Finite Group Databasebeta

← #24919481

Subgroups — #24919481

Order 10624.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128389C211yesmaximal, characteristic, complemented
2D4×D332C211yesmaximal, characteristic
3#19128067C211yesmaximal, characteristic, complemented
4#19128027C211yesmaximal, characteristic, complemented
5#19128016C211yesmaximal, characteristic, complemented
6#19127948C211yesmaximal, characteristic, complemented
7#19127931C211yesmaximal, characteristic, complemented
8#19127902C211yesmaximal, characteristic, complemented
9#19127853C211yesmaximal, characteristic, complemented
10#19127853C211yesmaximal, characteristic, complemented
11#19127801C211yesmaximal, characteristic, complemented
12#19127771C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#478×C83C211yesmaximal, characteristic, complemented
16C22×D332—21no—
17C2×D4×D83C2211yescharacteristic
18C2×D4×D83C2211yescharacteristic
19C2×#13050579C2211yescharacteristic, complemented
20C2×#13050579C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic
22C2×#13050573C2211yescharacteristic, complemented
23C2×C8×D83C2211yescharacteristic, complemented
24#14581789C2211yescharacteristic
25#14581787—21no—
26#14581783—21no—
27#14581777—21no—
28#14581772C2211yescharacteristic, complemented
29#14581772C2211yescharacteristic, complemented
30#14581767C2211yescharacteristic
31#14581767C2211yescharacteristic
32#14581766—21no—
33#14581762—21no—
34#14581762—21no—
35C4×Dic166C2211yescharacteristic, complemented
36C4×D332C2211yescharacteristic
37#14581750—21no—
38#14581750C2211yescharacteristic, complemented
39C4⋊C4×D83C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581715C2211yescharacteristic, complemented
42#14581714C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic, complemented
44#14581713C2211yescharacteristic
45#14581712C2211yescharacteristic, complemented
46#14581712C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic, complemented
48#14581712C2211yescharacteristic, complemented
49(C2×D83)⋊C8C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51#14581703C2211yescharacteristic, complemented
52#125×C83C2211yescharacteristic, complemented
53C4×D4×C83C2211yescharacteristic
54#107×C83C2211yescharacteristic, complemented
55#106×C83C2211yescharacteristic, complemented
56C4⋊C8×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60C23×D83—21no—
61C2×Dic166C2311yescharacteristic
62C2×D332—21no—
63C2×D332—21no—
64C2×D332—21no—
65C2×D332—21no—
66C2×D332—44no—
67C2×D332D411yescharacteristic
68C2×D332C2311yescharacteristic
69C2×D332C2311yescharacteristic
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72D4×D83—22no—
73D4×D83—22no—
74D4×D83—41no—
75D4×D83—41no—
76C2×C4×D83—21no—
77C2×C4×D83C2311yescharacteristic
78C2×C4×D83C2311yescharacteristic
79#13050579—41no—
80#13050579—41no—
81#13050578—22no—
82#13050573—41no—
83C2×C83⋊C8C2311yescharacteristic
84C2×C83⋊C8C2311yescharacteristic
85C8×D83—41no—
86C4⋊Dic83—21no—
87C4⋊Dic83—21no—
88C4⋊Dic83D411yescharacteristic
89C4⋊Dic83C2311yescharacteristic
90Dic83⋊C4—21no—
91Dic83⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4D411yescharacteristic
98(C2×D83)⋊C4D411yescharacteristic
99C4×Dic83—21no—
100C4×Dic83—21no—
101C2×D4×C83C2311yescharacteristic
102C22×C4×C83—21no—
103C2×C8×C83C2311yescharacteristic
104C2×C8×C83C2311yescharacteristic
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83C2311yescharacteristic
108C4⋊C4×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C42×C83C2311yescharacteristic
112C23×C83—21no—
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—42no—
118C22×D83—42no—
119C22×D83—81no—
120C22×D83—81no—
121C22×D83C2×D411yescharacteristic
122C22×D83C2×D411yescharacteristic
123D4×C83—21no—
124D4×C83#4112yes—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—42no—
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2411yescharacteristic
132C22⋊D83—42no—
133C22⋊D83—42no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83C2×D411yescharacteristic
137C2×Dic83C2×D411yescharacteristic
138D332—22no—
139D332—21no—
140D332—21no—
141D332—21no—
142D332—21no—
143D332—42no—
144D332—42no—
145D332—42no—
146D332—81no—
147C4×D83—21no—
148C4×D83—21no—
149C4×D83—21no—
150C4×D83—21no—
151C4×D83—42no—
152Dic166—21no—
153Dic166—21no—
154C8×C83—21no—
155C8×C83—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C22×C83—22no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83C22×D411yescharacteristic
162C2×D83—21no—
163C2×D83—21no—
164C2×D83—21no—
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—84no—
174C2×D83—84no—
175C2×D83AGL(1,ℤ/8)12yes—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83C2×#4111yescharacteristic
182C4×C83H(5,𝔽2)11yescharacteristic
183Dic83—21no—
184Dic83—21no—
185Dic83—41no—
186Dic83—41no—
187C2×C83—42no—
188C2×C83#54611yescharacteristic
189C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C83#57711yescharacteristic
191D83—22no—
192D83—41no—
193D83—81no—
194D83—81no—
195#3310—831nomaximal, Sylow
196C83#331011yescharacteristic, complemented, Sylow
197#552—831no—
198D42—831no—
199#490—831no—
200#482—831no—
201#480—831no—
202#478D8311yescharacteristic, complemented
203#468—831no—
204#465—831no—
205#460—831no—
206#450—831no—
207#450—831no—
208#440—831no—
209#435—831no—
210#418—831no—
211C2×#102—831no—
212C22×D4—831no—
213C22×D4—831no—
214C22×D4—1661no—
215C2×SD16—831no—
216C2×SD16—831no—
217C2×SD16—831no—
218C2×M16—831no—
219C22×C8—831no—
220#127—831no—
221#126—1661no—
222#125—1661no—
223#125C2×D8311yescharacteristic, complemented
224#123—1661no—
225C22⋊Q8—831no—
226C22⋊Q8—831no—
227#121—831no—
228#121—831no—
229#121—1661no—
230#120—1661no—
231#120—1661no—
232C4×Q8—831no—
233C4×D4—831no—
234C4×D4C2×D8311yescharacteristic
235#117—831no—
236#117—1661no—
237C2×C4⋊C4—831no—
238#107—831no—
239#107C2×D8311yescharacteristic, complemented
240#106C2×D8311yescharacteristic, complemented
241C4⋊C8C2×D8311yescharacteristic, complemented
242#103—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102C2×D8311yescharacteristic
251#102C2×D8311yescharacteristic
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97—831no—
255C24—1661no—
256C24—1661no—
257C2×Q8—831no—
258C2×D4—3324no—
259C2×D4—3321no—
260C2×D4—3321no—
261C2×D4—831no—
262C2×D4—831no—
263C2×D4—831no—
264C2×D4—1661no—
265C2×D4—1661no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4—1662no—
269C2×D4—1662no—
270C2×D4—1661no—
271C2×D4—1661no—
272C2×D4C22×D8311yescharacteristic
273C22×C4—21no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4—1661no—
277SD16—3321no—
278SD16—3321no—
279SD16—1662no—
280M16—3321no—
281C2×C8—3321no—
282C2×C8—831no—
283C2×C8—831no—
284C2×C8C22×D8311yescharacteristic
285C2×C8C22×D8311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—831no—
289C4⋊C4—831no—
290C4⋊C4—1661no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C4⋊C4C22×D8311yescharacteristic
295C4⋊C4C22×D8311yescharacteristic
296C4⋊C4C22×D8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—831no—
299C22⋊C4—831no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C42—1661no—
306C42—1661no—
307C42C22×D8311yescharacteristic
308C23—21no—
309C23—3322no—
310C23—3322no—
311C23—3322no—
312C23—3322no—
313C23—831no—
314C23—831no—
315C23—6641no—
316C23—6641no—
317C23—1661no—
318C23—1661no—
319Q8—1661no—
320Q8—1661no—
321D4—21no—
322D4—3322no—
323D4—3322no—
324D4—3322no—
325D4—3322no—
326D4—3322no—
327D4—6641no—
328D4—1662no—
329D4—1661no—
330D4—1661no—
331D4—1661no—
332D4—1661no—
333D4#1305059112yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—3322no—
339C2×C4—831no—
340C2×C4—831no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4D4×D8311yescharacteristic
348C2×C4D4×D8311yescharacteristic
349C2×C4C23×D8311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1661no—
353C8—1661no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—3321no—
358C22—3321no—
359C22—3321no—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22—832no—
364C22—6644no—
365C22—6644no—
366C22—1661no—
367C22—1661no—
368C22—1661no—
369C22—1661no—
370C22—1661no—
371C22C2×D4×D8311yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3321no—
378C4—3321no—
379C4—1661no—
380C4—1661no—
381C4C2×#1305059111yescharacteristic
382C4#1458183811yescharacteristic
383C2—42no—
384C2—3321no—
385C2—6641no—
386C2—6641no—
387C2—1662no—
388C2#1912834211yescharacteristic
389C2AGL(1,ℤ/8)×D8311yescharacteristic
390C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#478×C83C2
socleC22×C83C22×D4
prime radicalD4×D332C2
hypercenter#478D83
hypocenterC83#3310
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #19128389C2semidirect
3. #19128067C2semidirect
4. #19128027C2semidirect
5. #19128016C2semidirect
6. #19127948C2semidirect
7. #19127931C2semidirect
8. #19127902C2semidirect
9. #19127853C2semidirect
10. #19127853C2semidirect
11. #19127801C2semidirect
12. #19127771C2semidirect
13. #19127704C2semidirect
14. #102×D83C2semidirect
15. #478×C83C2semidirect
19. C2×#13050579C22semidirect
20. C2×#13050579C22semidirect
22. C2×#13050573C22semidirect
23. C2×C8×D83C22semidirect
28. #14581772C22semidirect
29. #14581772C22semidirect
35. C4×Dic166C22semidirect
38. #14581750C22semidirect
39. C4⋊C4×D83C22semidirect
40. #14581725C22semidirect
41. #14581715C22semidirect
42. #14581714C22semidirect
43. #14581714C22semidirect
45. #14581712C22semidirect
46. #14581712C22semidirect
47. #14581712C22semidirect
48. #14581712C22semidirect
49. (C2×D83)⋊C8C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. #14581703C22semidirect
52. #125×C83C22semidirect
54. #107×C83C22semidirect
55. #106×C83C22semidirect
56. C4⋊C8×C83C22semidirect
196. C83#3310semidirect
202. #478D83semidirect
223. #125C2×D83semidirect
239. #107C2×D83semidirect
240. #106C2×D83semidirect
241. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283892
D4×D3322
#191280672
#191280272
#191280162
#191279482
#191279312
#191279022
#191278532
#191278012
#191277712
#191277042
#102×D832
#478×C832
#331083