Finite Group Databasebeta

← C23×#3309 #15670928

Subgroups — C23×#3309 #15670928

Order 2944.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#553C211yesmaximal, characteristic, complemented
2C23×#550C211yesmaximal, characteristic, complemented
3C23×#490C211yesmaximal, characteristic, complemented
4C23×#482C211yesmaximal, characteristic, complemented
5C23×#480C211yesmaximal, characteristic, complemented
6C23×#479C211yesmaximal, characteristic, complemented
7C23×#469C211yesmaximal, characteristic, complemented
8C23×#466C211yesmaximal, characteristic, complemented
9C23×#459C211yesmaximal, characteristic, complemented
10C23×#449C211yesmaximal, characteristic, complemented
11C23×#449C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C23×#435C211yesmaximal, characteristic, complemented
14C23×#417C211yesmaximal, characteristic, complemented
15C23×#416C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×#41×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C2×D8×C23C2211yescharacteristic
20C2×D8×C23C2211yescharacteristic
21C2×M16×C23C2211yescharacteristic
22C22×C8×C23C2211yescharacteristic
23C23×#127—21no—
24C23×#127C2211yescharacteristic
25C23×#126—21no—
26C23×#125C2211yescharacteristic
27C23×#123—21no—
28C23×#121—21no—
29C23×#121—21no—
30C23×#121—21no—
31C23×#121C2211yescharacteristic
32C23×#121C2211yescharacteristic
33C23×#121C2211yescharacteristic
34C23×#121C2211yescharacteristic
35C4×Q8×C23C2211yescharacteristic
36C4×D4×C23—21no—
37C4×D4×C23—21no—
38C4×D4×C23C2211yescharacteristic
39C4×D4×C23C2211yescharacteristic
40C23×#117C2211yescharacteristic
41C2×C4⋊C4×C23C2211yescharacteristic
42C23×#107C2211yescharacteristic
43C23×#107C2211yescharacteristic
44C23×#106C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×#103C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#102C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic
58C23×#97C2211yescharacteristic
59#41×C23—22no—
60#41×C23—22no—
61#41×C23—41no—
62#41×C23—41no—
63C2×Q8×C23C2311yescharacteristic
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—21no—
71C2×D4×C23—21no—
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×C23C2311yescharacteristic
81C22×C4×C23C2311yescharacteristic
82SD16×C23—22no—
83D8×C23—41no—
84D8×C23—41no—
85M16×C23—41no—
86C2×C8×C23—41no—
87C2×C8×C23C2311yescharacteristic
88C2×C8×C23C2311yescharacteristic
89C2×C8×C23C2311yescharacteristic
90C2×C8×C23C2311yescharacteristic
91C4⋊C4×C23—21no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23D411yescharacteristic
96C4⋊C4×C23C2311yescharacteristic
97C4⋊C4×C23C2311yescharacteristic
98C4⋊C4×C23C2311yescharacteristic
99C4⋊C4×C23C2311yescharacteristic
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—21no—
114C23×C23C2×D411yescharacteristic
115C23×C23C2×D411yescharacteristic
116Q8×C23—21no—
117Q8×C23#4112yes—
118D4×C23—21no—
119D4×C23—21no—
120D4×C23—21no—
121D4×C23—21no—
122D4×C23—21no—
123D4×C23—21no—
124D4×C23—41no—
125D4×C23—42no—
126D4×C23—42no—
127D4×C23—42no—
128D4×C23—42no—
129D4×C23—81no—
130D4×C23—81no—
131D4×C23—81no—
132D4×C23—81no—
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—42no—
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#3309C2311yesmaximal, characteristic, direct product factor, Sylow
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—41no—
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×C23C22×D411yescharacteristic
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—22no—
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23C2×#4111yescharacteristic
182C4×C23H(5,𝔽2)11yescharacteristic
183#553C2×C2311yescharacteristic, complemented
184#550C2×C2311yescharacteristic, complemented
185#490C2×C2311yescharacteristic, complemented
186#482C2×C2311yescharacteristic, complemented
187#480C2×C2311yescharacteristic, complemented
188#479C2×C2311yescharacteristic, complemented
189#469C2×C2311yescharacteristic, complemented
190#466C2×C2311yescharacteristic, complemented
191#459C2×C2311yescharacteristic, complemented
192#449C2×C2311yescharacteristic, complemented
193#449C2×C2311yescharacteristic, complemented
194#440C2×C2311yescharacteristic, complemented
195#435C2×C2311yescharacteristic, complemented
196#417C2×C2311yescharacteristic, complemented
197#416C2×C2311yescharacteristic, complemented
198C2×C23—41no—
199C2×C23—41no—
200C2×C23—81no—
201C2×C23—81no—
202C2×C23—81no—
203C2×C23#54611yescharacteristic
204C2×C23#57511yescharacteristic
205C2×C23#57611yescharacteristic
206C2×#41C22×C2311yescharacteristic
207C2×#41C22×C2311yescharacteristic
208C2×SD16C22×C2311yescharacteristic
209C2×D8C22×C2311yescharacteristic
210C2×D8C22×C2311yescharacteristic
211C2×M16C22×C2311yescharacteristic
212C22×C8C22×C2311yescharacteristic
213#127—21no—
214#127C22×C2311yescharacteristic
215#126—21no—
216#125C22×C2311yescharacteristic
217#123—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#121C22×C2311yescharacteristic
222#121C22×C2311yescharacteristic
223#121C22×C2311yescharacteristic
224#121C22×C2311yescharacteristic
225C4×Q8C22×C2311yescharacteristic
226C4×D4—21no—
227C4×D4—21no—
228C4×D4C22×C2311yescharacteristic
229C4×D4C22×C2311yescharacteristic
230#117C22×C2311yescharacteristic
231C2×C4⋊C4C22×C2311yescharacteristic
232#107C22×C2311yescharacteristic
233#107C22×C2311yescharacteristic
234#106C22×C2311yescharacteristic
235C4⋊C8C22×C2311yescharacteristic
236#103C22×C2311yescharacteristic
237#103C22×C2311yescharacteristic
238#103C22×C2311yescharacteristic
239#102C22×C2311yescharacteristic
240#102C22×C2311yescharacteristic
241#102C22×C2311yescharacteristic
242#102C22×C2311yescharacteristic
243#102C22×C2311yescharacteristic
244#102C22×C2311yescharacteristic
245#102C22×C2311yescharacteristic
246C22⋊C8C22×C2311yescharacteristic
247C22⋊C8C22×C2311yescharacteristic
248#97C22×C2311yescharacteristic
249C23#330911yescharacteristic, direct product factor, Sylow
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C23×C2311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4D4×C2311yescharacteristic
264C2×D4C23×C2311yescharacteristic
265C2×D4C23×C2311yescharacteristic
266C2×D4C23×C2311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C23×C2311yescharacteristic
272C22×C4C23×C2311yescharacteristic
273SD16—22no—
274D8—41no—
275D8—41no—
276M16—41no—
277C2×C8—41no—
278C2×C8C23×C2311yescharacteristic
279C2×C8C23×C2311yescharacteristic
280C2×C8C23×C2311yescharacteristic
281C2×C8C23×C2311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4D4×C2311yescharacteristic
287C4⋊C4C23×C2311yescharacteristic
288C4⋊C4C23×C2311yescharacteristic
289C4⋊C4C23×C2311yescharacteristic
290C4⋊C4C23×C2311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4D4×C2311yescharacteristic
298C22⋊C4D4×C2311yescharacteristic
299C42—21no—
300C42—21no—
301C42C23×C2311yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23C2×D4×C2311yescharacteristic
306C23C2×D4×C2311yescharacteristic
307Q8—21no—
308Q8#41×C2312yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—41no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
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—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4C2×D4×C2311yescharacteristic
338C2×C4C2×D4×C2311yescharacteristic
339C2×C4C2×D4×C2311yescharacteristic
340C2×C4C2×D4×C2311yescharacteristic
341C2×C4C24×C2311yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×D4×C2311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4C2×#41×C2311yescharacteristic
372C4C23×H(5,𝔽2)11yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—81no—
378C2C23×#54611yescharacteristic
379C2C23×#57511yescharacteristic
380C2C23×#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×#550C2semidirect
3. C23×#490C2semidirect
4. C23×#482C2semidirect
5. C23×#480C2semidirect
6. C23×#479C2semidirect
7. C23×#469C2semidirect
8. C23×#466C2semidirect
9. C23×#459C2semidirect
10. C23×#449C2semidirect
11. C23×#449C2semidirect
12. C23×#440C2semidirect
13. C23×#435C2semidirect
14. C23×#417C2semidirect
15. C23×#416C2semidirect
155. #3309C23direct (N × H)
183. #553C2×C23semidirect
184. #550C2×C23semidirect
185. #490C2×C23semidirect
186. #482C2×C23semidirect
187. #480C2×C23semidirect
188. #479C2×C23semidirect
189. #469C2×C23semidirect
190. #466C2×C23semidirect
191. #459C2×C23semidirect
192. #449C2×C23semidirect
193. #449C2×C23semidirect
194. #440C2×C23semidirect
195. #435C2×C23semidirect
196. #417C2×C23semidirect
197. #416C2×C23semidirect
249. C23#3309direct (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×#5502
C23×#4902
C23×#4822
C23×#4802
C23×#4792
C23×#4692
C23×#4662
C23×#4592
C23×#4492
C23×#4402
C23×#4352
C23×#4172
C23×#4162
#330923