Finite Group Databasebeta

← C83×#3516 #24905201

Subgroups — C83×#3516 #24905201

Order 10624.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#557×C83C211yesmaximal, characteristic, complemented
2#556×C83C213yesmaximal, complemented
3#554×C83C213yesmaximal, complemented
4#553×C83C213yesmaximal, complemented
5#552×C83C213yesmaximal, complemented
6D4×Q8×C83C211yesmaximal, characteristic, complemented
7#548×C83C211yesmaximal, characteristic, complemented
8#548×C83C211yesmaximal, characteristic, complemented
9#542×C83C213yesmaximal, complemented
10#541×C83C213yesmaximal, complemented
11#529×C83C213yesmaximal, complemented
12#529×C83C213yesmaximal, complemented
13#520×C83C211yesmaximal, characteristic, complemented
14#519×C83C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C83C211yesmaximal, characteristic, complemented
16C2×#41×C83C2211yescharacteristic
17C22×Q8×C83C2211yescharacteristic
18C22×Q8×C83C2211yescharacteristic
19C4⋊Q8×C83C2213yes—
20C4⋊Q8×C83C2213yes—
21#126×C83C2216yes—
22#126×C83C2213yes—
23#126×C83C2213yes—
24#126×C83C2216yes—
25#125×C83C2213yes—
26#125×C83C2213yes—
27#125×C83C2216yes—
28#125×C83C2213yes—
29#124×C83C2213yes—
30#124×C83C2213yes—
31#124×C83C2213yes—
32#123×C83C2211yescharacteristic
33#123×C83C2213yes—
34#123×C83C2216yes—
35C22⋊Q8×C83C2213yes—
36C22⋊Q8×C83C2213yes—
37C22⋊Q8×C83C2213yes—
38C22⋊Q8×C83C2213yes—
39C22⋊Q8×C83C2213yes—
40C22⋊Q8×C83C2216yes—
41#121×C83C2213yes—
42#121×C83C2213yes—
43#121×C83C2213yes—
44C4×Q8×C83C2211yescharacteristic
45C4×Q8×C83C2213yes—
46C4×Q8×C83C2211yescharacteristic
47C4×Q8×C83C2213yes—
48C4×Q8×C83C2213yes—
49C4×Q8×C83C2211yescharacteristic
50C4×Q8×C83C2213yes—
51C4×Q8×C83C2211yescharacteristic
52C4×Q8×C83C2213yes—
53C4×D4×C83C2213yes—
54C4×D4×C83C2213yes—
55C4×D4×C83C2213yes—
56C4×D4×C83C2213yes—
57C4×D4×C83C2213yes—
58C4×D4×C83C2216yes—
59#117×C83C2213yes—
60#117×C83C2213yes—
61#117×C83C2213yes—
62#117×C83C2216yes—
63C2×C4⋊C4×C83C2213yes—
64C2×C4⋊C4×C83C2213yes—
65C2×C42×C83C2213yes—
66#41×C83—24no—
67C2×Q8×C83—24no—
68C2×Q8×C83—24no—
69C2×Q8×C83C2313yes—
70C2×Q8×C83C2313yes—
71C2×Q8×C83C2313yes—
72C2×Q8×C83C2311yescharacteristic
73C2×D4×C83C2313yes—
74C2×D4×C83C2311yescharacteristic
75C2×D4×C83C2313yes—
76C2×D4×C83C2313yes—
77C22×C4×C83C2311yescharacteristic
78C22×C4×C83C2313yes—
79C22×C4×C83C2313yes—
80C22×C4×C83C2313yes—
81C22×C4×C83C2313yes—
82C22×C4×C83C2316yes—
83C4⋊C4×C83C2313yes—
84C4⋊C4×C83C2313yes—
85C4⋊C4×C83C2313yes—
86C4⋊C4×C83C2313yes—
87C4⋊C4×C83C2316yes—
88C4⋊C4×C83C2311yescharacteristic
89C4⋊C4×C83C2313yes—
90C4⋊C4×C83C2313yes—
91C4⋊C4×C83C2313yes—
92C4⋊C4×C83C2313yes—
93C4⋊C4×C83C2316yes—
94C4⋊C4×C83C2313yes—
95C4⋊C4×C83C2313yes—
96C4⋊C4×C83C2316yes—
97C4⋊C4×C83C2313yes—
98C4⋊C4×C83C2311yescharacteristic
99C4⋊C4×C83C2313yes—
100C22⋊C4×C83C2313yes—
101C22⋊C4×C83C2311yescharacteristic
102C22⋊C4×C83C2313yes—
103C22⋊C4×C83C2313yes—
104C22⋊C4×C83C2313yes—
105C22⋊C4×C83C2316yes—
106C22⋊C4×C83C2311yescharacteristic
107C22⋊C4×C83C2313yes—
108C22⋊C4×C83C2311yescharacteristic
109C22⋊C4×C83C2316yes—
110C22⋊C4×C83C2313yes—
111C22⋊C4×C83C2313yes—
112C42×C83C2313yes—
113C42×C83C2313yes—
114C42×C83C2313yes—
115C42×C83C2313yes—
116C42×C83C2316yes—
117C42×C83C2313yes—
118C42×C83C2313yes—
119C23×C83C2411yescharacteristic
120C23×C83C2413yes—
121C23×C83C2411yescharacteristic
122Q8×C83—26no—
123Q8×C83—26no—
124Q8×C83—43no—
125Q8×C83#4114yes—
126D4×C83—26no—
127D4×C83—22no—
128D4×C83—26no—
129D4×C83—43no—
130C2×C4×C83—22no—
131C2×C4×C83—26no—
132C2×C4×C83—26no—
133C2×C4×C83—26no—
134C2×C4×C83—46no—
135C2×C4×C83—43no—
136C2×C4×C83C2411yescharacteristic
137C2×C4×C83C2411yescharacteristic
138C2×C4×C83C2413yes—
139C2×C4×C83C2411yescharacteristic
140C2×C4×C83C2413yes—
141C2×C4×C83C2411yescharacteristic
142C2×C4×C83C2413yes—
143C2×C4×C83C2413yes—
144C2×C4×C83C2413yes—
145C2×C4×C83C2413yes—
146C2×C4×C83C2411yescharacteristic
147C2×C4×C83C2413yes—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—23no—
152C22×C83—21no—
153C22×C83—23no—
154C22×C83—23no—
155C22×C83—21no—
156C22×C832−1+412yes—
157C22×C83C2511yescharacteristic
158C4×C83—21no—
159C4×C83—23no—
160C4×C83—23no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—23no—
164C4×C83—21no—
165C4×C83—23no—
166C4×C83—23no—
167C4×C83—23no—
168C4×C83—21no—
169C4×C83C2×#4116yes—
170C2×C83—22no—
171C2×C83—41no—
172C2×C83—43no—
173C2×C83C22×#4111yescharacteristic
174C2×C83C2×2−1+411yescharacteristic
175C2×C83#58511yescharacteristic
176#3516C8311yesmaximal, characteristic, direct product factor, Sylow
177C83#351611yescharacteristic, direct product factor, Sylow
178#557C2×C8311yescharacteristic, complemented
179#556C2×C8313yescomplemented
180#554C2×C8313yescomplemented
181#553C2×C8313yescomplemented
182#552C2×C8313yescomplemented
183D4×Q8C2×C8311yescharacteristic, complemented
184#548C2×C8311yescharacteristic, complemented
185#548C2×C8311yescharacteristic, complemented
186#542C2×C8313yescomplemented
187#541C2×C8313yescomplemented
188#529C2×C8313yescomplemented
189#529C2×C8313yescomplemented
190#520C2×C8311yescharacteristic, complemented
191#519C2×C8311yescharacteristic, complemented
192C2×C4×Q8C2×C8311yescharacteristic, complemented
193C2×#41C22×C8311yescharacteristic
194C22×Q8C22×C8311yescharacteristic
195C22×Q8C22×C8311yescharacteristic
196C4⋊Q8C22×C8313yes—
197C4⋊Q8C22×C8313yes—
198#126C22×C8316yes—
199#126C22×C8313yes—
200#126C22×C8313yes—
201#126C22×C8316yes—
202#125C22×C8313yes—
203#125C22×C8313yes—
204#125C22×C8316yes—
205#125C22×C8313yes—
206#124C22×C8313yes—
207#124C22×C8313yes—
208#124C22×C8313yes—
209#123C22×C8311yescharacteristic
210#123C22×C8313yes—
211#123C22×C8316yes—
212C22⋊Q8C22×C8313yes—
213C22⋊Q8C22×C8313yes—
214C22⋊Q8C22×C8313yes—
215C22⋊Q8C22×C8313yes—
216C22⋊Q8C22×C8313yes—
217C22⋊Q8C22×C8316yes—
218#121C22×C8313yes—
219#121C22×C8313yes—
220#121C22×C8313yes—
221C4×Q8C22×C8311yescharacteristic
222C4×Q8C22×C8313yes—
223C4×Q8C22×C8311yescharacteristic
224C4×Q8C22×C8313yes—
225C4×Q8C22×C8313yes—
226C4×Q8C22×C8311yescharacteristic
227C4×Q8C22×C8313yes—
228C4×Q8C22×C8311yescharacteristic
229C4×Q8C22×C8313yes—
230C4×D4C22×C8313yes—
231C4×D4C22×C8313yes—
232C4×D4C22×C8313yes—
233C4×D4C22×C8313yes—
234C4×D4C22×C8313yes—
235C4×D4C22×C8316yes—
236#117C22×C8313yes—
237#117C22×C8313yes—
238#117C22×C8313yes—
239#117C22×C8316yes—
240C2×C4⋊C4C22×C8313yes—
241C2×C4⋊C4C22×C8313yes—
242C2×C42C22×C8313yes—
243#41—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8C23×C8313yes—
247C2×Q8C23×C8313yes—
248C2×Q8C23×C8313yes—
249C2×Q8C23×C8311yescharacteristic
250C2×D4C23×C8313yes—
251C2×D4C23×C8311yescharacteristic
252C2×D4C23×C8313yes—
253C2×D4C23×C8313yes—
254C22×C4C23×C8311yescharacteristic
255C22×C4C23×C8313yes—
256C22×C4C23×C8313yes—
257C22×C4C23×C8313yes—
258C22×C4C23×C8313yes—
259C22×C4C23×C8316yes—
260C4⋊C4C23×C8313yes—
261C4⋊C4C23×C8313yes—
262C4⋊C4C23×C8313yes—
263C4⋊C4C23×C8313yes—
264C4⋊C4C23×C8316yes—
265C4⋊C4C23×C8311yescharacteristic
266C4⋊C4C23×C8313yes—
267C4⋊C4C23×C8313yes—
268C4⋊C4C23×C8313yes—
269C4⋊C4C23×C8313yes—
270C4⋊C4C23×C8316yes—
271C4⋊C4C23×C8313yes—
272C4⋊C4C23×C8313yes—
273C4⋊C4C23×C8316yes—
274C4⋊C4C23×C8313yes—
275C4⋊C4C23×C8311yescharacteristic
276C4⋊C4C23×C8313yes—
277C22⋊C4C23×C8313yes—
278C22⋊C4C23×C8311yescharacteristic
279C22⋊C4C23×C8313yes—
280C22⋊C4C23×C8313yes—
281C22⋊C4C23×C8313yes—
282C22⋊C4C23×C8316yes—
283C22⋊C4C23×C8311yescharacteristic
284C22⋊C4C23×C8313yes—
285C22⋊C4C23×C8311yescharacteristic
286C22⋊C4C23×C8316yes—
287C22⋊C4C23×C8313yes—
288C22⋊C4C23×C8313yes—
289C42C23×C8313yes—
290C42C23×C8313yes—
291C42C23×C8313yes—
292C42C23×C8313yes—
293C42C23×C8316yes—
294C42C23×C8313yes—
295C42C23×C8313yes—
296C23C24×C8311yescharacteristic
297C23C24×C8313yes—
298C23C24×C8311yescharacteristic
299Q8—26no—
300Q8—26no—
301Q8—43no—
302Q8#41×C8314yes—
303D4—26no—
304D4—22no—
305D4—26no—
306D4—43no—
307C2×C4—22no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—46no—
312C2×C4—43no—
313C2×C4C24×C8311yescharacteristic
314C2×C4C24×C8311yescharacteristic
315C2×C4C24×C8313yes—
316C2×C4C24×C8311yescharacteristic
317C2×C4C24×C8313yes—
318C2×C4C24×C8311yescharacteristic
319C2×C4C24×C8313yes—
320C2×C4C24×C8313yes—
321C2×C4C24×C8313yes—
322C2×C4C24×C8313yes—
323C2×C4C24×C8311yescharacteristic
324C2×C4C24×C8313yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C222−1+4×C8312yes—
334C22C25×C8311yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—21no—
342C4—23no—
343C4—23no—
344C4—23no—
345C4—21no—
346C4C2×#41×C8316yes—
347C2—22no—
348C2—41no—
349C2—43no—
350C2C22×#41×C8311yescharacteristic
351C2C2×2−1+4×C8311yescharacteristic
352C2#585×C8311yescharacteristic
subgroupG/H
center, socleC22×C83C25
commutator, Frattini, cosocleC22C25×C83
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. #557×C83C2semidirect
2. #556×C83C2semidirect
3. #554×C83C2semidirect
4. #553×C83C2semidirect
5. #552×C83C2semidirect
6. D4×Q8×C83C2semidirect
7. #548×C83C2semidirect
8. #548×C83C2semidirect
9. #542×C83C2semidirect
10. #541×C83C2semidirect
11. #529×C83C2semidirect
12. #529×C83C2semidirect
13. #520×C83C2semidirect
14. #519×C83C2semidirect
15. C2×C4×Q8×C83C2semidirect
176. #3516C83direct (N × H)
177. C83#3516direct (N × H)
178. #557C2×C83semidirect
179. #556C2×C83semidirect
180. #554C2×C83semidirect
181. #553C2×C83semidirect
182. #552C2×C83semidirect
183. D4×Q8C2×C83semidirect
184. #548C2×C83semidirect
185. #548C2×C83semidirect
186. #542C2×C83semidirect
187. #541C2×C83semidirect
188. #529C2×C83semidirect
189. #529C2×C83semidirect
190. #520C2×C83semidirect
191. #519C2×C83semidirect
192. C2×C4×Q8C2×C83semidirect

Maximal subgroups

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

subgroupindex
#557×C832
#556×C832
#554×C832
#553×C832
#552×C832
D4×Q8×C832
#548×C832
#542×C832
#541×C832
#529×C832
#520×C832
#519×C832
C2×C4×Q8×C832
#351683