Finite Group Databasebeta

← #24917816

Subgroups — #24917816

Order 10624.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128331C211yesmaximal, characteristic, complemented
2#19128269C211yesmaximal, characteristic
3#19128067C211yesmaximal, characteristic, complemented
4#19128064C211yesmaximal, characteristic, complemented
5#19128062C211yesmaximal, characteristic, complemented
6#19128060C211yesmaximal, characteristic, complemented
7#19128057C211yesmaximal, characteristic, complemented
8#19128048C211yesmaximal, characteristic, complemented
9#19128046C211yesmaximal, characteristic, complemented
10#19127704C211yesmaximal, characteristic, complemented
11#19127701C211yesmaximal, characteristic, complemented
12#19127696C211yesmaximal, characteristic, complemented
13#102×D83C211yesmaximal, characteristic, complemented
14#19127655C211yesmaximal, characteristic, complemented
15#488×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050573C2211yescharacteristic
19C2×C8×D83C2211yescharacteristic
20#14581792C2211yescharacteristic, complemented
21#14581791—21no—
22#14581791C2211yescharacteristic, complemented
23#14581789C2211yescharacteristic
24#14581788C2211yescharacteristic
25#14581783C2211yescharacteristic, complemented
26#14581782—21no—
27#14581781C2211yescharacteristic
28#14581772—21no—
29#14581770—21no—
30#14581765—21no—
31#14581765—21no—
32#14581762—21no—
33#14581762—21no—
34#14581750C2211yescharacteristic
35#14581749C2211yescharacteristic
36C4⋊C4×D83C2211yescharacteristic
37#14581716C2211yescharacteristic, complemented
38#14581716C2211yescharacteristic, complemented
39#14581714C2211yescharacteristic
40#14581714C2211yescharacteristic
41#14581714C2211yescharacteristic, complemented
42#14581714C2211yescharacteristic, complemented
43#14581713C2211yescharacteristic
44#14581713C2211yescharacteristic
45#14581712C2211yescharacteristic
46#14581712C2211yescharacteristic
47#14581712C2211yescharacteristic, complemented
48#14581712C2211yescharacteristic, complemented
49#14581703C2211yescharacteristic, complemented
50#14581702C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52#125×C83C2211yescharacteristic, complemented
53#124×C83C2211yescharacteristic
54#103×C83C2211yescharacteristic, complemented
55#103×C83C2211yescharacteristic, complemented
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic
58C4×C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050591—22no—
61#13050591—22no—
62C2×Dic166—21no—
63C2×Dic166—21no—
64C2×D332—21no—
65C2×D332—21no—
66C2×D332—21no—
67C2×D332C2311yescharacteristic
68C2×D332C2311yescharacteristic
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72D4×D83—22no—
73D4×D83—22no—
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76C2×C4×D83C2311yescharacteristic
77#13050573—22no—
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C8×D83—22no—
81C4⋊Dic83—21no—
82C4⋊Dic83—21no—
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4D411yescharacteristic
89Dic83⋊C4D411yescharacteristic
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4D411yescharacteristic
94(C2×D83)⋊C4D411yescharacteristic
95C4×Dic83—21no—
96C4×Dic83C2311yescharacteristic
97C2×Q8×C83C2311yescharacteristic
98C2×D4×C83C2311yescharacteristic
99C2×C8×C83C2311yescharacteristic
100C2×C8×C83C2311yescharacteristic
101C4⋊C4×C83—21no—
102C4⋊C4×C83—21no—
103C4⋊C4×C83C2311yescharacteristic
104C4⋊C4×C83C2311yescharacteristic
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C42×C83C2311yescharacteristic
108C23×C83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—42no—
112C22×D83—81no—
113C22×D83—81no—
114C22×D83C2×D411yescharacteristic
115Q8×C83—21no—
116Q8×C83—21no—
117D4×C83—21no—
118D4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83C2×D411yescharacteristic
123C2×C4×C83C2×D411yescharacteristic
124C2×C4×C83C2411yescharacteristic
125C22⋊D83—42no—
126C22⋊D83—81no—
127C22⋊D83—81no—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83C2×D411yescharacteristic
131C2×Dic83C2×D411yescharacteristic
132C2×Dic83C2×D411yescharacteristic
133D332—21no—
134D332—21no—
135D332—21no—
136D332—21no—
137D332—42no—
138D332—81no—
139D332—81no—
140C4×D83—42no—
141C4×D83—81no—
142C4×D83#4112yes—
143C4×D83#4112yes—
144Dic166—81no—
145Dic166—81no—
146C8×C83—21no—
147C8×C83—21no—
148C83⋊C8—21no—
149C83⋊C8—21no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83C22×D411yescharacteristic
154C2×D83—21no—
155C2×D83—21no—
156C2×D83—41no—
157C2×D83—41no—
158C2×D83—41no—
159C2×D83—41no—
160C2×D83—41no—
161C2×D83—41no—
162C2×D83—84no—
163C2×D83#12412yes—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—41no—
169C4×C83C2×#4111yescharacteristic
170C4×C83C2×#4111yescharacteristic
171Dic83—21no—
172Dic83—21no—
173Dic83—41no—
174Dic83—41no—
175Dic83#12412yes—
176C2×C83—81no—
177C2×C83C2×#12411yescharacteristic
178C2×C83#57611yescharacteristic
179C2×C83#57711yescharacteristic
180D83—22no—
181D83—81no—
182D83—81no—
183#3136—831nomaximal, Sylow
184C83#313611yescharacteristic, complemented, Sylow
185#544—831no—
186#535—831no—
187#490—831no—
188#490—831no—
189#489—831no—
190#489—831no—
191#488—831no—
192#488D8311yescharacteristic, complemented
193#486—831no—
194#486—831no—
195#418—831no—
196#417—831no—
197#416—831no—
198C2×#102—831no—
199#405—831no—
200C2×#41—831no—
201C22×D4—831no—
202C2×M16—831no—
203C22×C8—831no—
204C4⋊Q8—831no—
205C4⋊Q8—831no—
206C4⋊Q8—1661no—
207#127—831no—
208#127—831no—
209#125—831no—
210#125—1661no—
211#125C2×D8311yescharacteristic, complemented
212#124—831no—
213#124C2×D8311yescharacteristic
214C22⋊Q8—1661no—
215C22⋊Q8—1661no—
216#121—1661no—
217#121—1661no—
218#120—1661no—
219#120—1661no—
220#117—831no—
221#117—831no—
222C2×C4⋊C4—831no—
223#103—831no—
224#103—831no—
225#103C2×D8311yescharacteristic, complemented
226#103C2×D8311yescharacteristic, complemented
227#102—831no—
228#102—831no—
229#102—831no—
230#102—831no—
231#102—831no—
232#102—831no—
233#102—831no—
234#102—831no—
235#102—831no—
236#102—831no—
237#102C2×D8311yescharacteristic
238#102C2×D8311yescharacteristic
239#97—831no—
240#97—831no—
241C4×C8—831no—
242C4×C8C2×D8311yescharacteristic, complemented
243C24—1661no—
244#41—1662no—
245#41—1662no—
246C2×Q8—1661no—
247C2×Q8—1661no—
248C2×Q8C22×D8311yescharacteristic
249C2×D4—831no—
250C2×D4—831no—
251C2×D4—1661no—
252C2×D4—1662no—
253C2×D4—1662no—
254C2×D4—1661no—
255C2×D4—1661no—
256C2×D4—1661no—
257C2×D4—1661no—
258C2×D4—1661no—
259C2×D4C22×D8311yescharacteristic
260C22×C4—831no—
261C22×C4—1661no—
262C22×C4—1661no—
263M16—1662no—
264C2×C8—831no—
265C2×C8—831no—
266C2×C8—1662no—
267C2×C8C22×D8311yescharacteristic
268C2×C8C22×D8311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—831no—
272C4⋊C4—831no—
273C4⋊C4—831no—
274C4⋊C4—831no—
275C4⋊C4—1661no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4C22×D8311yescharacteristic
281C4⋊C4C22×D8311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—831no—
285C22⋊C4—831no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C42—831no—
290C42—1661no—
291C42C22×D8311yescharacteristic
292C23—21no—
293C23—3322no—
294C23—831no—
295C23—6641no—
296C23—6641no—
297C23—1661no—
298C23—1661no—
299Q8—21no—
300Q8—21no—
301Q8—6641no—
302Q8—6641no—
303D4—21no—
304D4—21no—
305D4—3322no—
306D4—3322no—
307D4—6641no—
308D4—6641no—
309D4—6641no—
310D4—6641no—
311D4—1661no—
312D4—1661no—
313D4—1661no—
314D4—1661no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—3322no—
319C2×C4—831no—
320C2×C4—832no—
321C2×C4—832no—
322C2×C4—831no—
323C2×C4—831no—
324C2×C4—6641no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4D4×D8311yescharacteristic
328C2×C4D4×D8311yescharacteristic
329C2×C4C23×D8311yescharacteristic
330C8—21no—
331C8—21no—
332C8—1661no—
333C8—1661no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—3321no—
338C22—3321no—
339C22—3321no—
340C22—3321no—
341C22—3321no—
342C22—3321no—
343C22—832no—
344C22—6644no—
345C22—1661no—
346C22—1661no—
347C22C2×D4×D8311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3321no—
354C4—3321no—
355C4—832no—
356C4—1661no—
357C4—1661no—
358C4#41×D8311yescharacteristic
359C4#41×D8311yescharacteristic
360C2—81no—
361C2—6641no—
362C2—6641no—
363C2—1662no—
364C2#124×D8311yescharacteristic
365C2#1912850311yescharacteristic
366C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#488×C83C2
socleC22×C83C22×D4
prime radical#19128269C2
hypercenter#488D83
hypocenterC83#3136
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #19128331C2semidirect
3. #19128067C2semidirect
4. #19128064C2semidirect
5. #19128062C2semidirect
6. #19128060C2semidirect
7. #19128057C2semidirect
8. #19128048C2semidirect
9. #19128046C2semidirect
10. #19127704C2semidirect
11. #19127701C2semidirect
12. #19127696C2semidirect
13. #102×D83C2semidirect
14. #19127655C2semidirect
15. #488×C83C2semidirect
20. #14581792C22semidirect
22. #14581791C22semidirect
25. #14581783C22semidirect
37. #14581716C22semidirect
38. #14581716C22semidirect
41. #14581714C22semidirect
42. #14581714C22semidirect
47. #14581712C22semidirect
48. #14581712C22semidirect
49. #14581703C22semidirect
50. #14581702C22semidirect
51. C8×Dic83C22semidirect
52. #125×C83C22semidirect
54. #103×C83C22semidirect
55. #103×C83C22semidirect
58. C4×C8×C83C22semidirect
184. C83#3136semidirect
192. #488D83semidirect
211. #125C2×D83semidirect
225. #103C2×D83semidirect
226. #103C2×D83semidirect
242. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283312
#191282692
#191280672
#191280642
#191280622
#191280602
#191280572
#191280482
#191280462
#191277042
#191277012
#191276962
#102×D832
#191276552
#488×C832
#313683