Finite Group Databasebeta

← C23×#3322 #15670941

Subgroups — C23×#3322 #15670941

Order 2944.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#553C211yesmaximal, characteristic, complemented
2C23×#546C211yesmaximal, characteristic, complemented
3C23×#488C211yesmaximal, characteristic, complemented
4C23×#485C211yesmaximal, characteristic, complemented
5C23×#480C211yesmaximal, characteristic, complemented
6C23×#478C211yesmaximal, characteristic, complemented
7C23×#469C211yesmaximal, characteristic, complemented
8C23×#468C211yesmaximal, characteristic, complemented
9C23×#463C211yesmaximal, characteristic, complemented
10C23×#449C211yesmaximal, characteristic, complemented
11C23×#447C211yesmaximal, characteristic, complemented
12C4×D8×C23C211yesmaximal, characteristic, complemented
13C23×#436C211yesmaximal, characteristic, complemented
14C23×#418C211yesmaximal, characteristic, complemented
15C23×#417C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic, complemented
19C2×D8×C23C2211yescharacteristic
20C2×D8×C23C2211yescharacteristic
21C2×M16×C23C2211yescharacteristic
22C2×M16×C23C2211yescharacteristic, complemented
23C23×#126—21no—
24C23×#125C2211yescharacteristic, complemented
25C23×#124C2211yescharacteristic
26C23×#123—21no—
27C23×#123—21no—
28C23×C22⋊Q8C2211yescharacteristic
29C23×#121—21no—
30C23×#121—21no—
31C23×#121C2211yescharacteristic
32C23×#121C2211yescharacteristic
33C23×#121C2211yescharacteristic, complemented
34C23×#120—21no—
35C4×D4×C23—21no—
36C4×D4×C23C2211yescharacteristic
37C4×D4×C23C2211yescharacteristic
38C4×D4×C23C2211yescharacteristic, complemented
39C23×#117C2211yescharacteristic
40C2×C4⋊C4×C23C2211yescharacteristic, complemented
41C2×C22⋊C4×C23—21no—
42C23×#107C2211yescharacteristic, complemented
43C23×#107C2211yescharacteristic, 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
58C4×C8×C23C2211yescharacteristic, complemented
59C24×C23—21no—
60#41×C23—22no—
61#41×C23—41no—
62C2×Q8×C23C2311yescharacteristic
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—41no—
70C2×D4×C23D411yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C2×D4×C23C2311yescharacteristic
73C2×D4×C23C2311yescharacteristic
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23C2311yescharacteristic
79C22×C4×C23C2311yescharacteristic
80SD16×C23—41no—
81D8×C23—22no—
82D8×C23—41no—
83M16×C23—41no—
84M16×C23—41no—
85C2×C8×C23C2311yescharacteristic
86C2×C8×C23C2311yescharacteristic
87C2×C8×C23C2311yescharacteristic
88C2×C8×C23C2311yescharacteristic
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×C23—21no—
107C22⋊C4×C23D411yescharacteristic
108C22⋊C4×C23D411yescharacteristic
109C42×C23—21no—
110C42×C23C2311yescharacteristic
111C23×C23—21no—
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—42no—
115C23×C23—42no—
116C23×C23—81no—
117C23×C23C2×D411yescharacteristic
118C23×C23C2×D411yescharacteristic
119Q8×C23—21no—
120Q8×C23—21no—
121D4×C23—21no—
122D4×C23—21no—
123D4×C23—21no—
124D4×C23—21no—
125D4×C23—21no—
126D4×C23—41no—
127D4×C23—42no—
128D4×C23—42no—
129D4×C23—42no—
130D4×C23—81no—
131D4×C23—81no—
132D4×C23#4112yes—
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—21no—
141C2×C4×C23—21no—
142C2×C4×C23—42no—
143C2×C4×C23—81no—
144C2×C4×C23—81no—
145C2×C4×C23—81no—
146C2×C4×C23C2×D411yescharacteristic
147C2×C4×C23C2×D411yescharacteristic
148C2×C4×C23C2×D411yescharacteristic
149C2×C4×C23C2×D411yescharacteristic
150C2×C4×C23C2411yescharacteristic
151C8×C23—21no—
152C8×C23—21no—
153C8×C23—21no—
154C8×C23—21no—
155#3322C2311yesmaximal, characteristic, direct product factor, Sylow
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—22no—
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—41no—
164C22×C23—41no—
165C22×C23—41no—
166C22×C23—41no—
167C22×C23—41no—
168C22×C23—41no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23—84no—
172C22×C23C22×D411yescharacteristic
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23C2×#4111yescharacteristic
183C4×C23H(5,𝔽2)11yescharacteristic
184#553C2×C2311yescharacteristic, complemented
185#546C2×C2311yescharacteristic, complemented
186#488C2×C2311yescharacteristic, complemented
187#485C2×C2311yescharacteristic, complemented
188#480C2×C2311yescharacteristic, complemented
189#478C2×C2311yescharacteristic, complemented
190#469C2×C2311yescharacteristic, complemented
191#468C2×C2311yescharacteristic, complemented
192#463C2×C2311yescharacteristic, complemented
193#449C2×C2311yescharacteristic, complemented
194#447C2×C2311yescharacteristic, complemented
195C4×D8C2×C2311yescharacteristic, complemented
196#436C2×C2311yescharacteristic, complemented
197#418C2×C2311yescharacteristic, complemented
198#417C2×C2311yescharacteristic, complemented
199C2×C23—41no—
200C2×C23—41no—
201C2×C23—42no—
202C2×C23—81no—
203C2×C23—81no—
204C2×C23#54611yescharacteristic
205C2×C23#57511yescharacteristic
206C2×C23#57611yescharacteristic
207C2×#41C22×C2311yescharacteristic
208C22×D4C22×C2311yescharacteristic
209C2×SD16C22×C2311yescharacteristic, complemented
210C2×D8C22×C2311yescharacteristic
211C2×D8C22×C2311yescharacteristic
212C2×M16C22×C2311yescharacteristic
213C2×M16C22×C2311yescharacteristic, complemented
214#126—21no—
215#125C22×C2311yescharacteristic, complemented
216#124C22×C2311yescharacteristic
217#123—21no—
218#123—21no—
219C22⋊Q8C22×C2311yescharacteristic
220#121—21no—
221#121—21no—
222#121C22×C2311yescharacteristic
223#121C22×C2311yescharacteristic
224#121C22×C2311yescharacteristic, complemented
225#120—21no—
226C4×D4—21no—
227C4×D4C22×C2311yescharacteristic
228C4×D4C22×C2311yescharacteristic
229C4×D4C22×C2311yescharacteristic, complemented
230#117C22×C2311yescharacteristic
231C2×C4⋊C4C22×C2311yescharacteristic, complemented
232C2×C22⋊C4—21no—
233#107C22×C2311yescharacteristic, complemented
234#107C22×C2311yescharacteristic, complemented
235#106C22×C2311yescharacteristic, complemented
236C4⋊C8C22×C2311yescharacteristic, complemented
237#103C22×C2311yescharacteristic, complemented
238#103C22×C2311yescharacteristic, complemented
239#103C22×C2311yescharacteristic, complemented
240#102C22×C2311yescharacteristic
241#102C22×C2311yescharacteristic
242#102C22×C2311yescharacteristic
243#102C22×C2311yescharacteristic
244#102C22×C2311yescharacteristic
245#102C22×C2311yescharacteristic
246#102C22×C2311yescharacteristic, complemented
247C22⋊C8C22×C2311yescharacteristic
248C22⋊C8C22×C2311yescharacteristic, complemented
249C4×C8C22×C2311yescharacteristic, complemented
250C23#332211yescharacteristic, direct product factor, Sylow
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C23×C2311yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4D4×C2311yescharacteristic
263C2×D4C23×C2311yescharacteristic
264C2×D4C23×C2311yescharacteristic
265C2×D4C23×C2311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C23×C2311yescharacteristic
271C22×C4C23×C2311yescharacteristic
272SD16—41no—
273D8—22no—
274D8—41no—
275M16—41no—
276M16—41no—
277C2×C8C23×C2311yescharacteristic
278C2×C8C23×C2311yescharacteristic
279C2×C8C23×C2311yescharacteristic
280C2×C8C23×C2311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4D4×C2311yescharacteristic
286C4⋊C4C23×C2311yescharacteristic
287C4⋊C4C23×C2311yescharacteristic
288C4⋊C4C23×C2311yescharacteristic
289C4⋊C4C23×C2311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4D4×C2311yescharacteristic
300C22⋊C4D4×C2311yescharacteristic
301C42—21no—
302C42C23×C2311yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23C2×D4×C2311yescharacteristic
310C23C2×D4×C2311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—41no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324D4#41×C2312yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C2×D4×C2311yescharacteristic
339C2×C4C2×D4×C2311yescharacteristic
340C2×C4C2×D4×C2311yescharacteristic
341C2×C4C2×D4×C2311yescharacteristic
342C2×C4C24×C2311yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22C22×D4×C2311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4C2×#41×C2311yescharacteristic
374C4C23×H(5,𝔽2)11yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—81no—
380C2C23×#54611yescharacteristic
381C2C23×#57511yescharacteristic
382C2C23×#57611yescharacteristic
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×#553C2semidirect
2. C23×#546C2semidirect
3. C23×#488C2semidirect
4. C23×#485C2semidirect
5. C23×#480C2semidirect
6. C23×#478C2semidirect
7. C23×#469C2semidirect
8. C23×#468C2semidirect
9. C23×#463C2semidirect
10. C23×#449C2semidirect
11. C23×#447C2semidirect
12. C4×D8×C23C2semidirect
13. C23×#436C2semidirect
14. C23×#418C2semidirect
15. C23×#417C2semidirect
18. C2×SD16×C23C22semidirect
22. C2×M16×C23C22semidirect
24. C23×#125C22semidirect
33. C23×#121C22semidirect
38. C4×D4×C23C22semidirect
40. C2×C4⋊C4×C23C22semidirect
42. C23×#107C22semidirect
43. C23×#107C22semidirect
44. C23×#106C22semidirect
45. C23×C4⋊C8C22semidirect
46. C23×#103C22semidirect
47. C23×#103C22semidirect
48. C23×#103C22semidirect
55. C23×#102C22semidirect
57. C23×C22⋊C8C22semidirect
58. C4×C8×C23C22semidirect
155. #3322C23direct (N × H)
184. #553C2×C23semidirect
185. #546C2×C23semidirect
186. #488C2×C23semidirect
187. #485C2×C23semidirect
188. #480C2×C23semidirect
189. #478C2×C23semidirect
190. #469C2×C23semidirect
191. #468C2×C23semidirect
192. #463C2×C23semidirect
193. #449C2×C23semidirect
194. #447C2×C23semidirect
195. C4×D8C2×C23semidirect
196. #436C2×C23semidirect
197. #418C2×C23semidirect
198. #417C2×C23semidirect
209. C2×SD16C22×C23semidirect
213. C2×M16C22×C23semidirect
215. #125C22×C23semidirect
224. #121C22×C23semidirect
229. C4×D4C22×C23semidirect
231. C2×C4⋊C4C22×C23semidirect
233. #107C22×C23semidirect
234. #107C22×C23semidirect
235. #106C22×C23semidirect
236. C4⋊C8C22×C23semidirect
237. #103C22×C23semidirect
238. #103C22×C23semidirect
239. #103C22×C23semidirect
246. #102C22×C23semidirect
248. C22⋊C8C22×C23semidirect
249. C4×C8C22×C23semidirect
250. C23#3322direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5532
C23×#5462
C23×#4882
C23×#4852
C23×#4802
C23×#4782
C23×#4692
C23×#4682
C23×#4632
C23×#4492
C23×#4472
C4×D8×C232
C23×#4362
C23×#4182
C23×#4172
#332223