Finite Group Databasebeta

← #24918564

Subgroups — #24918564

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#19128323C211yesmaximal, characteristic, complemented
2#19128269C211yesmaximal, characteristic, complemented
3#19128087C211yesmaximal, characteristic, complemented
4#19128086C211yesmaximal, characteristic, complemented
5#19128062C211yesmaximal, characteristic, complemented
6#19128046C211yesmaximal, characteristic, complemented
7#19127913C211yesmaximal, characteristic, complemented
8#19127901C211yesmaximal, characteristic, complemented
9#19127897C211yesmaximal, characteristic, complemented
10#19127862C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127849C211yesmaximal, characteristic, complemented
13#19127836C211yesmaximal, characteristic, complemented
14#19127761C211yesmaximal, characteristic, complemented
15#464×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050577C2211yescharacteristic
19C2×#13050577C2211yescharacteristic, complemented
20C2×#13050577C2211yescharacteristic, complemented
21C2×D664C2211yescharacteristic
22C2×D664C2211yescharacteristic, complemented
23C2×#13050575C2211yescharacteristic
24#14581791C2211yescharacteristic, complemented
25#14581789C2211yescharacteristic
26#14581788C2211yescharacteristic
27#14581786—21no—
28#14581781C2211yescharacteristic
29#14581781C2211yescharacteristic, complemented
30#14581780—21no—
31#14581777—21no—
32#14581770—21no—
33#14581768C2211yescharacteristic
34#14581767C2211yescharacteristic
35#14581765—21no—
36#14581765—21no—
37#14581762—21no—
38#14581762—21no—
39C4×D332C2211yescharacteristic, 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×Dic166—21no—
63C2×D332—21no—
64C2×D332—21no—
65C2×D332—21no—
66C2×D332—21no—
67C2×D332C2311yescharacteristic
68C2×D332C2311yescharacteristic
69C2×D332C2311yescharacteristic
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73D4×D83—41no—
74D4×D83—41no—
75C2×C4×D83—21no—
76C2×C4×D83—21no—
77C2×C4×D83C2311yescharacteristic
78#13050577—41no—
79#13050577—41no—
80#13050577—41no—
81D664—41no—
82D664—41no—
83#13050575—41no—
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89Dic83⋊C4D411yescharacteristic
90Dic83⋊C4D411yescharacteristic
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)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98(C2×D83)⋊C4D411yescharacteristic
99(C2×D83)⋊C4D411yescharacteristic
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—21no—
118C22×D83—42no—
119C22×D83—81no—
120C22×D83—81no—
121C22×D83C2×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125Q8×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2411yescharacteristic
134C22⋊D83—42no—
135C22⋊D83—81no—
136C22⋊D83—81no—
137C2×Dic83—21no—
138C2×Dic83C2×D411yescharacteristic
139C2×Dic83C2×D411yescharacteristic
140C2×Dic83C2×D411yescharacteristic
141D332—21no—
142D332—21no—
143D332—21no—
144D332—21no—
145D332—21no—
146D332—21no—
147D332—42no—
148D332—81no—
149D332—81no—
150D332—81no—
151C4×D83—21no—
152C4×D83—21no—
153C4×D83—42no—
154C4×D83—81no—
155Dic166—81no—
156C8×C83—21no—
157C8×C83—21no—
158C83⋊C8—21no—
159C83⋊C8—21no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83C22×D411yescharacteristic
164C2×D83—21no—
165C2×D83—21no—
166C2×D83—21no—
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—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83H(5,𝔽2)11yescharacteristic
183C4×C83H(5,𝔽2)11yescharacteristic
184Dic83—21no—
185Dic83—21no—
186Dic83—21no—
187Dic83—41no—
188C2×C83—81no—
189C2×C83#53511yescharacteristic
190C2×C83#57611yescharacteristic
191C2×C83#57711yescharacteristic
192D83—41no—
193D83—81no—
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#464D8311yescharacteristic, complemented
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×Q16C2×D8311yescharacteristic, complemented
216C2×SD16—831no—
217C2×SD16—831no—
218C2×SD16—831no—
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—1661no—
230#124C2×D8311yescharacteristic
231#123—1661no—
232C22⋊Q8—831no—
233C22⋊Q8—1661no—
234#121—831no—
235#121—1661no—
236#121—1661no—
237#120—1661no—
238#120—1661no—
239C4×Q8C2×D8311yescharacteristic, complemented
240C4×D4—831no—
241#117—831no—
242C4⋊C8—831no—
243C4⋊C8C2×D8311yescharacteristic, complemented
244#103—831no—
245#103C2×D8311yescharacteristic, complemented
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251#102C2×D8311yescharacteristic
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97—831no—
255C4×C8—831no—
256C24—1661no—
257#41—3321no—
258#41—3321no—
259C2×Q8—1661no—
260C2×Q8C22×D8311yescharacteristic
261C2×Q8C22×D8311yescharacteristic
262C2×D4—3321no—
263C2×D4—3321no—
264C2×D4—831no—
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×D4—1661no—
274C2×D4C22×D8311yescharacteristic
275C22×C4—831no—
276C22×C4—1661no—
277C22×C4—1661no—
278Q16—41no—
279SD16—41no—
280SD16—3321no—
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—21no—
299C22⋊C4—831no—
300C22⋊C4—831no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C42—21no—
309C42—831no—
310C42C22×D8311yescharacteristic
311C23—21no—
312C23—3322no—
313C23—831no—
314C23—6641no—
315C23—6641no—
316C23—1661no—
317C23—1661no—
318C23—1661no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—6641no—
324D4—21no—
325D4—21no—
326D4—3322no—
327D4—3322no—
328D4—6641no—
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—3322no—
343C2×C4—831no—
344C2×C4—831no—
345C2×C4—831no—
346C2×C4—6641no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4D4×D8311yescharacteristic
351C2×C4D4×D8311yescharacteristic
352C2×C4C23×D8311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1661no—
356C8—1661no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—6644no—
370C22—1661no—
371C22—1661no—
372C22—1661no—
373C22C2×D4×D8311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3321no—
380C4—1661no—
381C4—1661no—
382C4—1661no—
383C4#1458183711yescharacteristic
384C4#1458183811yescharacteristic
385C2—81no—
386C2—3321no—
387C2—6641no—
388C2—6641no—
389C2—6641no—
390C2#1912826711yescharacteristic
391C2#1912850311yescharacteristic
392C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#464×C83C2
socleC22×C83C22×D4
hypercenter#464D83
hypocenterC83#3224
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. #19128323C2semidirect
2. #19128269C2semidirect
3. #19128087C2semidirect
4. #19128086C2semidirect
5. #19128062C2semidirect
6. #19128046C2semidirect
7. #19127913C2semidirect
8. #19127901C2semidirect
9. #19127897C2semidirect
10. #19127862C2semidirect
11. #19127850C2semidirect
12. #19127849C2semidirect
13. #19127836C2semidirect
14. #19127761C2semidirect
15. #464×C83C2semidirect
19. C2×#13050577C22semidirect
20. C2×#13050577C22semidirect
22. C2×D664C22semidirect
24. #14581791C22semidirect
29. #14581781C22semidirect
39. C4×D332C22semidirect
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
197. C83#3224semidirect
204. #464D83semidirect
215. C2×Q16C2×D83semidirect
239. C4×Q8C2×D83semidirect
243. C4⋊C8C2×D83semidirect
245. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283232
#191282692
#191280872
#191280862
#191280622
#191280462
#191279132
#191279012
#191278972
#191278622
#191278502
#191278492
#191278362
#191277612
#464×C832
#322483