Finite Group Databasebeta

← #24918532

Subgroups — #24918532

Order 10624.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128309C211yesmaximal, characteristic, complemented
2#19128269C211yesmaximal, characteristic
3#19128094C211yesmaximal, characteristic, complemented
4#19128072C211yesmaximal, characteristic, complemented
5#19128064C211yesmaximal, characteristic, complemented
6#19128057C211yesmaximal, characteristic, complemented
7#19127920C211yesmaximal, characteristic, complemented
8#19127902C211yesmaximal, characteristic, complemented
9#19127901C211yesmaximal, characteristic, complemented
10#19127866C211yesmaximal, characteristic, complemented
11#19127862C211yesmaximal, characteristic, complemented
12#19127853C211yesmaximal, characteristic, complemented
13#19127850C211yesmaximal, characteristic, complemented
14#19127761C211yesmaximal, characteristic, complemented
15#464×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050579C2211yescharacteristic
20C2×#13050579C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic
22C2×#13050577C2211yescharacteristic
23C2×#13050577C2211yescharacteristic, complemented
24#14581790C2211yescharacteristic, complemented
25#14581789C2211yescharacteristic
26#14581788C2211yescharacteristic
27#14581787—21no—
28#14581786—21no—
29#14581784—21no—
30#14581783C2211yescharacteristic, complemented
31#14581782—21no—
32#14581781C2211yescharacteristic
33#14581772C2211yescharacteristic
34#14581768C2211yescharacteristic
35#14581765—21no—
36#14581765—21no—
37#14581762—21no—
38#14581762—21no—
39C4×Dic166C2211yescharacteristic, complemented
40#14581749C2211yescharacteristic
41C4⋊(C83⋊C8)C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic
44#14581714C2211yescharacteristic, complemented
45#14581713C2211yescharacteristic
46#14581712C2211yescharacteristic
47#14581712C2211yescharacteristic, complemented
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50#14581702C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52C2×Q16×C83C2211yescharacteristic, complemented
53C2×SD16×C83C2211yescharacteristic
54#124×C83C2211yescharacteristic
55C4×Q8×C83C2211yescharacteristic, complemented
56C4⋊C8×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60#13050591—41no—
61#13050591—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—21no—
64C2×D332—21no—
65C2×D332—21no—
66C2×D332C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71D4×D83—41no—
72D4×D83—41no—
73C2×C4×D83—21no—
74C2×C4×D83C2311yescharacteristic
75#13050580—41no—
76#13050579—41no—
77#13050579—41no—
78#13050578—41no—
79#13050577—41no—
80#13050577—41no—
81C2×C83⋊C8C2311yescharacteristic
82C2×C83⋊C8C2311yescharacteristic
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83C2311yescharacteristic
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91Dic83⋊C4D411yescharacteristic
92Dic83⋊C4D411yescharacteristic
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×Dic83C2311yescharacteristic
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83C2311yescharacteristic
104Q16×C83—41no—
105SD16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C42×C83—21no—
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—42no—
118C22×D83—81no—
119C22×D83—81no—
120C22×D83C2×D411yescharacteristic
121Q8×C83—21no—
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C22⋊D83—42no—
134C22⋊D83—81no—
135C22⋊D83—81no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83C2×D411yescharacteristic
139C2×Dic83C2×D411yescharacteristic
140C2×Dic83C2×D411yescharacteristic
141D332—21no—
142D332—21no—
143D332—21no—
144D332—21no—
145D332—42no—
146D332—81no—
147D332—81no—
148C4×D83—21no—
149C4×D83—21no—
150C4×D83—42no—
151Dic166—21no—
152Dic166—21no—
153C8×C83—21no—
154C8×C83—21no—
155C83⋊C8—21no—
156C83⋊C8—21no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83C22×D411yescharacteristic
161C2×D83—21no—
162C2×D83—21no—
163C2×D83—21no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—41no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—84no—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83H(5,𝔽2)11yescharacteristic
177C4×C83H(5,𝔽2)11yescharacteristic
178Dic83—21no—
179Dic83—21no—
180Dic83—21no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—81no—
184C2×C83#53511yescharacteristic
185C2×C83#57711yescharacteristic
186C2×C83#57711yescharacteristic
187D83—41no—
188D83—81no—
189D83—81no—
190#3222—831nomaximal, Sylow
191C83#322211yescharacteristic, complemented, Sylow
192#541—831no—
193#535—831no—
194#497—831no—
195#492—831no—
196#490—831no—
197#488—831no—
198#464—831no—
199#464D8311yescharacteristic, complemented
200#460—831no—
201#460—831no—
202#452—831no—
203#452—831no—
204#450—831no—
205#450—831no—
206#433—831no—
207C2×#41—831no—
208C22×D4—831no—
209C2×Q16—831no—
210C2×Q16C2×D8311yescharacteristic, complemented
211C2×SD16—831no—
212C2×SD16—831no—
213C2×SD16—831no—
214C2×SD16—831no—
215C2×SD16—831no—
216C2×SD16C2×D8311yescharacteristic
217C4⋊Q8—831no—
218#127—831no—
219#127—831no—
220#126—1661no—
221#126—1661no—
222#125—831no—
223#125—1661no—
224#125—1661no—
225#124—831no—
226#124C2×D8311yescharacteristic
227C22⋊Q8—831no—
228C22⋊Q8—831no—
229#121—1661no—
230#121—1661no—
231#120—1661no—
232#120—1661no—
233C4×Q8—831no—
234C4×Q8C2×D8311yescharacteristic, complemented
235#117—831no—
236C4⋊C8—831no—
237C4⋊C8C2×D8311yescharacteristic, complemented
238#103—831no—
239#103C2×D8311yescharacteristic, complemented
240#102—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102C2×D8311yescharacteristic
246C22⋊C8—831no—
247C22⋊C8—831no—
248#97—831no—
249C4×C8—831no—
250C24—1661no—
251#41—3321no—
252#41—3321no—
253C2×Q8—831no—
254C2×Q8C22×D8311yescharacteristic
255C2×Q8C22×D8311yescharacteristic
256C2×D4—3321no—
257C2×D4—3321no—
258C2×D4—831no—
259C2×D4—831no—
260C2×D4—1661no—
261C2×D4—1661no—
262C2×D4—1661no—
263C2×D4—1661no—
264C2×D4—1661no—
265C2×D4—1661no—
266C2×D4C22×D8311yescharacteristic
267C22×C4—831no—
268C22×C4—1661no—
269Q16—41no—
270Q16—3321no—
271SD16—41no—
272SD16—3321no—
273SD16—3321no—
274SD16—3321no—
275SD16—3321no—
276SD16—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8C22×D8311yescharacteristic
280C2×C8C22×D8311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—831no—
283C4⋊C4—831no—
284C4⋊C4—831no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4—1661no—
288C4⋊C4—1661no—
289C4⋊C4—1661no—
290C4⋊C4—1661no—
291C4⋊C4—1661no—
292C4⋊C4C22×D8311yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—831no—
296C22⋊C4—831no—
297C22⋊C4—1661no—
298C22⋊C4—1661no—
299C22⋊C4—1661no—
300C22⋊C4—1661no—
301C42—21no—
302C42—831no—
303C42—1661no—
304C42C22×D8311yescharacteristic
305C23—21no—
306C23—3322no—
307C23—831no—
308C23—6641no—
309C23—6641no—
310C23—1661no—
311C23—1661no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—1661no—
317Q8—1661no—
318D4—21no—
319D4—21no—
320D4—3322no—
321D4—3322no—
322D4—6641no—
323D4—6641no—
324D4—6641no—
325D4—6641no—
326D4—1661no—
327D4—1661no—
328D4—1661no—
329D4—1661no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3322no—
334C2×C4—831no—
335C2×C4—831no—
336C2×C4—831no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4D4×D8311yescharacteristic
342C2×C4D4×D8311yescharacteristic
343C2×C4C23×D8311yescharacteristic
344C8—21no—
345C8—21no—
346C8—1661no—
347C8—1661no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—3321no—
352C22—3321no—
353C22—3321no—
354C22—3321no—
355C22—3321no—
356C22—3321no—
357C22—6644no—
358C22—1661no—
359C22—1661no—
360C22—1661no—
361C22C2×D4×D8311yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3321no—
368C4—3321no—
369C4—1661no—
370C4—1661no—
371C4—1661no—
372C4#1458183711yescharacteristic
373C4#1458183811yescharacteristic
374C2—81no—
375C2—3321no—
376C2—6641no—
377C2—6641no—
378C2#1912826711yescharacteristic
379C2#1912850811yescharacteristic
380C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#464×C83C2
socleC22×C83C22×D4
prime radical#19128269C2
hypercenter#464D83
hypocenterC83#3222
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #19128309C2semidirect
3. #19128094C2semidirect
4. #19128072C2semidirect
5. #19128064C2semidirect
6. #19128057C2semidirect
7. #19127920C2semidirect
8. #19127902C2semidirect
9. #19127901C2semidirect
10. #19127866C2semidirect
11. #19127862C2semidirect
12. #19127853C2semidirect
13. #19127850C2semidirect
14. #19127761C2semidirect
15. #464×C83C2semidirect
18. C2×#13050580C22semidirect
20. C2×#13050579C22semidirect
23. C2×#13050577C22semidirect
24. #14581790C22semidirect
30. #14581783C22semidirect
39. C4×Dic166C22semidirect
41. C4⋊(C83⋊C8)C22semidirect
42. #14581716C22semidirect
44. #14581714C22semidirect
47. #14581712C22semidirect
50. #14581702C22semidirect
51. C8×Dic83C22semidirect
52. C2×Q16×C83C22semidirect
55. C4×Q8×C83C22semidirect
56. C4⋊C8×C83C22semidirect
57. #103×C83C22semidirect
191. C83#3222semidirect
199. #464D83semidirect
210. C2×Q16C2×D83semidirect
234. C4×Q8C2×D83semidirect
237. C4⋊C8C2×D83semidirect
239. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283092
#191282692
#191280942
#191280722
#191280642
#191280572
#191279202
#191279022
#191279012
#191278662
#191278622
#191278532
#191278502
#191277612
#464×C832
#322283