Finite Group Databasebeta

← #24918551

Subgroups — #24918551

Order 10624.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128270C211yesmaximal, characteristic, complemented
2#19128091C211yesmaximal, characteristic, complemented
3#19128083C211yesmaximal, characteristic, complemented
4#19128063C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19127923C211yesmaximal, characteristic, complemented
7#19127916C211yesmaximal, characteristic, complemented
8#19127901C211yesmaximal, characteristic, complemented
9#19127896C211yesmaximal, characteristic, complemented
10#19127861C211yesmaximal, characteristic, complemented
11#19127851C211yesmaximal, characteristic, complemented
12#19127844C211yesmaximal, characteristic, complemented
13#19127834C211yesmaximal, characteristic, complemented
14#19127762C211yesmaximal, characteristic, complemented
15#542×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050578C2211yescharacteristic, complemented
20C2×#13050577C2211yescharacteristic
21C2×#13050577C2211yescharacteristic
22C2×#13050577C2211yescharacteristic, complemented
23C2×#13050575C2211yescharacteristic
24C2×#13050575C2211yescharacteristic
25C2×#13050575C2211yescharacteristic, complemented
26#14581789C2211yescharacteristic
27#14581789C2211yescharacteristic
28#14581780C2211yescharacteristic
29#14581780C2211yescharacteristic
30#14581767—21no—
31#14581767—21no—
32#14581762—21no—
33#14581762—21no—
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35C4⋊(C83⋊C8)C2211yescharacteristic, complemented
36#14581715C2211yescharacteristic, complemented
37#14581715C2211yescharacteristic, complemented
38#14581712C2211yescharacteristic
39#14581712C2211yescharacteristic
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic, complemented
43#14581712C2211yescharacteristic, complemented
44C22⋊(C83⋊C8)C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46#14581703C2211yescharacteristic, complemented
47C4×C83⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C83C2211yescharacteristic, complemented
49#126×C83—21no—
50#124×C83—21no—
51#124×C83C2211yescharacteristic, complemented
52#123×C83—21no—
53C22⋊Q8×C83—21no—
54C22⋊Q8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic, complemented
57C4×D4×C83C2211yescharacteristic, complemented
58#117×C83C2211yescharacteristic
59C23×D83—21no—
60#13050590—41no—
61#13050590—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—21no—
64C2×D332—21no—
65C2×D332—21no—
66C2×D332—21no—
67C2×D332—21no—
68C2×D332—41no—
69C2×D332—41no—
70C2×D332C2311yescharacteristic
71C2×D332C2311yescharacteristic
72C2×D332C2311yescharacteristic
73C2×C22⋊D83—21no—
74C2×C4×D83—21no—
75#13050580—41no—
76#13050578—41no—
77#13050577—41no—
78#13050577—41no—
79#13050577—41no—
80#13050575—41no—
81#13050575—41no—
82#13050575—41no—
83C2×C83⋊C8C2311yescharacteristic
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C2×C83⋊C8C2311yescharacteristic
87(C2×D83)⋊C4—21no—
88(C2×D83)⋊C4—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91C2×Q8×C83—21no—
92C2×Q8×C83C2311yescharacteristic
93C2×D4×C83—21no—
94C2×D4×C83C2311yescharacteristic
95C22×C4×C83—21no—
96C22×C4×C83C2311yescharacteristic
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83—21no—
100C4⋊C4×C83D411yescharacteristic
101C4⋊C4×C83D411yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C4⋊C4×C83C2311yescharacteristic
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83—21no—
108C22⋊C4×C83—21no—
109C22⋊C4×C83D411yescharacteristic
110C22⋊C4×C83D411yescharacteristic
111C42×C83—21no—
112C42×C83C2311yescharacteristic
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C22×D83—21no—
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—42no—
120C22×D83—81no—
121C22×D83—81no—
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—81no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—81no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—81no—
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2411yescharacteristic
140C22⋊D83—42no—
141C2×Dic83—21no—
142D332—21no—
143D332—21no—
144D332—21no—
145D332—21no—
146D332—21no—
147D332—21no—
148D332—42no—
149D332—81no—
150D332—81no—
151D332—81no—
152D332—81no—
153C4×D83—42no—
154Dic166—21no—
155Dic166—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C83⋊C8—21no—
159C83⋊C8—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83C22×D411yescharacteristic
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—84no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—21no—
182C4×C83—41no—
183C4×C83—41no—
184C4×C83—41no—
185C4×C83H(5,𝔽2)11yescharacteristic
186C4×C83H(5,𝔽2)11yescharacteristic
187Dic83—41no—
188C2×C83—41no—
189C2×C83—81no—
190C2×C83#53511yescharacteristic
191C2×C83#57611yescharacteristic
192C2×C83#57711yescharacteristic
193D83—81no—
194D83—81no—
195D83—81no—
196#3224—831nomaximal, Sylow
197C83#322411yescharacteristic, complemented, Sylow
198#542D8311yescharacteristic, complemented
199#535—831no—
200#496—831no—
201#495—831no—
202#489—831no—
203#486—831no—
204#464—831no—
205#463—831no—
206#460—831no—
207#459—831no—
208#452—831no—
209#450—831no—
210#449—831no—
211#447—831no—
212#433—831no—
213C2×#41—831no—
214C22×D4—831no—
215C2×Q16—831no—
216C2×SD16—831no—
217C2×SD16—831no—
218C2×SD16—831no—
219C2×SD16—831no—
220C2×D8—831no—
221C2×D8—831no—
222C2×D8—831no—
223C4⋊Q8C2×D8311yescharacteristic, complemented
224#127—831no—
225#127—831no—
226#126—21no—
227#124—21no—
228#124—831no—
229#124—831no—
230#124C2×D8311yescharacteristic, complemented
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8C2×D8311yescharacteristic
234#121—1661no—
235#121—1661no—
236#121C2×D8311yescharacteristic
237#120—1661no—
238#120—1661no—
239C4×Q8C2×D8311yescharacteristic, complemented
240C4×D4C2×D8311yescharacteristic, complemented
241#117C2×D8311yescharacteristic
242C4⋊C8—831no—
243C4⋊C8—831no—
244#103—831no—
245#103—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251#102—831no—
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97—831no—
255C4×C8—831no—
256C24—1661no—
257#41—3321no—
258#41—3321no—
259C2×Q8—21no—
260C2×Q8—831no—
261C2×Q8C22×D8311yescharacteristic
262C2×D4—21no—
263C2×D4—3321no—
264C2×D4—3321no—
265C2×D4—831no—
266C2×D4—831no—
267C2×D4—831no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4—1661no—
272C2×D4—1661no—
273C2×D4—1661no—
274C2×D4C22×D8311yescharacteristic
275C22×C4—21no—
276C22×C4—1661no—
277C22×C4C22×D8311yescharacteristic
278Q16—3321no—
279SD16—3321no—
280SD16—3321no—
281SD16—3321no—
282SD16—3321no—
283D8—3321no—
284D8—3321no—
285D8—3321no—
286C2×C8—831no—
287C2×C8—831no—
288C2×C8—831no—
289C2×C8—831no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22⋊D8311yescharacteristic
294C4⋊C4C22⋊D8311yescharacteristic
295C4⋊C4C22×D8311yescharacteristic
296C4⋊C4C22×D8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4C22⋊D8311yescharacteristic
307C22⋊C4C22⋊D8311yescharacteristic
308C42—21no—
309C42C22×D8311yescharacteristic
310C42C22×D8311yescharacteristic
311C23—21no—
312C23—3322no—
313C23—6641no—
314C23—6641no—
315C23—1661no—
316C23—1661no—
317C23—1661no—
318C23C2×C22⋊D8311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—1661no—
323Q8—1661no—
324D4—21no—
325D4—21no—
326D4—81no—
327D4—3322no—
328D4—3322no—
329D4—6641no—
330D4—6641no—
331D4—6641no—
332D4—6641no—
333D4—1661no—
334D4—1661no—
335D4—1661no—
336D4—1661no—
337D4—1661no—
338D4—1661no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—3322no—
346C2×C4—1661no—
347C2×C4C2×C22⋊D8311yescharacteristic
348C2×C4C2×C22⋊D8311yescharacteristic
349C2×C4C2×C22⋊D8311yescharacteristic
350C2×C4C2×C22⋊D8311yescharacteristic
351C2×C4C2×C22⋊D8311yescharacteristic
352C2×C4C23×D8311yescharacteristic
353C8—1661no—
354C8—1661no—
355C8—1661no—
356C8—1661no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—3321no—
370C22—3321no—
371C22—3321no—
372C22—6644no—
373C22C22×C22⋊D8311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—3321no—
383C4#1458183811yescharacteristic
384C4#1458183811yescharacteristic
385C2—41no—
386C2—81no—
387C2—6641no—
388C2—6641no—
389C2—6641no—
390C2#1912826511yescharacteristic
391C2#1912850411yescharacteristic
392C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#542×C83C2
socleC22×C83C22×D4
hypercenter#542D83
hypocenterC83#3224
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #19128270C2semidirect
2. #19128091C2semidirect
3. #19128083C2semidirect
4. #19128063C2semidirect
5. #19128049C2semidirect
6. #19127923C2semidirect
7. #19127916C2semidirect
8. #19127901C2semidirect
9. #19127896C2semidirect
10. #19127861C2semidirect
11. #19127851C2semidirect
12. #19127844C2semidirect
13. #19127834C2semidirect
14. #19127762C2semidirect
15. #542×C83C2semidirect
18. C2×#13050580C22semidirect
19. C2×#13050578C22semidirect
22. C2×#13050577C22semidirect
25. C2×#13050575C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. C4⋊(C83⋊C8)C22semidirect
36. #14581715C22semidirect
37. #14581715C22semidirect
42. #14581712C22semidirect
43. #14581712C22semidirect
46. #14581703C22semidirect
47. C4×C83⋊C8C22semidirect
48. C4⋊Q8×C83C22semidirect
51. #124×C83C22semidirect
56. C4×Q8×C83C22semidirect
57. C4×D4×C83C22semidirect
197. C83#3224semidirect
198. #542D83semidirect
223. C4⋊Q8C2×D83semidirect
230. #124C2×D83semidirect
239. C4×Q8C2×D83semidirect
240. C4×D4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282702
#191280912
#191280832
#191280632
#191280492
#191279232
#191279162
#191279012
#191278962
#191278612
#191278512
#191278442
#191278342
#191277622
#542×C832
#322483