Finite Group Databasebeta

← #24919538

Subgroups — #24919538

Order 10624.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128347C211yesmaximal, characteristic
2#19128063C211yesmaximal, characteristic, complemented
3#19128036C211yesmaximal, characteristic, complemented
4#19128014C211yesmaximal, characteristic, complemented
5#19127978C211yesmaximal, characteristic, complemented
6#19127949C211yesmaximal, characteristic, complemented
7#19127940C211yesmaximal, characteristic, complemented
8#19127923C211yesmaximal, characteristic, complemented
9#19127861C211yesmaximal, characteristic, complemented
10#19127851C211yesmaximal, characteristic, complemented
11#19127799C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127700C211yesmaximal, characteristic, complemented
14C2×#14581712C211yesmaximal, characteristic, complemented
15#554×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050577C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21C2×#13050574C2211yescharacteristic, complemented
22C22×C83⋊C8C2211yescharacteristic
23#14581780C2211yescharacteristic
24#14581777—21no—
25#14581771C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581765—21no—
28#14581760—21no—
29C4×D332C2211yescharacteristic
30C22⋊C4×D83—21no—
31#14581725C2211yescharacteristic, complemented
32#14581725C2211yescharacteristic, complemented
33#14581725C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35#14581716C2211yescharacteristic, complemented
36#14581715C2211yescharacteristic, complemented
37#14581715C2211yescharacteristic, complemented
38#14581715C2211yescharacteristic, complemented
39#14581712C2211yescharacteristic
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic, complemented
47#14581703C2211yescharacteristic, complemented
48C4⋊Q8×C83C2211yescharacteristic, complemented
49#125×C83—21no—
50C22⋊Q8×C83—21no—
51C22⋊Q8×C83—21no—
52C22⋊Q8×C83C2211yescharacteristic
53C22⋊Q8×C83C2211yescharacteristic, complemented
54C4×Q8×C83C2211yescharacteristic, complemented
55C4×D4×C83—21no—
56C4×D4×C83C2211yescharacteristic
57C2×C4⋊C4×C83C2211yescharacteristic
58C2×C4⋊C4×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050590—22no—
61#13050590—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—22no—
64C2×D332—21no—
65C2×D332—41no—
66C2×D332C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72#13050580—41no—
73#13050577—22no—
74#13050577—41no—
75#13050574—41no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C2×C83⋊C8C2311yescharacteristic
81C4⋊Dic83C2311yescharacteristic
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87(C2×D83)⋊C4—21no—
88C2×Q8×C83—21no—
89C2×Q8×C83C2311yescharacteristic
90C2×D4×C83D411yescharacteristic
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83D411yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C4⋊C4×C83C2311yescharacteristic
104C4⋊C4×C83C2311yescharacteristic
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83D411yescharacteristic
108C22⋊C4×C83D411yescharacteristic
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83C2×D411yescharacteristic
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—81no—
118Q8×C83—21no—
119Q8×C83—21no—
120Q8×C83—81no—
121D4×C83—22no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—81no—
131C2×C4×C83—81no—
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2411yescharacteristic
137C22⋊D83—42no—
138C22⋊D83—81no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141D332—21no—
142D332—21no—
143D332—21no—
144D332—42no—
145D332#4112yes—
146C4×D83—42no—
147C4×D83—81no—
148Dic166—21no—
149Dic166—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83#13712yes—
160C22×C83C22×D411yescharacteristic
161C2×D83—22no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—41no—
167C2×D83—84no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83C2×#4111yescharacteristic
177C4×C83H(5,𝔽2)11yescharacteristic
178Dic83—41no—
179Dic83—41no—
180C2×C83—22no—
181C2×C83—41no—
182C2×C83#54611yescharacteristic
183C2×C83C2×#13711yescharacteristic
184C2×C83#57611yescharacteristic
185D83—42no—
186D83—81no—
187#3314—831nomaximal, Sylow
188C83#331411yescharacteristic, complemented, Sylow
189#554D8311yescharacteristic, complemented
190#546—831no—
191#489—831no—
192#484—831no—
193#480—831no—
194#474—831no—
195#468—831no—
196#467—831no—
197#464—831no—
198#452—831no—
199#450—831no—
200#440—831no—
201#435—831no—
202#417—831no—
203C2×#102—831no—
204C2×#41—831no—
205C22×D4—831no—
206C2×Q16—831no—
207C2×SD16—831no—
208C2×SD16—831no—
209C2×M16—831no—
210C22×C8—831no—
211C4⋊Q8C2×D8311yescharacteristic, complemented
212#125—21no—
213#124—831no—
214#123—1661no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—831no—
218C22⋊Q8C2×D8311yescharacteristic
219C22⋊Q8C2×D8311yescharacteristic, complemented
220#121—831no—
221#121—1661no—
222#120—1661no—
223C4×Q8C2×D8311yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—831no—
226C4×D4C2×D8311yescharacteristic
227C2×C4⋊C4C2×D8311yescharacteristic
228C2×C4⋊C4C2×D8311yescharacteristic, complemented
229C2×C22⋊C4—1661no—
230#107—831no—
231#107—831no—
232#107—831no—
233C4⋊C8—831no—
234#103—831no—
235#103—831no—
236#103—831no—
237#103—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244C22⋊C8—831no—
245C22⋊C8—831no—
246#97—831no—
247C24—1661no—
248#41—3321no—
249#41—1662no—
250C2×Q8—21no—
251C2×Q8—831no—
252C2×Q8C22×D8311yescharacteristic
253C2×D4—3321no—
254C2×D4—831no—
255C2×D4—831no—
256C2×D4—1662no—
257C2×D4—1661no—
258C2×D4—1661no—
259C2×D4—1661no—
260C2×D4C22⋊D8311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4C22×D8311yescharacteristic
266C22×C4C22×D8311yescharacteristic
267Q16—3321no—
268SD16—3321no—
269SD16—1662no—
270M16—3321no—
271C2×C8—3321no—
272C2×C8—831no—
273C2×C8—831no—
274C2×C8—831no—
275C2×C8—831no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—831no—
283C4⋊C4—1661no—
284C4⋊C4C22⋊D8311yescharacteristic
285C4⋊C4C22×D8311yescharacteristic
286C4⋊C4C22×D8311yescharacteristic
287C4⋊C4C22×D8311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4C22⋊D8311yescharacteristic
296C22⋊C4C22⋊D8311yescharacteristic
297C42—21no—
298C42C22×D8311yescharacteristic
299C23—3322no—
300C23—3322no—
301C23—6641no—
302C23—1661no—
303C23—1661no—
304C23C2×C22⋊D8311yescharacteristic
305C23C2×C22⋊D8311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—1661no—
310Q8—1661no—
311D4—22no—
312D4—3322no—
313D4—3322no—
314D4—832no—
315D4—6641no—
316D4—1661no—
317D4—1661no—
318D4—1661no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—3322no—
330C2×C4—6641no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4C2×C22⋊D8311yescharacteristic
334C2×C4C2×C22⋊D8311yescharacteristic
335C2×C4C2×C22⋊D8311yescharacteristic
336C2×C4C2×C22⋊D8311yescharacteristic
337C2×C4C23×D8311yescharacteristic
338C8—1661no—
339C8—1661no—
340C8—1661no—
341C8—1661no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—3321no—
348C22—3321no—
349C22—3321no—
350C22—3321no—
351C22—3321no—
352C22—6644no—
353C22—1662no—
354C22#1458182312yes—
355C22C22×C22⋊D8311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—3321no—
365C4—3321no—
366C4#41×D8311yescharacteristic
367C4#1458183811yescharacteristic
368C2—22no—
369C2—41no—
370C2—3322no—
371C2—6641no—
372C2#1912833911yescharacteristic
373C2C2×#1458182311yescharacteristic
374C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#554×C83C2
socleC22×C83C22×D4
prime radical#19128347C2
hypercenter#554D83
hypocenterC83#3314
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #19128063C2semidirect
3. #19128036C2semidirect
4. #19128014C2semidirect
5. #19127978C2semidirect
6. #19127949C2semidirect
7. #19127940C2semidirect
8. #19127923C2semidirect
9. #19127861C2semidirect
10. #19127851C2semidirect
11. #19127799C2semidirect
12. #19127773C2semidirect
13. #19127700C2semidirect
14. C2×#14581712C2semidirect
15. #554×C83C2semidirect
18. C2×#13050580C22semidirect
21. C2×#13050574C22semidirect
31. #14581725C22semidirect
32. #14581725C22semidirect
33. #14581725C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581716C22semidirect
36. #14581715C22semidirect
37. #14581715C22semidirect
38. #14581715C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
48. C4⋊Q8×C83C22semidirect
53. C22⋊Q8×C83C22semidirect
54. C4×Q8×C83C22semidirect
58. C2×C4⋊C4×C83C22semidirect
188. C83#3314semidirect
189. #554D83semidirect
211. C4⋊Q8C2×D83semidirect
219. C22⋊Q8C2×D83semidirect
223. C4×Q8C2×D83semidirect
228. C2×C4⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283472
#191280632
#191280362
#191280142
#191279782
#191279492
#191279402
#191279232
#191278612
#191278512
#191277992
#191277732
#191277002
C2×#145817122
#554×C832
#331483