Finite Group Databasebeta

← C23×#3306 #15670925

Subgroups — C23×#3306 #15670925

Order 2944.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#547C211yesmaximal, characteristic, complemented
2C23×#546C211yesmaximal, characteristic, complemented
3C23×#489C211yesmaximal, characteristic, complemented
4C23×#483C211yesmaximal, characteristic, complemented
5C23×#481C211yesmaximal, characteristic, complemented
6C23×#476C211yesmaximal, characteristic, complemented
7C23×#469C211yesmaximal, characteristic, complemented
8C23×#465C211yesmaximal, characteristic, complemented
9C23×#463C211yesmaximal, characteristic, complemented
10C23×#449C211yesmaximal, characteristic, complemented
11C23×#447C211yesmaximal, characteristic, complemented
12C23×#442C211yesmaximal, characteristic, complemented
13C23×#435C211yesmaximal, characteristic, complemented
14C23×#417C211yesmaximal, characteristic, complemented
15C2×C23×#102C211yesmaximal, characteristic, complemented
16C2×#41×C23—21no—
17C2×#41×C23C2211yescharacteristic
18C22×D4×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic, complemented
20C2×D8×C23C2211yescharacteristic
21C2×D8×C23C2211yescharacteristic
22C2×M16×C23C2211yescharacteristic, complemented
23C22×C8×C23C2211yescharacteristic
24C23×C4⋊Q8C2211yescharacteristic, complemented
25C23×#124C2211yescharacteristic
26C23×#123—21no—
27C23×#123—21no—
28C23×#123—21no—
29C23×C22⋊Q8—21no—
30C23×C22⋊Q8C2211yescharacteristic
31C23×#121—21no—
32C23×#121C2211yescharacteristic
33C23×#121C2211yescharacteristic
34C23×#121C2211yescharacteristic, complemented
35C23×#120—21no—
36C4×D4×C23C2211yescharacteristic
37C4×D4×C23C2211yescharacteristic
38C4×D4×C23C2211yescharacteristic, complemented
39C2×C4⋊C4×C23C2211yescharacteristic
40C2×C4⋊C4×C23C2211yescharacteristic, complemented
41C2×C22⋊C4×C23—21no—
42C23×#106C2211yescharacteristic, complemented
43C23×#106C2211yescharacteristic, complemented
44C23×#106C2211yescharacteristic, complemented
45C23×C4⋊C8C2211yescharacteristic, complemented
46C23×#103C2211yescharacteristic, complemented
47C23×#103C2211yescharacteristic, complemented
48C23×#103C2211yescharacteristic, complemented
49C23×#102C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic, complemented
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic, complemented
58C23×#97C2211yescharacteristic, complemented
59C24×C23—21no—
60#41×C23—22no—
61#41×C23—41no—
62#41×C23—44no—
63C2×Q8×C23—21no—
64C2×Q8×C23C2311yescharacteristic
65C2×D4×C23—22no—
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23—41no—
72C2×D4×C23D411yescharacteristic
73C2×D4×C23C2311yescharacteristic
74C2×D4×C23C2311yescharacteristic
75C2×D4×C23C2311yescharacteristic
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23C2311yescharacteristic
82C22×C4×C23C2311yescharacteristic
83SD16×C23—41no—
84D8×C23—22no—
85D8×C23—41no—
86M16×C23—41no—
87C2×C8×C23—41no—
88C2×C8×C23C2311yescharacteristic
89C2×C8×C23C2311yescharacteristic
90C2×C8×C23C2311yescharacteristic
91C2×C8×C23C2311yescharacteristic
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×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23D411yescharacteristic
110C22⋊C4×C23D411yescharacteristic
111C42×C23C2311yescharacteristic
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—42no—
116C23×C23—42no—
117C23×C23—81no—
118C23×C23C2×D411yescharacteristic
119C23×C23C2×D411yescharacteristic
120Q8×C23—21no—
121Q8×C23—21no—
122Q8×C23—42no—
123D4×C23—22no—
124D4×C23—21no—
125D4×C23—21no—
126D4×C23—21no—
127D4×C23—21no—
128D4×C23—21no—
129D4×C23—42no—
130D4×C23—42no—
131D4×C23—42no—
132D4×C23—42no—
133D4×C23—81no—
134D4×C23#4112yescomplemented
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
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—42no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—81no—
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#3306C2311yesmaximal, 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×C23AGL(1,ℤ/8)12yes—
175C22×C23C22×D411yescharacteristic
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—41no—
184C4×C23—41no—
185C4×C23C2×#4111yescharacteristic
186C4×C23H(5,𝔽2)11yescharacteristic
187#547C2×C2311yescharacteristic, complemented
188#546C2×C2311yescharacteristic, complemented
189#489C2×C2311yescharacteristic, complemented
190#483C2×C2311yescharacteristic, complemented
191#481C2×C2311yescharacteristic, complemented
192#476C2×C2311yescharacteristic, complemented
193#469C2×C2311yescharacteristic, complemented
194#465C2×C2311yescharacteristic, complemented
195#463C2×C2311yescharacteristic, complemented
196#449C2×C2311yescharacteristic, complemented
197#447C2×C2311yescharacteristic, complemented
198#442C2×C2311yescharacteristic, complemented
199#435C2×C2311yescharacteristic, complemented
200#417C2×C2311yescharacteristic, complemented
201C2×#102C2×C2311yescharacteristic, complemented
202C2×C23—22no—
203C2×C23—41no—
204C2×C23—42no—
205C2×C23—81no—
206C2×C23—81no—
207C2×C23#54611yescharacteristic
208C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C23#57711yescharacteristic
210C2×#41—21no—
211C2×#41C22×C2311yescharacteristic
212C22×D4C22×C2311yescharacteristic
213C2×SD16C22×C2311yescharacteristic, complemented
214C2×D8C22×C2311yescharacteristic
215C2×D8C22×C2311yescharacteristic
216C2×M16C22×C2311yescharacteristic, complemented
217C22×C8C22×C2311yescharacteristic
218C4⋊Q8C22×C2311yescharacteristic, complemented
219#124C22×C2311yescharacteristic
220#123—21no—
221#123—21no—
222#123—21no—
223C22⋊Q8—21no—
224C22⋊Q8C22×C2311yescharacteristic
225#121—21no—
226#121C22×C2311yescharacteristic
227#121C22×C2311yescharacteristic
228#121C22×C2311yescharacteristic, complemented
229#120—21no—
230C4×D4C22×C2311yescharacteristic
231C4×D4C22×C2311yescharacteristic
232C4×D4C22×C2311yescharacteristic, complemented
233C2×C4⋊C4C22×C2311yescharacteristic
234C2×C4⋊C4C22×C2311yescharacteristic, complemented
235C2×C22⋊C4—21no—
236#106C22×C2311yescharacteristic, complemented
237#106C22×C2311yescharacteristic, complemented
238#106C22×C2311yescharacteristic, complemented
239C4⋊C8C22×C2311yescharacteristic, complemented
240#103C22×C2311yescharacteristic, complemented
241#103C22×C2311yescharacteristic, complemented
242#103C22×C2311yescharacteristic, complemented
243#102C22×C2311yescharacteristic
244#102C22×C2311yescharacteristic
245#102C22×C2311yescharacteristic
246#102C22×C2311yescharacteristic
247#102C22×C2311yescharacteristic
248#102C22×C2311yescharacteristic
249#102C22×C2311yescharacteristic, complemented
250C22⋊C8C22×C2311yescharacteristic
251C22⋊C8C22×C2311yescharacteristic, complemented
252#97C22×C2311yescharacteristic, complemented
253C23#330611yescharacteristic, direct product factor, Sylow
254C24—21no—
255#41—22no—
256#41—41no—
257#41—44no—
258C2×Q8—21no—
259C2×Q8C23×C2311yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4D4×C2311yescharacteristic
268C2×D4C23×C2311yescharacteristic
269C2×D4C23×C2311yescharacteristic
270C2×D4C23×C2311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4C23×C2311yescharacteristic
277C22×C4C23×C2311yescharacteristic
278SD16—41no—
279D8—22no—
280D8—41no—
281M16—41no—
282C2×C8—41no—
283C2×C8C23×C2311yescharacteristic
284C2×C8C23×C2311yescharacteristic
285C2×C8C23×C2311yescharacteristic
286C2×C8C23×C2311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4D4×C2311yescharacteristic
292C4⋊C4C23×C2311yescharacteristic
293C4⋊C4C23×C2311yescharacteristic
294C4⋊C4C23×C2311yescharacteristic
295C4⋊C4C23×C2311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4D4×C2311yescharacteristic
305C22⋊C4D4×C2311yescharacteristic
306C42C23×C2311yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23C2×D4×C2311yescharacteristic
314C23C2×D4×C2311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—81no—
329D4#41×C2312yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4C2×D4×C2311yescharacteristic
345C2×C4C2×D4×C2311yescharacteristic
346C2×C4C2×D4×C2311yescharacteristic
347C2×C4C2×D4×C2311yescharacteristic
348C2×C4C24×C2311yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—84no—
368C22C23×AGL(1,ℤ/8)12yes—
369C22C22×D4×C2311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4C2×#41×C2311yescharacteristic
380C4C23×H(5,𝔽2)11yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—81no—
386C2C23×#54611yescharacteristic
387C2C2×C23×AGL(1,ℤ/8)11yescharacteristic
388C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
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×#547C2semidirect
2. C23×#546C2semidirect
3. C23×#489C2semidirect
4. C23×#483C2semidirect
5. C23×#481C2semidirect
6. C23×#476C2semidirect
7. C23×#469C2semidirect
8. C23×#465C2semidirect
9. C23×#463C2semidirect
10. C23×#449C2semidirect
11. C23×#447C2semidirect
12. C23×#442C2semidirect
13. C23×#435C2semidirect
14. C23×#417C2semidirect
15. C2×C23×#102C2semidirect
19. C2×SD16×C23C22semidirect
22. C2×M16×C23C22semidirect
24. C23×C4⋊Q8C22semidirect
34. C23×#121C22semidirect
38. C4×D4×C23C22semidirect
40. C2×C4⋊C4×C23C22semidirect
42. C23×#106C22semidirect
43. C23×#106C22semidirect
44. C23×#106C22semidirect
45. C23×C4⋊C8C22semidirect
46. C23×#103C22semidirect
47. C23×#103C22semidirect
48. C23×#103C22semidirect
55. C23×#102C22semidirect
57. C23×C22⋊C8C22semidirect
58. C23×#97C22semidirect
134. D4×C23#41semidirect
158. #3306C23direct (N × H)
187. #547C2×C23semidirect
188. #546C2×C23semidirect
189. #489C2×C23semidirect
190. #483C2×C23semidirect
191. #481C2×C23semidirect
192. #476C2×C23semidirect
193. #469C2×C23semidirect
194. #465C2×C23semidirect
195. #463C2×C23semidirect
196. #449C2×C23semidirect
197. #447C2×C23semidirect
198. #442C2×C23semidirect
199. #435C2×C23semidirect
200. #417C2×C23semidirect
201. C2×#102C2×C23semidirect
213. C2×SD16C22×C23semidirect
216. C2×M16C22×C23semidirect
218. C4⋊Q8C22×C23semidirect
228. #121C22×C23semidirect
232. C4×D4C22×C23semidirect
234. C2×C4⋊C4C22×C23semidirect
236. #106C22×C23semidirect
237. #106C22×C23semidirect
238. #106C22×C23semidirect
239. C4⋊C8C22×C23semidirect
240. #103C22×C23semidirect
241. #103C22×C23semidirect
242. #103C22×C23semidirect
249. #102C22×C23semidirect
251. C22⋊C8C22×C23semidirect
252. #97C22×C23semidirect
253. C23#3306direct (N × H)
329. D4#41×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5472
C23×#5462
C23×#4892
C23×#4832
C23×#4812
C23×#4762
C23×#4692
C23×#4652
C23×#4632
C23×#4492
C23×#4472
C23×#4422
C23×#4352
C23×#4172
C2×C23×#1022
#330623