Finite Group Databasebeta

← #24918561

Subgroups — #24918561

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#19128324C211yesmaximal, characteristic, complemented
2#19128268C211yesmaximal, characteristic, complemented
3#19128087C211yesmaximal, characteristic, complemented
4#19128082C211yesmaximal, characteristic, complemented
5#19128059C211yesmaximal, characteristic, complemented
6#19128046C211yesmaximal, characteristic, complemented
7#19127921C211yesmaximal, characteristic, complemented
8#19127915C211yesmaximal, characteristic, complemented
9#19127899C211yesmaximal, characteristic, complemented
10#19127895C211yesmaximal, characteristic, complemented
11#19127861C211yesmaximal, characteristic, complemented
12#19127848C211yesmaximal, characteristic, complemented
13#19127835C211yesmaximal, characteristic, complemented
14#19127759C211yesmaximal, characteristic, complemented
15#450×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C2×#13050580C2211yescharacteristic, complemented
18C2×#13050579C2211yescharacteristic, complemented
19C2×#13050577C2211yescharacteristic, complemented
20C2×D664C2211yescharacteristic, complemented
21C2×#13050575C2211yescharacteristic
22C2×#13050575C2211yescharacteristic
23#14581791C2211yescharacteristic, complemented
24#14581788C2211yescharacteristic
25#14581788C2211yescharacteristic
26#14581787—21no—
27#14581781C2211yescharacteristic, complemented
28#14581779C2211yescharacteristic
29#14581779C2211yescharacteristic
30#14581778—21no—
31#14581774—21no—
32#14581772—21no—
33#14581766C2211yescharacteristic
34#14581763—21no—
35#14581763—21no—
36#14581761—21no—
37#14581761—21no—
38Q8×Dic83C2211yescharacteristic, complemented
39#14581754C2211yescharacteristic, complemented
40#14581751C2211yescharacteristic
41Dic83⋊C8C2211yescharacteristic, complemented
42Dic83⋊C8C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic, complemented
46#14581713C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581712C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic
50#14581702C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52C22×D4×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic
54C2×SD16×C83C2211yescharacteristic
55C22⋊Q8×C83C2211yescharacteristic
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic
59#13050590—41no—
60#13050590—41no—
61C2×Dic166—21no—
62C2×Dic166C2311yescharacteristic
63C2×D332C2311yescharacteristic
64C2×D332C2311yescharacteristic
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73#13050580—41no—
74#13050579—41no—
75#13050577—41no—
76D664—41no—
77#13050575—41no—
78#13050575—41no—
79C2×C83⋊C8C2311yescharacteristic
80C2×C83⋊C8C2311yescharacteristic
81C4⋊Dic83—21no—
82C4⋊Dic83D411yescharacteristic
83C4⋊Dic83D411yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86C22⋊Dic83—21no—
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83D411yescharacteristic
91C22⋊Dic83D411yescharacteristic
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96C4×Dic83—21no—
97C4×Dic83C2311yescharacteristic
98C4×Dic83C2311yescharacteristic
99C24×C83—21no—
100C2×Q8×C83C2311yescharacteristic
101C2×D4×C83—21no—
102C2×D4×C83—41no—
103C2×D4×C83—41no—
104C2×D4×C83C2311yescharacteristic
105C2×D4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107SD16×C83—41no—
108SD16×C83—41no—
109C2×C8×C83C2311yescharacteristic
110C2×C8×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83C2311yescharacteristic
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—42no—
117C23×C83—81no—
118C23×C83—81no—
119C23×C83C2×D411yescharacteristic
120C22×D83—21no—
121C22×D83—21no—
122Q8×C83—21no—
123Q8×C83—21no—
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—42no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2411yescharacteristic
135C22⋊D83—42no—
136C22⋊D83—81no—
137C22⋊D83—81no—
138C22⋊D83—81no—
139C22⋊D83—81no—
140C22⋊D83—81no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145C2×Dic83C2×D411yescharacteristic
146C2×Dic83C2×D411yescharacteristic
147D332—21no—
148D332—21no—
149D332—21no—
150D332—21no—
151C4×D83—42no—
152C4×D83—81no—
153Dic166—21no—
154Dic166—21no—
155Dic166—81no—
156C8×C83—21no—
157C8×C83—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×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—84no—
170C22×C83C22×D411yescharacteristic
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C4×C83—21no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83H(5,𝔽2)11yescharacteristic
181C4×C83H(5,𝔽2)11yescharacteristic
182Dic83—21no—
183Dic83—21no—
184Dic83—21no—
185Dic83—21no—
186Dic83—41no—
187Dic83—41no—
188C2×C83—41no—
189C2×C83—81no—
190C2×C83—81no—
191C2×C83#53511yescharacteristic
192C2×C83#57611yescharacteristic
193C2×C83#57711yescharacteristic
194D83—81no—
195D83—81no—
196#3224—831nomaximal, Sylow
197C83#322411yescharacteristic, complemented, Sylow
198#542—831no—
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#450D8311yescharacteristic, complemented
210#449—831no—
211#447—831no—
212#433—831no—
213C2×#41—831no—
214C22×D4C2×D8311yescharacteristic
215C2×Q16—831no—
216C2×SD16—831no—
217C2×SD16—831no—
218C2×SD16C2×D8311yescharacteristic
219C2×SD16C2×D8311yescharacteristic
220C2×D8—831no—
221C2×D8—831no—
222C2×D8—831no—
223C4⋊Q8—831no—
224#127—831no—
225#127—831no—
226#126—1661no—
227#124—831no—
228#124—831no—
229#124—831no—
230#124—1661no—
231#123—1661no—
232C22⋊Q8—1661no—
233C22⋊Q8C2×D8311yescharacteristic
234#121—831no—
235#121—1661no—
236#121—1661no—
237#120—1661no—
238#120—1661no—
239C4×Q8—831no—
240C4×D4—831no—
241#117—831no—
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#102C2×D8311yescharacteristic
251#102C2×D8311yescharacteristic
252C22⋊C8—831no—
253C22⋊C8C2×D8311yescharacteristic
254#97—831no—
255C4×C8—831no—
256C24—21no—
257#41—3321no—
258#41—3321no—
259C2×Q8—831no—
260C2×Q8—1661no—
261C2×Q8C22×D8311yescharacteristic
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—831no—
266C2×D4—831no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4—1661no—
272C2×D4—1661no—
273C2×D4C22×D8311yescharacteristic
274C2×D4C22×D8311yescharacteristic
275C22×C4—1661no—
276C22×C4—1661no—
277C22×C4C22×D8311yescharacteristic
278Q16—3321no—
279SD16—41no—
280SD16—41no—
281SD16—3321no—
282SD16—3321no—
283D8—3321no—
284D8—3321no—
285D8—3321no—
286C2×C8—831no—
287C2×C8—831no—
288C2×C8C22×D8311yescharacteristic
289C2×C8C22×D8311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—831no—
292C4⋊C4—831no—
293C4⋊C4—831no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4C22×D8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—831no—
299C22⋊C4—831no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C42—831no—
309C42—831no—
310C42—1661no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—1661no—
317C23—1661no—
318C23D4×D8311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—6641no—
322Q8—1661no—
323Q8—1661no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—3322no—
330D4—6641no—
331D4—6641no—
332D4—6641no—
333D4—6641no—
334D4—6641no—
335D4—1661no—
336D4—1661no—
337D4—1661no—
338D4—1661no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3322no—
344C2×C4—831no—
345C2×C4—831no—
346C2×C4—831no—
347C2×C4—831no—
348C2×C4—6641no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4D4×D8311yescharacteristic
352C2×C4C23×D8311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1661no—
356C8—1661no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—3321no—
368C22—3321no—
369C22—3321no—
370C22—3321no—
371C22—3321no—
372C22—3321no—
373C22C2×D4×D8311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—3321no—
378C4—3321no—
379C4—1661no—
380C4—1661no—
381C4—1661no—
382C4—1661no—
383C4#1458183711yescharacteristic
384C4#1458183711yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—6641no—
389C2—6641no—
390C2#1912826611yescharacteristic
391C2#1912850511yescharacteristic
392C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
hypercenter#450D83
hypocenterC83#3224
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #19128324C2semidirect
2. #19128268C2semidirect
3. #19128087C2semidirect
4. #19128082C2semidirect
5. #19128059C2semidirect
6. #19128046C2semidirect
7. #19127921C2semidirect
8. #19127915C2semidirect
9. #19127899C2semidirect
10. #19127895C2semidirect
11. #19127861C2semidirect
12. #19127848C2semidirect
13. #19127835C2semidirect
14. #19127759C2semidirect
15. #450×C83C2semidirect
17. C2×#13050580C22semidirect
18. C2×#13050579C22semidirect
19. C2×#13050577C22semidirect
20. C2×D664C22semidirect
23. #14581791C22semidirect
27. #14581781C22semidirect
38. Q8×Dic83C22semidirect
39. #14581754C22semidirect
41. Dic83⋊C8C22semidirect
42. Dic83⋊C8C22semidirect
43. #14581717C22semidirect
44. #14581715C22semidirect
45. #14581714C22semidirect
48. #14581712C22semidirect
50. #14581702C22semidirect
51. C8×Dic83C22semidirect
197. C83#3224semidirect
209. #450D83semidirect

Maximal subgroups

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

subgroupindex
#191283242
#191282682
#191280872
#191280822
#191280592
#191280462
#191279212
#191279152
#191278992
#191278952
#191278612
#191278482
#191278352
#191277592
#450×C832
#322483