Finite Group Databasebeta

← C23×#3524 #15671143

Subgroups — C23×#3524 #15671143

Order 2944.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#559C213yesmaximal, complemented
2C23×#557C211yesmaximal, characteristic, complemented
3C23×#555C213yesmaximal, complemented
4C23×#553C213yesmaximal, complemented
5C23×#552C213yesmaximal, complemented
6C23×#550C211yesmaximal, characteristic, complemented
7C23×#548C211yesmaximal, characteristic, complemented
8C23×#548C211yesmaximal, characteristic, complemented
9C23×#542C213yesmaximal, complemented
10C23×#540C213yesmaximal, complemented
11C23×#529C213yesmaximal, complemented
12C23×#529C213yesmaximal, complemented
13C23×#520C211yesmaximal, characteristic, complemented
14C23×#519C211yesmaximal, characteristic, complemented
15C4×#41×C23C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×#41×C23C2211yescharacteristic
18C22×Q8×C23C2211yescharacteristic
19C23×C4⋊Q8C2213yes—
20C23×#127C2213yes—
21C23×#126C2216yes—
22C23×#126C2211yescharacteristic
23C23×#126C2213yes—
24C23×#125C2213yes—
25C23×#125C2213yes—
26C23×#125C2213yes—
27C23×#124C2213yes—
28C23×#124C2213yes—
29C23×#124C2213yes—
30C23×#124C2216yes—
31C23×#123C2216yes—
32C23×#123C2213yes—
33C23×#123C2213yes—
34C23×#123C2216yes—
35C23×C22⋊Q8C2213yes—
36C23×C22⋊Q8C2213yes—
37C23×C22⋊Q8C2213yes—
38C23×#121C2213yes—
39C23×#121C2213yes—
40C23×#121C2213yes—
41C23×#121C2213yes—
42C23×#121C2216yes—
43C23×#121C2213yes—
44C4×Q8×C23C2211yescharacteristic
45C4×Q8×C23C2211yescharacteristic
46C4×Q8×C23C2211yescharacteristic
47C4×Q8×C23C2213yes—
48C4×Q8×C23C2213yes—
49C4×Q8×C23C2211yescharacteristic
50C4×Q8×C23C2213yes—
51C4×D4×C23C2213yes—
52C4×D4×C23C2213yes—
53C4×D4×C23C2213yes—
54C4×D4×C23C2213yes—
55C4×D4×C23C2216yes—
56C4×D4×C23C2213yes—
57C4×D4×C23C2213yes—
58C4×D4×C23C2213yes—
59C23×#117C2213yes—
60C23×#117C2216yes—
61C23×#117C2213yes—
62C23×#117C2213yes—
63C23×#117C2213yes—
64C2×C4⋊C4×C23C2213yes—
65C2×C42×C23C2213yes—
66#41×C23—24no—
67#41×C23—24no—
68C2×Q8×C23—24no—
69C2×Q8×C23C2313yes—
70C2×Q8×C23C2313yes—
71C2×Q8×C23C2311yescharacteristic
72C2×D4×C23C2313yes—
73C2×D4×C23C2313yes—
74C2×D4×C23C2313yes—
75C2×D4×C23C2313yes—
76C2×D4×C23C2316yes—
77C2×D4×C23C2313yes—
78C22×C4×C23C2313yes—
79C22×C4×C23C2313yes—
80C22×C4×C23C2313yes—
81C22×C4×C23C2313yes—
82C22×C4×C23C2313yes—
83C22×C4×C23C2316yes—
84C4⋊C4×C23C2313yes—
85C4⋊C4×C23C2313yes—
86C4⋊C4×C23C2313yes—
87C4⋊C4×C23C2311yescharacteristic
88C4⋊C4×C23C2313yes—
89C4⋊C4×C23C2313yes—
90C4⋊C4×C23C2313yes—
91C4⋊C4×C23C2313yes—
92C4⋊C4×C23C2313yes—
93C4⋊C4×C23C2316yes—
94C4⋊C4×C23C2311yescharacteristic
95C4⋊C4×C23C2313yes—
96C4⋊C4×C23C2313yes—
97C4⋊C4×C23C2311yescharacteristic
98C4⋊C4×C23C2313yes—
99C22⋊C4×C23C2311yescharacteristic
100C22⋊C4×C23C2313yes—
101C22⋊C4×C23C2313yes—
102C22⋊C4×C23C2313yes—
103C22⋊C4×C23C2316yes—
104C22⋊C4×C23C2313yes—
105C22⋊C4×C23C2316yes—
106C22⋊C4×C23C2311yescharacteristic
107C22⋊C4×C23C2313yes—
108C22⋊C4×C23C2311yescharacteristic
109C22⋊C4×C23C2313yes—
110C22⋊C4×C23C2316yes—
111C22⋊C4×C23C2313yes—
112C42×C23C2311yescharacteristic
113C42×C23C2313yes—
114C42×C23C2313yes—
115C42×C23C2313yes—
116C42×C23C2316yes—
117C42×C23C2313yes—
118C42×C23C2313yes—
119C23×C23C2411yescharacteristic
120C23×C23C2413yes—
121C23×C23C2413yes—
122Q8×C23—26no—
123Q8×C23—43no—
124Q8×C23#4114yes—
125D4×C23—26no—
126D4×C23—26no—
127D4×C23—43no—
128D4×C23—43no—
129D4×C23—43no—
130D4×C23—46no—
131C2×C4×C23—26no—
132C2×C4×C23—26no—
133C2×C4×C23—26no—
134C2×C4×C23—43no—
135C2×C4×C23—46no—
136C2×C4×C23—43no—
137C2×C4×C23C2411yescharacteristic
138C2×C4×C23C2411yescharacteristic
139C2×C4×C23C2411yescharacteristic
140C2×C4×C23C2413yes—
141C2×C4×C23C2411yescharacteristic
142C2×C4×C23C2413yes—
143C2×C4×C23C2411yescharacteristic
144C2×C4×C23C2413yes—
145C2×C4×C23C2413yes—
146C2×C4×C23C2413yes—
147C2×C4×C23C2411yescharacteristic
148C2×C4×C23C2413yes—
149#3524C2311yesmaximal, characteristic, direct product factor, Sylow
150C22×C23—21no—
151C22×C23—23no—
152C22×C23—23no—
153C22×C23—23no—
154C22×C23—23no—
155C22×C23—23no—
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—23no—
159C22×C23C2511yescharacteristic
160C4×C23—21no—
161C4×C23—21no—
162C4×C23—23no—
163C4×C23—21no—
164C4×C23—23no—
165C4×C23—21no—
166C4×C23—23no—
167C4×C23—23no—
168C4×C23—21no—
169C4×C23—23no—
170C4×C23—21no—
171C4×C23C2×#4116yes—
172#559C2×C2313yescomplemented
173#557C2×C2311yescharacteristic, complemented
174#555C2×C2313yescomplemented
175#553C2×C2313yescomplemented
176#552C2×C2313yescomplemented
177#550C2×C2311yescharacteristic, complemented
178#548C2×C2311yescharacteristic, complemented
179#548C2×C2311yescharacteristic, complemented
180#542C2×C2313yescomplemented
181#540C2×C2313yescomplemented
182#529C2×C2313yescomplemented
183#529C2×C2313yescomplemented
184#520C2×C2311yescharacteristic, complemented
185#519C2×C2311yescharacteristic, complemented
186C4×#41C2×C2311yescharacteristic, complemented
187C2×C23—41no—
188C2×C23—43no—
189C2×C23—43no—
190C2×C23C22×#4111yescharacteristic
191C2×C23#58511yescharacteristic
192C2×C23#58511yescharacteristic
193C2×#41C22×C2311yescharacteristic
194C2×#41C22×C2311yescharacteristic
195C22×Q8C22×C2311yescharacteristic
196C4⋊Q8C22×C2313yes—
197#127C22×C2313yes—
198#126C22×C2311yescharacteristic
199#126C22×C2313yes—
200#126C22×C2316yes—
201#125C22×C2313yes—
202#125C22×C2313yes—
203#125C22×C2313yes—
204#124C22×C2313yes—
205#124C22×C2313yes—
206#124C22×C2313yes—
207#124C22×C2316yes—
208#123C22×C2316yes—
209#123C22×C2313yes—
210#123C22×C2313yes—
211#123C22×C2316yes—
212C22⋊Q8C22×C2313yes—
213C22⋊Q8C22×C2313yes—
214C22⋊Q8C22×C2313yes—
215#121C22×C2313yes—
216#121C22×C2313yes—
217#121C22×C2313yes—
218#121C22×C2313yes—
219#121C22×C2316yes—
220#121C22×C2313yes—
221C4×Q8C22×C2311yescharacteristic
222C4×Q8C22×C2311yescharacteristic
223C4×Q8C22×C2313yes—
224C4×Q8C22×C2311yescharacteristic
225C4×Q8C22×C2313yes—
226C4×Q8C22×C2311yescharacteristic
227C4×Q8C22×C2313yes—
228C4×D4C22×C2313yes—
229C4×D4C22×C2313yes—
230C4×D4C22×C2313yes—
231C4×D4C22×C2313yes—
232C4×D4C22×C2313yes—
233C4×D4C22×C2313yes—
234C4×D4C22×C2313yes—
235C4×D4C22×C2316yes—
236#117C22×C2313yes—
237#117C22×C2313yes—
238#117C22×C2313yes—
239#117C22×C2313yes—
240#117C22×C2316yes—
241C2×C4⋊C4C22×C2313yes—
242C2×C42C22×C2313yes—
243C23#352411yescharacteristic, direct product factor, Sylow
244#41—24no—
245#41—24no—
246C2×Q8—24no—
247C2×Q8C23×C2313yes—
248C2×Q8C23×C2313yes—
249C2×Q8C23×C2311yescharacteristic
250C2×D4C23×C2313yes—
251C2×D4C23×C2313yes—
252C2×D4C23×C2313yes—
253C2×D4C23×C2313yes—
254C2×D4C23×C2316yes—
255C2×D4C23×C2313yes—
256C22×C4C23×C2313yes—
257C22×C4C23×C2313yes—
258C22×C4C23×C2313yes—
259C22×C4C23×C2313yes—
260C22×C4C23×C2313yes—
261C22×C4C23×C2316yes—
262C4⋊C4C23×C2313yes—
263C4⋊C4C23×C2313yes—
264C4⋊C4C23×C2313yes—
265C4⋊C4C23×C2311yescharacteristic
266C4⋊C4C23×C2313yes—
267C4⋊C4C23×C2313yes—
268C4⋊C4C23×C2313yes—
269C4⋊C4C23×C2313yes—
270C4⋊C4C23×C2313yes—
271C4⋊C4C23×C2316yes—
272C4⋊C4C23×C2311yescharacteristic
273C4⋊C4C23×C2313yes—
274C4⋊C4C23×C2313yes—
275C4⋊C4C23×C2311yescharacteristic
276C4⋊C4C23×C2313yes—
277C22⋊C4C23×C2311yescharacteristic
278C22⋊C4C23×C2313yes—
279C22⋊C4C23×C2313yes—
280C22⋊C4C23×C2313yes—
281C22⋊C4C23×C2316yes—
282C22⋊C4C23×C2313yes—
283C22⋊C4C23×C2316yes—
284C22⋊C4C23×C2311yescharacteristic
285C22⋊C4C23×C2313yes—
286C22⋊C4C23×C2311yescharacteristic
287C22⋊C4C23×C2313yes—
288C22⋊C4C23×C2316yes—
289C22⋊C4C23×C2313yes—
290C42C23×C2311yescharacteristic
291C42C23×C2313yes—
292C42C23×C2313yes—
293C42C23×C2313yes—
294C42C23×C2316yes—
295C42C23×C2313yes—
296C42C23×C2313yes—
297C23C24×C2311yescharacteristic
298C23C24×C2313yes—
299C23C24×C2313yes—
300Q8—26no—
301Q8—43no—
302Q8#41×C2314yes—
303D4—26no—
304D4—26no—
305D4—43no—
306D4—43no—
307D4—43no—
308D4—46no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—43no—
313C2×C4—46no—
314C2×C4—43no—
315C2×C4C24×C2311yescharacteristic
316C2×C4C24×C2311yescharacteristic
317C2×C4C24×C2311yescharacteristic
318C2×C4C24×C2313yes—
319C2×C4C24×C2311yescharacteristic
320C2×C4C24×C2313yes—
321C2×C4C24×C2311yescharacteristic
322C2×C4C24×C2313yes—
323C2×C4C24×C2313yes—
324C2×C4C24×C2313yes—
325C2×C4C24×C2311yescharacteristic
326C2×C4C24×C2313yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—21no—
334C22—21no—
335C22—23no—
336C22C25×C2311yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—23no—
344C4—23no—
345C4—21no—
346C4—23no—
347C4—21no—
348C4C2×#41×C2316yes—
349C2—41no—
350C2—43no—
351C2—43no—
352C2C22×#41×C2311yescharacteristic
353C2C23×#58511yescharacteristic
354C2C23×#58511yescharacteristic
subgroupG/H
center, socleC22×C23C25
commutator, Frattini, cosocleC22C25×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#559C2semidirect
2. C23×#557C2semidirect
3. C23×#555C2semidirect
4. C23×#553C2semidirect
5. C23×#552C2semidirect
6. C23×#550C2semidirect
7. C23×#548C2semidirect
8. C23×#548C2semidirect
9. C23×#542C2semidirect
10. C23×#540C2semidirect
11. C23×#529C2semidirect
12. C23×#529C2semidirect
13. C23×#520C2semidirect
14. C23×#519C2semidirect
15. C4×#41×C23C2semidirect
149. #3524C23direct (N × H)
172. #559C2×C23semidirect
173. #557C2×C23semidirect
174. #555C2×C23semidirect
175. #553C2×C23semidirect
176. #552C2×C23semidirect
177. #550C2×C23semidirect
178. #548C2×C23semidirect
179. #548C2×C23semidirect
180. #542C2×C23semidirect
181. #540C2×C23semidirect
182. #529C2×C23semidirect
183. #529C2×C23semidirect
184. #520C2×C23semidirect
185. #519C2×C23semidirect
186. C4×#41C2×C23semidirect
243. C23#3524direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5592
C23×#5572
C23×#5552
C23×#5532
C23×#5522
C23×#5502
C23×#5482
C23×#5422
C23×#5402
C23×#5292
C23×#5202
C23×#5192
C4×#41×C232
#352423