Finite Group Databasebeta

← #24919483

Subgroups — #24919483

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
2#19128336C211yesmaximal, characteristic
3#19128023C211yesmaximal, characteristic, complemented
4#19128013C211yesmaximal, characteristic, complemented
5#19127998C211yesmaximal, characteristic, complemented
6#19127946C211yesmaximal, characteristic, complemented
7#19127929C211yesmaximal, characteristic, complemented
8#19127898C211yesmaximal, characteristic, complemented
9#19127853C211yesmaximal, characteristic, complemented
10#19127853C211yesmaximal, characteristic, complemented
11#19127803C211yesmaximal, characteristic, complemented
12#19127774C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#490×C83C211yesmaximal, characteristic, complemented
16C2×D4×D83—21no—
17C2×D4×D83C2211yescharacteristic
18C2×D4×D83C2211yescharacteristic
19C2×#13050579C2211yescharacteristic
20C2×#13050578C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic, complemented
22C2×#13050573C2211yescharacteristic, complemented
23C2×C8×D83C2211yescharacteristic, complemented
24#14581787—21no—
25#14581783—21no—
26#14581777—21no—
27#14581772C2211yescharacteristic, complemented
28#14581772C2211yescharacteristic, complemented
29#14581763—21no—
30#14581763C2211yescharacteristic
31#14581763C2211yescharacteristic
32#14581760—21no—
33#14581760—21no—
34C4×Dic166C2211yescharacteristic, complemented
35C4×D332C2211yescharacteristic
36C4×D332C2211yescharacteristic
37#14581750—21no—
38#14581750C2211yescharacteristic, complemented
39C4⋊C4×D83C2211yescharacteristic, complemented
40#14581726C2211yescharacteristic, complemented
41#14581725C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43C4⋊(C83⋊C8)C2211yescharacteristic, complemented
44#14581717C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic, complemented
47#14581713C2211yescharacteristic, complemented
48#14581712C2211yescharacteristic
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51(C2×D83)⋊C8C2211yescharacteristic, complemented
52#127×C83C2211yescharacteristic
53#125×C83C2211yescharacteristic, complemented
54#102×C83C2211yescharacteristic, complemented
55#102×C83C2211yescharacteristic, complemented
56#102×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic, complemented
58#97×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C23×D83—21no—
61C2×Dic166C2311yescharacteristic
62C2×D332D411yescharacteristic
63C2×D332C2311yescharacteristic
64C2×C22⋊D83—21no—
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68D4×D83—22no—
69D4×D83—22no—
70D4×D83—41no—
71D4×D83—41no—
72D4×D83—44no—
73C2×C4×D83—21no—
74C2×C4×D83—21no—
75C2×C4×D83C2311yescharacteristic
76C2×C4×D83C2311yescharacteristic
77#13050579—22no—
78#13050578—41no—
79#13050578—41no—
80#13050573—41no—
81C2×C83⋊C8C2311yescharacteristic
82C2×C83⋊C8C2311yescharacteristic
83C8×D83—41no—
84C4⋊Dic83—21no—
85C4⋊Dic83—21no—
86C4⋊Dic83D411yescharacteristic
87C4⋊Dic83C2311yescharacteristic
88C4⋊Dic83C2311yescharacteristic
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92C22⋊Dic83—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×C83—21no—
102C2×D4×C83—21no—
103C2×D4×C83C2311yescharacteristic
104C2×D4×C83C2311yescharacteristic
105C2×C8×C83C2311yescharacteristic
106C2×C8×C83C2311yescharacteristic
107C4⋊C4×C83—21no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C4⋊C4×C83C2311yescharacteristic
111C42×C83C2311yescharacteristic
112C23×C83—21no—
113C23×C83—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—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—42no—
128D4×C83—81no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2411yescharacteristic
134C22⋊D83—42no—
135C22⋊D83—42no—
136C22⋊D83—42no—
137C22⋊D83—42no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83C2×D411yescharacteristic
142C2×Dic83C2×D411yescharacteristic
143D332—22no—
144D332—21no—
145D332#4112yes—
146C4×D83—21no—
147C4×D83—21no—
148C4×D83—21no—
149C4×D83—21no—
150C4×D83—42no—
151C4×D83—42no—
152Dic166—21no—
153Dic166—21no—
154C8×C83—21no—
155C8×C83—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83C22×D411yescharacteristic
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—21no—
168C2×D83—22no—
169C2×D83—21no—
170C2×D83—21no—
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×C83C2×#4111yescharacteristic
181C4×C83H(5,𝔽2)11yescharacteristic
182Dic83—21no—
183Dic83—21no—
184Dic83—41no—
185Dic83—41no—
186Dic83—41no—
187C2×C83—81no—
188C2×C83—81no—
189C2×C83#54611yescharacteristic
190C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C83#57711yescharacteristic
192D83—22no—
193D83—41no—
194D83—42no—
195#3310—831nomaximal, Sylow
196C83#331011yescharacteristic, complemented, Sylow
197#552—831no—
198D42—831no—
199#490D8311yescharacteristic, complemented
200#482—831no—
201#480—831no—
202#478—831no—
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#127C2×D8311yescharacteristic
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×D4—831no—
235#117—831no—
236#117—1661no—
237C2×C4⋊C4—831no—
238#107—831no—
239#107—831no—
240#106—831no—
241C4⋊C8—831no—
242#103—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248#102C2×D8311yescharacteristic, complemented
249#102C2×D8311yescharacteristic, complemented
250#102C2×D8311yescharacteristic, complemented
251#102C2×D8311yescharacteristic, complemented
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97C2×D8311yescharacteristic, complemented
255C24—1661no—
256C24—1661no—
257C2×Q8—831no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—3324no—
261C2×D4—3321no—
262C2×D4—3321no—
263C2×D4—831no—
264C2×D4—831no—
265C2×D4—1661no—
266C2×D4—1661no—
267C2×D4—1662no—
268C2×D4—1662no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4C22×D8311yescharacteristic
272C2×D4C22×D8311yescharacteristic
273C22×C4—831no—
274C22×C4—831no—
275C22×C4—1661no—
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—831no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C4⋊C4—1661no—
295C4⋊C4C22×D8311yescharacteristic
296C4⋊C4C22×D8311yescharacteristic
297C22⋊C4—831no—
298C22⋊C4—831no—
299C22⋊C4—1661no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C42—1661no—
306C42—1661no—
307C42C22×D8311yescharacteristic
308C23—21no—
309C23—21no—
310C23—3322no—
311C23—3322no—
312C23—3322no—
313C23—3322no—
314C23—831no—
315C23—831no—
316C23—6641no—
317C23—6641no—
318C23—1661no—
319Q8—1661no—
320Q8—1661no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—3322no—
328D4—3322no—
329D4—3322no—
330D4—3322no—
331D4—832no—
332D4—1662no—
333D4—1661no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—3322no—
337C2×C4—3322no—
338C2×C4—831no—
339C2×C4—831no—
340C2×C4—1661no—
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—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3321no—
361C22—3321no—
362C22—832no—
363C22—6644no—
364C22—6644no—
365C22—1661no—
366C22—1662no—
367C22—1661no—
368C22—1661no—
369C22—1661no—
370C22—1661no—
371C22C2×D4×D8311yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—3321no—
377C4—3321no—
378C4—3321no—
379C4—1661no—
380C4—1661no—
381C4#41×D8311yescharacteristic
382C4#1458183811yescharacteristic
383C2—81no—
384C2—81no—
385C2—3322no—
386C2—3321no—
387C2—1662no—
388C2#1912834411yescharacteristic
389C2AGL(1,ℤ/8)×D8311yescharacteristic
390C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#490×C83C2
socleC22×C83C22×D4
prime radical#19128336C2
hypercenter#490D83
hypocenterC83#3310
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #19128389C2semidirect
3. #19128023C2semidirect
4. #19128013C2semidirect
5. #19127998C2semidirect
6. #19127946C2semidirect
7. #19127929C2semidirect
8. #19127898C2semidirect
9. #19127853C2semidirect
10. #19127853C2semidirect
11. #19127803C2semidirect
12. #19127774C2semidirect
13. #19127704C2semidirect
14. #102×D83C2semidirect
15. #490×C83C2semidirect
20. C2×#13050578C22semidirect
21. C2×#13050578C22semidirect
22. C2×#13050573C22semidirect
23. C2×C8×D83C22semidirect
27. #14581772C22semidirect
28. #14581772C22semidirect
34. C4×Dic166C22semidirect
38. #14581750C22semidirect
39. C4⋊C4×D83C22semidirect
40. #14581726C22semidirect
41. #14581725C22semidirect
42. #14581723C22semidirect
43. C4⋊(C83⋊C8)C22semidirect
44. #14581717C22semidirect
46. #14581713C22semidirect
47. #14581713C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. (C2×D83)⋊C8C22semidirect
53. #125×C83C22semidirect
54. #102×C83C22semidirect
55. #102×C83C22semidirect
56. #102×C83C22semidirect
57. #102×C83C22semidirect
58. #97×C83C22semidirect
196. C83#3310semidirect
199. #490D83semidirect
223. #125C2×D83semidirect
248. #102C2×D83semidirect
249. #102C2×D83semidirect
250. #102C2×D83semidirect
251. #102C2×D83semidirect
254. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283892
#191283362
#191280232
#191280132
#191279982
#191279462
#191279292
#191278982
#191278532
#191278032
#191277742
#191277042
#102×D832
#490×C832
#331083