Finite Group Databasebeta

← #3524×C233 #44449556

Subgroups — #3524×C233 #44449556

Order 29824.

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
1#559×C233C213yesmaximal, complemented
2#557×C233C211yesmaximal, characteristic, complemented
3#555×C233C213yesmaximal, complemented
4#553×C233C213yesmaximal, complemented
5#552×C233C213yesmaximal, complemented
6#550×C233C211yesmaximal, characteristic, complemented
7#548×C233C211yesmaximal, characteristic, complemented
8#548×C233C211yesmaximal, characteristic, complemented
9#542×C233C213yesmaximal, complemented
10#540×C233C213yesmaximal, complemented
11#529×C233C213yesmaximal, complemented
12#529×C233C213yesmaximal, complemented
13#520×C233C211yesmaximal, characteristic, complemented
14#519×C233C211yesmaximal, characteristic, complemented
15C4×#41×C233C211yesmaximal, characteristic, complemented
16C2×#41×C233C2211yescharacteristic
17C2×#41×C233C2211yescharacteristic
18C22×Q8×C233C2211yescharacteristic
19C4⋊Q8×C233C2213yes—
20#127×C233C2213yes—
21#126×C233C2211yescharacteristic
22#126×C233C2213yes—
23#126×C233C2216yes—
24#125×C233C2213yes—
25#125×C233C2213yes—
26#125×C233C2213yes—
27#124×C233C2213yes—
28#124×C233C2213yes—
29#124×C233C2213yes—
30#124×C233C2216yes—
31#123×C233C2216yes—
32#123×C233C2213yes—
33#123×C233C2213yes—
34#123×C233C2216yes—
35C22⋊Q8×C233C2213yes—
36C22⋊Q8×C233C2213yes—
37C22⋊Q8×C233C2213yes—
38#121×C233C2213yes—
39#121×C233C2213yes—
40#121×C233C2213yes—
41#121×C233C2213yes—
42#121×C233C2216yes—
43#121×C233C2213yes—
44C4×Q8×C233C2211yescharacteristic
45C4×Q8×C233C2211yescharacteristic
46C4×Q8×C233C2213yes—
47C4×Q8×C233C2211yescharacteristic
48C4×Q8×C233C2213yes—
49C4×Q8×C233C2211yescharacteristic
50C4×Q8×C233C2213yes—
51C4×D4×C233C2213yes—
52C4×D4×C233C2213yes—
53C4×D4×C233C2213yes—
54C4×D4×C233C2213yes—
55C4×D4×C233C2213yes—
56C4×D4×C233C2213yes—
57C4×D4×C233C2213yes—
58C4×D4×C233C2216yes—
59#117×C233C2213yes—
60#117×C233C2213yes—
61#117×C233C2213yes—
62#117×C233C2213yes—
63#117×C233C2216yes—
64C2×C4⋊C4×C233C2213yes—
65C2×C42×C233C2213yes—
66#41×C233—24no—
67#41×C233—24no—
68C2×Q8×C233—24no—
69C2×Q8×C233C2313yes—
70C2×Q8×C233C2313yes—
71C2×Q8×C233C2311yescharacteristic
72C2×D4×C233C2313yes—
73C2×D4×C233C2313yes—
74C2×D4×C233C2313yes—
75C2×D4×C233C2313yes—
76C2×D4×C233C2316yes—
77C2×D4×C233C2313yes—
78C22×C4×C233C2313yes—
79C22×C4×C233C2313yes—
80C22×C4×C233C2313yes—
81C22×C4×C233C2313yes—
82C22×C4×C233C2313yes—
83C22×C4×C233C2316yes—
84C4⋊C4×C233C2313yes—
85C4⋊C4×C233C2313yes—
86C4⋊C4×C233C2313yes—
87C4⋊C4×C233C2311yescharacteristic
88C4⋊C4×C233C2313yes—
89C4⋊C4×C233C2313yes—
90C4⋊C4×C233C2313yes—
91C4⋊C4×C233C2313yes—
92C4⋊C4×C233C2313yes—
93C4⋊C4×C233C2316yes—
94C4⋊C4×C233C2311yescharacteristic
95C4⋊C4×C233C2313yes—
96C4⋊C4×C233C2313yes—
97C4⋊C4×C233C2311yescharacteristic
98C4⋊C4×C233C2313yes—
99C22⋊C4×C233C2311yescharacteristic
100C22⋊C4×C233C2313yes—
101C22⋊C4×C233C2313yes—
102C22⋊C4×C233C2313yes—
103C22⋊C4×C233C2316yes—
104C22⋊C4×C233C2313yes—
105C22⋊C4×C233C2316yes—
106C22⋊C4×C233C2311yescharacteristic
107C22⋊C4×C233C2313yes—
108C22⋊C4×C233C2311yescharacteristic
109C22⋊C4×C233C2313yes—
110C22⋊C4×C233C2316yes—
111C22⋊C4×C233C2313yes—
112C42×C233C2311yescharacteristic
113C42×C233C2313yes—
114C42×C233C2313yes—
115C42×C233C2313yes—
116C42×C233C2316yes—
117C42×C233C2313yes—
118C42×C233C2313yes—
119C23×C233C2411yescharacteristic
120C23×C233C2413yes—
121C23×C233C2413yes—
122Q8×C233—26no—
123Q8×C233—43no—
124Q8×C233#4114yes—
125D4×C233—26no—
126D4×C233—26no—
127D4×C233—43no—
128D4×C233—43no—
129D4×C233—43no—
130D4×C233—46no—
131C2×C4×C233—26no—
132C2×C4×C233—26no—
133C2×C4×C233—26no—
134C2×C4×C233—43no—
135C2×C4×C233—46no—
136C2×C4×C233—43no—
137C2×C4×C233C2411yescharacteristic
138C2×C4×C233C2411yescharacteristic
139C2×C4×C233C2411yescharacteristic
140C2×C4×C233C2413yes—
141C2×C4×C233C2411yescharacteristic
142C2×C4×C233C2413yes—
143C2×C4×C233C2411yescharacteristic
144C2×C4×C233C2413yes—
145C2×C4×C233C2413yes—
146C2×C4×C233C2413yes—
147C2×C4×C233C2411yescharacteristic
148C2×C4×C233C2413yes—
149C22×C233—21no—
150C22×C233—23no—
151C22×C233—23no—
152C22×C233—23no—
153C22×C233—23no—
154C22×C233—23no—
155C22×C233—21no—
156C22×C233—21no—
157C22×C233—23no—
158C22×C233C2511yescharacteristic
159C4×C233—21no—
160C4×C233—21no—
161C4×C233—23no—
162C4×C233—21no—
163C4×C233—23no—
164C4×C233—21no—
165C4×C233—23no—
166C4×C233—23no—
167C4×C233—21no—
168C4×C233—23no—
169C4×C233—21no—
170C4×C233C2×#4116yes—
171C2×C233—41no—
172C2×C233—43no—
173C2×C233—43no—
174C2×C233C22×#4111yescharacteristic
175C2×C233#58511yescharacteristic
176C2×C233#58511yescharacteristic
177C233#352411yescharacteristic, direct product factor, Sylow
178#3524C23311yesmaximal, characteristic, direct product factor, Sylow
179#559C2×C23313yescomplemented
180#557C2×C23311yescharacteristic, complemented
181#555C2×C23313yescomplemented
182#553C2×C23313yescomplemented
183#552C2×C23313yescomplemented
184#550C2×C23311yescharacteristic, complemented
185#548C2×C23311yescharacteristic, complemented
186#548C2×C23311yescharacteristic, complemented
187#542C2×C23313yescomplemented
188#540C2×C23313yescomplemented
189#529C2×C23313yescomplemented
190#529C2×C23313yescomplemented
191#520C2×C23311yescharacteristic, complemented
192#519C2×C23311yescharacteristic, complemented
193C4×#41C2×C23311yescharacteristic, complemented
194C2×#41C22×C23311yescharacteristic
195C2×#41C22×C23311yescharacteristic
196C22×Q8C22×C23311yescharacteristic
197C4⋊Q8C22×C23313yes—
198#127C22×C23313yes—
199#126C22×C23311yescharacteristic
200#126C22×C23313yes—
201#126C22×C23316yes—
202#125C22×C23313yes—
203#125C22×C23313yes—
204#125C22×C23313yes—
205#124C22×C23313yes—
206#124C22×C23313yes—
207#124C22×C23313yes—
208#124C22×C23316yes—
209#123C22×C23316yes—
210#123C22×C23313yes—
211#123C22×C23313yes—
212#123C22×C23316yes—
213C22⋊Q8C22×C23313yes—
214C22⋊Q8C22×C23313yes—
215C22⋊Q8C22×C23313yes—
216#121C22×C23313yes—
217#121C22×C23313yes—
218#121C22×C23313yes—
219#121C22×C23313yes—
220#121C22×C23316yes—
221#121C22×C23313yes—
222C4×Q8C22×C23311yescharacteristic
223C4×Q8C22×C23311yescharacteristic
224C4×Q8C22×C23313yes—
225C4×Q8C22×C23311yescharacteristic
226C4×Q8C22×C23313yes—
227C4×Q8C22×C23311yescharacteristic
228C4×Q8C22×C23313yes—
229C4×D4C22×C23313yes—
230C4×D4C22×C23313yes—
231C4×D4C22×C23313yes—
232C4×D4C22×C23313yes—
233C4×D4C22×C23313yes—
234C4×D4C22×C23313yes—
235C4×D4C22×C23313yes—
236C4×D4C22×C23316yes—
237#117C22×C23313yes—
238#117C22×C23313yes—
239#117C22×C23313yes—
240#117C22×C23313yes—
241#117C22×C23316yes—
242C2×C4⋊C4C22×C23313yes—
243C2×C42C22×C23313yes—
244#41—24no—
245#41—24no—
246C2×Q8—24no—
247C2×Q8C23×C23313yes—
248C2×Q8C23×C23313yes—
249C2×Q8C23×C23311yescharacteristic
250C2×D4C23×C23313yes—
251C2×D4C23×C23313yes—
252C2×D4C23×C23313yes—
253C2×D4C23×C23313yes—
254C2×D4C23×C23316yes—
255C2×D4C23×C23313yes—
256C22×C4C23×C23313yes—
257C22×C4C23×C23313yes—
258C22×C4C23×C23313yes—
259C22×C4C23×C23313yes—
260C22×C4C23×C23313yes—
261C22×C4C23×C23316yes—
262C4⋊C4C23×C23313yes—
263C4⋊C4C23×C23313yes—
264C4⋊C4C23×C23313yes—
265C4⋊C4C23×C23311yescharacteristic
266C4⋊C4C23×C23313yes—
267C4⋊C4C23×C23313yes—
268C4⋊C4C23×C23313yes—
269C4⋊C4C23×C23313yes—
270C4⋊C4C23×C23313yes—
271C4⋊C4C23×C23316yes—
272C4⋊C4C23×C23311yescharacteristic
273C4⋊C4C23×C23313yes—
274C4⋊C4C23×C23313yes—
275C4⋊C4C23×C23311yescharacteristic
276C4⋊C4C23×C23313yes—
277C22⋊C4C23×C23311yescharacteristic
278C22⋊C4C23×C23313yes—
279C22⋊C4C23×C23313yes—
280C22⋊C4C23×C23313yes—
281C22⋊C4C23×C23316yes—
282C22⋊C4C23×C23313yes—
283C22⋊C4C23×C23316yes—
284C22⋊C4C23×C23311yescharacteristic
285C22⋊C4C23×C23313yes—
286C22⋊C4C23×C23311yescharacteristic
287C22⋊C4C23×C23313yes—
288C22⋊C4C23×C23316yes—
289C22⋊C4C23×C23313yes—
290C42C23×C23311yescharacteristic
291C42C23×C23313yes—
292C42C23×C23313yes—
293C42C23×C23313yes—
294C42C23×C23316yes—
295C42C23×C23313yes—
296C42C23×C23313yes—
297C23C24×C23311yescharacteristic
298C23C24×C23313yes—
299C23C24×C23313yes—
300Q8—26no—
301Q8—43no—
302Q8#41×C23314yes—
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×C23311yescharacteristic
316C2×C4C24×C23311yescharacteristic
317C2×C4C24×C23311yescharacteristic
318C2×C4C24×C23313yes—
319C2×C4C24×C23311yescharacteristic
320C2×C4C24×C23313yes—
321C2×C4C24×C23311yescharacteristic
322C2×C4C24×C23313yes—
323C2×C4C24×C23313yes—
324C2×C4C24×C23313yes—
325C2×C4C24×C23311yescharacteristic
326C2×C4C24×C23313yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—21no—
334C22—21no—
335C22—23no—
336C22C25×C23311yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—23no—
344C4—23no—
345C4—21no—
346C4—23no—
347C4—21no—
348C4C2×#41×C23316yes—
349C2—41no—
350C2—43no—
351C2—43no—
352C2C22×#41×C23311yescharacteristic
353C2#585×C23311yescharacteristic
354C2#585×C23311yescharacteristic
subgroupG/H
center, socleC22×C233C25
commutator, Frattini, cosocleC22C25×C233
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C233 ◁ G

Complements

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

NH ≅ G/Ntype
1. #559×C233C2semidirect
2. #557×C233C2semidirect
3. #555×C233C2semidirect
4. #553×C233C2semidirect
5. #552×C233C2semidirect
6. #550×C233C2semidirect
7. #548×C233C2semidirect
8. #548×C233C2semidirect
9. #542×C233C2semidirect
10. #540×C233C2semidirect
11. #529×C233C2semidirect
12. #529×C233C2semidirect
13. #520×C233C2semidirect
14. #519×C233C2semidirect
15. C4×#41×C233C2semidirect
177. C233#3524direct (N × H)
178. #3524C233direct (N × H)
179. #559C2×C233semidirect
180. #557C2×C233semidirect
181. #555C2×C233semidirect
182. #553C2×C233semidirect
183. #552C2×C233semidirect
184. #550C2×C233semidirect
185. #548C2×C233semidirect
186. #548C2×C233semidirect
187. #542C2×C233semidirect
188. #540C2×C233semidirect
189. #529C2×C233semidirect
190. #529C2×C233semidirect
191. #520C2×C233semidirect
192. #519C2×C233semidirect
193. C4×#41C2×C233semidirect

Maximal subgroups

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

subgroupindex
#559×C2332
#557×C2332
#555×C2332
#553×C2332
#552×C2332
#550×C2332
#548×C2332
#542×C2332
#540×C2332
#529×C2332
#520×C2332
#519×C2332
C4×#41×C2332
#3524233