Finite Group Databasebeta

← #24919676

Subgroups — #24919676

Order 10624.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128389C211yesmaximal, characteristic, complemented
2#19128346C211yesmaximal, characteristic
3#19128058C211yesmaximal, characteristic, complemented
4#19128044C211yesmaximal, characteristic, complemented
5#19128021C211yesmaximal, characteristic, complemented
6#19127960C211yesmaximal, characteristic, complemented
7#19127948C211yesmaximal, characteristic, complemented
8#19127924C211yesmaximal, characteristic, complemented
9#19127864C211yesmaximal, characteristic, complemented
10#19127853C211yesmaximal, characteristic, complemented
11C4×#13050578C211yesmaximal, characteristic, complemented
12#19127778C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14#19127699C211yesmaximal, characteristic, complemented
15#478×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×Dic332C2211yescharacteristic, complemented
19C2×#13050579C2211yescharacteristic
20C2×#13050578C2211yescharacteristic
21C2×#13050573C2211yescharacteristic
22C2×#13050573C2211yescharacteristic, complemented
23#14581787—21no—
24#14581783—21no—
25#14581780C2211yescharacteristic
26#14581777—21no—
27#14581776—21no—
28#14581772C2211yescharacteristic
29#14581772C2211yescharacteristic
30#14581772C2211yescharacteristic, complemented
31#14581767C2211yescharacteristic
32#14581766—21no—
33#14581762—21no—
34C4×Dic166C2211yescharacteristic, complemented
35C4×D332C2211yescharacteristic
36#14581750—21no—
37#14581750C2211yescharacteristic
38C4⋊C4×D83C2211yescharacteristic, complemented
39C2×(C2×D83)⋊C4—21no—
40#14581723C2211yescharacteristic, complemented
41#14581717C2211yescharacteristic, complemented
42#14581715C2211yescharacteristic, complemented
43#14581715C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581712C2211yescharacteristic
48#14581712C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51C4×C83⋊C8C2211yescharacteristic, 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—
60#13050589—22no—
61#13050589—41no—
62C2×Dic166C2311yescharacteristic
63C2×Dic166C2311yescharacteristic
64C2×D332—21no—
65C2×D332D411yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69D4×D83—22no—
70D4×D83—41no—
71C22×Dic83—21no—
72C2×C4×D83—21no—
73C2×C4×D83C2311yescharacteristic
74C2×C4×D83C2311yescharacteristic
75Dic332—41no—
76#13050579—41no—
77#13050578—22no—
78#13050573—41no—
79#13050573—41no—
80C2×C83⋊C8C2311yescharacteristic
81C2×C83⋊C8C2311yescharacteristic
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83D411yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊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)⋊C4D411yescharacteristic
97(C2×D83)⋊C4D411yescharacteristic
98C4×Dic83—21no—
99C4×Dic83—21no—
100C2×D4×C83C2311yescharacteristic
101C22×C4×C83—21no—
102C2×C8×C83C2311yescharacteristic
103C2×C8×C83C2311yescharacteristic
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83C2311yescharacteristic
107C4⋊C4×C83C2311yescharacteristic
108C4⋊C4×C83C2311yescharacteristic
109C22⋊C4×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C22×D83—21no—
113C22×D83—42no—
114C22×D83—42no—
115C22×D83—81no—
116C22×D83C2×D411yescharacteristic
117C22×D83C2×D411yescharacteristic
118D4×C83—21no—
119D4×C83#4112yes—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—81no—
124C2×C4×C83C2×D411yescharacteristic
125C2×C4×C83C2×D411yescharacteristic
126C2×C4×C83C2411yescharacteristic
127C22⋊D83—42no—
128C22⋊D83—42no—
129C2×Dic83—21no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—42no—
133C2×Dic83C2×D411yescharacteristic
134C2×Dic83C2×D411yescharacteristic
135D332—21no—
136D332—21no—
137D332—41no—
138D332—81no—
139C4×D83—21no—
140C4×D83—21no—
141C4×D83—21no—
142C4×D83—21no—
143C4×D83—42no—
144Dic166—21no—
145Dic166—21no—
146Dic166—21no—
147Dic166—21no—
148C8×C83—21no—
149C8×C83—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—22no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83C22×D411yescharacteristic
156C2×D83—21no—
157C2×D83—21no—
158C2×D83—21no—
159C2×D83—21no—
160C2×D83—21no—
161C2×D83—21no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—84no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—41no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83C2×#4111yescharacteristic
172C4×C83H(5,𝔽2)11yescharacteristic
173Dic83—21no—
174Dic83—21no—
175Dic83—41no—
176Dic83—41no—
177Dic83—41no—
178C2×C83—42no—
179C2×C83#54611yescharacteristic
180C2×C83#57511yescharacteristic
181C2×C83#57711yescharacteristic
182D83—41no—
183D83—41no—
184D83—81no—
185#3323—831nomaximal, Sylow
186C83#332311yescharacteristic, complemented, Sylow
187#552—831no—
188#546—831no—
189#488—831no—
190#485—831no—
191#481—831no—
192#478D8311yescharacteristic, complemented
193#470—831no—
194#468—831no—
195#464—831no—
196#452—831no—
197#450—831no—
198C4×SD16—831no—
199#436—831no—
200#418—831no—
201#417—831no—
202C2×#41—831no—
203C22×D4—831no—
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16—831no—
207C2×M16—831no—
208C2×M16—831no—
209#126—1661no—
210#125—1661no—
211#125C2×D8311yescharacteristic, complemented
212#124—831no—
213#123—1661no—
214#123—1661no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8—831no—
218#121—831no—
219#121—1661no—
220#120—1661no—
221C4×Q8—831no—
222C4×D4—831no—
223C4×D4C2×D8311yescharacteristic
224#117—831no—
225#117—1661no—
226C2×C4⋊C4—831no—
227C2×C22⋊C4—1661no—
228#107C2×D8311yescharacteristic, complemented
229#106—831no—
230#106C2×D8311yescharacteristic, complemented
231C4⋊C8C2×D8311yescharacteristic, complemented
232#103—831no—
233#103—831no—
234#103—831no—
235#103—831no—
236#102—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102C2×D8311yescharacteristic
241#102C2×D8311yescharacteristic
242C22⋊C8—831no—
243C22⋊C8—831no—
244C4×C8—831no—
245C24—1661no—
246#41—3321no—
247#41—1662no—
248C2×Q8—831no—
249C2×Q8—831no—
250C2×D4—3321no—
251C2×D4—831no—
252C2×D4—831no—
253C2×D4—1661no—
254C2×D4—1661no—
255C2×D4—1661no—
256C2×D4—1662no—
257C2×D4C22×D8311yescharacteristic
258C22×C4—21no—
259C22×C4—831no—
260C22×C4—831no—
261C22×C4—1661no—
262C22×C4—1661no—
263Q16—3321no—
264SD16—3321no—
265SD16—1662no—
266M16—3321no—
267M16—3321no—
268C2×C8—831no—
269C2×C8—831no—
270C2×C8C22×D8311yescharacteristic
271C2×C8C22×D8311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—831no—
275C4⋊C4—831no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4—1661no—
281C4⋊C4C22×D8311yescharacteristic
282C4⋊C4C22×D8311yescharacteristic
283C4⋊C4C22×D8311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—831no—
286C22⋊C4—831no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C42—1661no—
295C42—1661no—
296C42C22×D8311yescharacteristic
297C23—21no—
298C23—3322no—
299C23—3322no—
300C23—831no—
301C23—831no—
302C23—6641no—
303C23—1661no—
304Q8—1661no—
305Q8—1661no—
306Q8—1661no—
307Q8—1661no—
308D4—21no—
309D4—3321no—
310D4—3322no—
311D4—3322no—
312D4—6641no—
313D4—1661no—
314D4—1661no—
315D4#1305059112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—3322no—
321C2×C4—3322no—
322C2×C4—831no—
323C2×C4—831no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4D4×D8311yescharacteristic
332C2×C4D4×D8311yescharacteristic
333C2×C4C23×D8311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1661no—
337C8—1661no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—3321no—
342C22—3321no—
343C22—3321no—
344C22—6644no—
345C22—1661no—
346C22—1661no—
347C22—1661no—
348C22—1661no—
349C22—1661no—
350C22—1661no—
351C22C2×D4×D8311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3321no—
358C4—3321no—
359C4—3321no—
360C4—1661no—
361C4—1661no—
362C4C2×#1305059111yescharacteristic
363C4#1458183811yescharacteristic
364C2—42no—
365C2—3321no—
366C2—3321no—
367C2—6641no—
368C2#1912834211yescharacteristic
369C2#1912849811yescharacteristic
370C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#478×C83C2
socleC22×C83C22×D4
prime radical#19128346C2
hypercenter#478D83
hypocenterC83#3323
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. #19128058C2semidirect
4. #19128044C2semidirect
5. #19128021C2semidirect
6. #19127960C2semidirect
7. #19127948C2semidirect
8. #19127924C2semidirect
9. #19127864C2semidirect
10. #19127853C2semidirect
11. C4×#13050578C2semidirect
12. #19127778C2semidirect
13. #19127704C2semidirect
14. #19127699C2semidirect
15. #478×C83C2semidirect
18. C2×Dic332C22semidirect
22. C2×#13050573C22semidirect
30. #14581772C22semidirect
34. C4×Dic166C22semidirect
38. C4⋊C4×D83C22semidirect
40. #14581723C22semidirect
41. #14581717C22semidirect
42. #14581715C22semidirect
43. #14581715C22semidirect
44. #14581715C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. C4×C83⋊C8C22semidirect
52. #125×C83C22semidirect
54. #107×C83C22semidirect
55. #106×C83C22semidirect
56. C4⋊C8×C83C22semidirect
186. C83#3323semidirect
192. #478D83semidirect
211. #125C2×D83semidirect
228. #107C2×D83semidirect
230. #106C2×D83semidirect
231. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283892
#191283462
#191280582
#191280442
#191280212
#191279602
#191279482
#191279242
#191278642
#191278532
C4×#130505782
#191277782
#191277042
#191276992
#478×C832
#332383