#3296C2311yesmaximal, characteristic, direct product factor, Sylow
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—22no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—41no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23—84no—
165C22×C23#13512yes—
166C22×C23C22×D411yescharacteristic
167C4×C23—21no—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—41no—
175C4×C23—41no—
176C4×C23—41no—
177C4×C23C2×#4111yescharacteristic
178C4×C23H(5,𝔽2)11yescharacteristic
179#552C2×C2311yescharacteristic, complemented
180#546C2×C2311yescharacteristic
181#488C2×C2311yescharacteristic, complemented
182#483C2×C2311yescharacteristic, complemented
183#482C2×C2311yescharacteristic, complemented
184#478C2×C2311yescharacteristic, complemented
185#467C2×C2311yescharacteristic, complemented
186#465C2×C2311yescharacteristic, complemented
187#464C2×C2311yescharacteristic, complemented
188#452C2×C2311yescharacteristic, complemented
189#450C2×C2311yescharacteristic, complemented
190C4×SD16C2×C2311yescharacteristic, complemented
191C8×D4C2×C2311yescharacteristic, complemented
192#416C2×C2311yescharacteristic, complemented
193C2×#102C2×C2311yescharacteristic, complemented
194C2×C23—22no—
195C2×C23—41no—
196C2×C23—42no—
197C2×C23—81no—
198C2×C23#54611yescharacteristic
199C2×C23C2×#13511yescharacteristic
200C2×C23#57711yescharacteristic
201C2×#41C22×C2311yescharacteristic
202C22×D4C22×C2311yescharacteristic
203C2×Q16C22×C2311yescharacteristic, complemented
204C2×SD16C22×C2311yescharacteristic
205C2×SD16C22×C2311yescharacteristic
206C22×C8C22×C2311yescharacteristic
207C22×C8C22×C2311yescharacteristic, complemented
208#126—21no—
209#125—21no—
210#125C22×C2311yescharacteristic, complemented
211#124C22×C2311yescharacteristic
212#123—21no—
213#123—21no—
214C22⋊Q8C22×C2311yescharacteristic
215C22⋊Q8C22×C2311yescharacteristic
216C22⋊Q8C22×C2311yescharacteristic, complemented
217#121—21no—
218#121C22×C2311yescharacteristic
219#120—21no—
220C4×Q8C22×C2311yescharacteristic, complemented
221C4×D4C22×C2311yescharacteristic
222C4×D4C22×C2311yescharacteristic
223#117—21no—
224#117C22×C2311yescharacteristic, complemented
225C2×C4⋊C4C22×C2311yescharacteristic
226C2×C22⋊C4—21no—
227#107C22×C2311yescharacteristic, complemented
228#106C22×C2311yescharacteristic, complemented
229#106C22×C2311yescharacteristic, complemented
230C4⋊C8C22×C2311yescharacteristic, complemented
231#103C22×C2311yescharacteristic, complemented
232#103C22×C2311yescharacteristic, complemented
233#103C22×C2311yescharacteristic, complemented
234#103C22×C2311yescharacteristic, complemented
235#102C22×C2311yescharacteristic
236#102C22×C2311yescharacteristic
237#102C22×C2311yescharacteristic
238#102C22×C2311yescharacteristic
239#102C22×C2311yescharacteristic
240#102C22×C2311yescharacteristic
241C22⋊C8C22×C2311yescharacteristic
242C22⋊C8C22×C2311yescharacteristic, complemented
243C4×C8C22×C2311yescharacteristic, complemented
244C23#329611yescharacteristic, direct product factor, Sylow
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8C23×C2311yescharacteristic
249C2×Q8C23×C2311yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4D4×C2311yescharacteristic
256C2×D4C23×C2311yescharacteristic
257C2×D4C23×C2311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C23×C2311yescharacteristic
262C22×C4C23×C2311yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C23×C2311yescharacteristic
269C2×C8C23×C2311yescharacteristic
270C2×C8C23×C2311yescharacteristic
271C2×C8C23×C2311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4D4×C2311yescharacteristic
280C4⋊C4C23×C2311yescharacteristic
281C4⋊C4C23×C2311yescharacteristic
282C4⋊C4C23×C2311yescharacteristic
283C4⋊C4C23×C2311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4D4×C2311yescharacteristic
293C22⋊C4D4×C2311yescharacteristic
294C42—21no—
295C42—21no—
296C42C23×C2311yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×D4×C2311yescharacteristic
303C23C2×D4×C2311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#41×C2312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4C2×D4×C2311yescharacteristic
330C2×C4C2×D4×C2311yescharacteristic
331C2×C4C2×D4×C2311yescharacteristic
332C2×C4C2×D4×C2311yescharacteristic
333C2×C4C24×C2311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22C23×#13512yes—
351C22C22×D4×C2311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4C2×#41×C2311yescharacteristic
363C4C23×H(5,𝔽2)11yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2C23×#54611yescharacteristic
369C2C2×C23×#13511yescharacteristic
370C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#546C2
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×#488C2semidirect
4. C23×#483C2semidirect
5. C23×#482C2semidirect
6. C23×#478C2semidirect
7. C23×#467C2semidirect
8. C23×#465C2semidirect
9. C23×#464C2semidirect
10. C23×#452C2semidirect
11. C23×#450C2semidirect
12. C4×SD16×C23C2semidirect
13. C8×D4×C23C2semidirect
14. C23×#416C2semidirect
15. C2×C23×#102C2semidirect
18. C2×Q16×C23C22semidirect
22. C22×C8×C23C22semidirect
25. C23×#125C22semidirect
31. C23×C22⋊Q8C22semidirect
35. C4×Q8×C23C22semidirect
39. C23×#117C22semidirect
42. C23×#107C22semidirect
43. C23×#106C22semidirect
44. C23×#106C22semidirect
45. C23×C4⋊C8C22semidirect
46. C23×#103C22semidirect
47. C23×#103C22semidirect
48. C23×#103C22semidirect
49. C23×#103C22semidirect
57. C23×C22⋊C8C22semidirect
58. C4×C8×C23C22semidirect
152. #3296C23direct (N × H)
179. #552C2×C23semidirect
181. #488C2×C23semidirect
182. #483C2×C23semidirect
183. #482C2×C23semidirect
184. #478C2×C23semidirect
185. #467C2×C23semidirect
186. #465C2×C23semidirect
187. #464C2×C23semidirect
188. #452C2×C23semidirect
189. #450C2×C23semidirect
190. C4×SD16C2×C23semidirect
191. C8×D4C2×C23semidirect
192. #416C2×C23semidirect
193. C2×#102C2×C23semidirect
203. C2×Q16C22×C23semidirect
207. C22×C8C22×C23semidirect
210. #125C22×C23semidirect
216. C22⋊Q8C22×C23semidirect
220. C4×Q8C22×C23semidirect
224. #117C22×C23semidirect
227. #107C22×C23semidirect
228. #106C22×C23semidirect
229. #106C22×C23semidirect
230. C4⋊C8C22×C23semidirect
231. #103C22×C23semidirect
232. #103C22×C23semidirect
233. #103C22×C23semidirect
234. #103C22×C23semidirect
242. C22⋊C8C22×C23semidirect
243. C4×C8C22×C23semidirect
244. C23#3296direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5522
C23×#5462
C23×#4882
C23×#4832
C23×#4822
C23×#4782
C23×#4672
C23×#4652
C23×#4642
C23×#4522
C23×#4502
C4×SD16×C232
C8×D4×C232
C23×#4162
C2×C23×#1022
#329623