Finite Group Databasebeta

← #24918552

Subgroups — #24918552

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#19128320C211yesmaximal, characteristic, complemented
2#19128264C211yesmaximal, characteristic, complemented
3#19128083C211yesmaximal, characteristic, complemented
4#19128061C211yesmaximal, characteristic, complemented
5#19128047C211yesmaximal, characteristic, complemented
6#19127921C211yesmaximal, characteristic, complemented
7#19127914C211yesmaximal, characteristic, complemented
8#19127899C211yesmaximal, characteristic, complemented
9#19127894C211yesmaximal, characteristic, complemented
10#19127860C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127843C211yesmaximal, characteristic, complemented
13#19127833C211yesmaximal, characteristic, complemented
14#19127763C211yesmaximal, characteristic, complemented
15#496×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050578C2211yescharacteristic, complemented
20C2×#13050577C2211yescharacteristic
21C2×#13050575C2211yescharacteristic
22#14581792C2211yescharacteristic, complemented
23#14581788C2211yescharacteristic
24#14581785—21no—
25#14581780C2211yescharacteristic
26#14581779C2211yescharacteristic
27#14581778—21no—
28#14581776—21no—
29#14581769—21no—
30#14581768C2211yescharacteristic
31#14581764—21no—
32#14581764—21no—
33#14581763C2211yescharacteristic
34#14581760—21no—
35#14581760—21no—
36Q8×Dic83C2211yescharacteristic, complemented
37D4×Dic83C2211yescharacteristic, complemented
38#14581749C2211yescharacteristic
39Dic83⋊C8C2211yescharacteristic, complemented
40Dic83⋊C8C2211yescharacteristic, complemented
41#14581717C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic
44#14581714C2211yescharacteristic
45#14581713C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581713C2211yescharacteristic, complemented
48#14581713C2211yescharacteristic, complemented
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51C4×C83⋊C8C2211yescharacteristic, complemented
52C2×SD16×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic, complemented
54C2×D8×C83C2211yescharacteristic
55C2×D8×C83C2211yescharacteristic, complemented
56#127×C83C2211yescharacteristic
57#124×C83C2211yescharacteristic, complemented
58#97×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050589—41no—
61#13050589—41no—
62C2×Dic166—21no—
63C2×Dic166C2311yescharacteristic
64C2×D332C2311yescharacteristic
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70D4×D83—41no—
71D4×D83—41no—
72C22×Dic83—21no—
73C22×Dic83—21no—
74C2×C4×D83C2311yescharacteristic
75#13050580—41no—
76#13050578—41no—
77#13050577—41no—
78#13050575—41no—
79C2×C83⋊C8C2311yescharacteristic
80C2×C83⋊C8C2311yescharacteristic
81C4⋊Dic83—21no—
82C4⋊Dic83—21no—
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4D411yescharacteristic
87Dic83⋊C4D411yescharacteristic
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4D411yescharacteristic
96(C2×D83)⋊C4D411yescharacteristic
97C4×Dic83—21no—
98C4×Dic83C2311yescharacteristic
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—21no—
101C2×D4×C83—21no—
102C2×D4×C83C2311yescharacteristic
103C2×D4×C83C2311yescharacteristic
104C2×D4×C83C2311yescharacteristic
105SD16×C83—41no—
106SD16×C83—41no—
107D8×C83—41no—
108D8×C83—41no—
109C2×C8×C83C2311yescharacteristic
110C2×C8×C83C2311yescharacteristic
111C22⋊C4×C83—21no—
112C22⋊C4×C83—21no—
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C22×D83—21no—
118C22×D83—42no—
119C22×D83—81no—
120C22×D83—81no—
121C22×D83C2×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—21no—
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—21no—
129D4×C83—21no—
130D4×C83—81no—
131D4×C83—81no—
132C2×C4×C83—21no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2411yescharacteristic
136C22⋊D83—42no—
137C22⋊D83—42no—
138C22⋊D83—81no—
139C22⋊D83—81no—
140C22⋊D83—81no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—42no—
145C2×Dic83—81no—
146C2×Dic83C2×D411yescharacteristic
147C2×Dic83C2×D411yescharacteristic
148C2×Dic83C2×D411yescharacteristic
149D332—21no—
150D332—21no—
151C4×D83—21no—
152C4×D83—21no—
153Dic166—21no—
154Dic166—21no—
155Dic166—81no—
156C8×C83—21no—
157C8×C83—21no—
158C83⋊C8—21no—
159C83⋊C8—21no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83C22×D411yescharacteristic
170C2×D83—21no—
171C2×D83—21no—
172C2×D83—21no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—84no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83H(5,𝔽2)11yescharacteristic
181C4×C83H(5,𝔽2)11yescharacteristic
182Dic83—21no—
183Dic83—21no—
184Dic83—21no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188C2×C83—81no—
189C2×C83—81no—
190C2×C83—81no—
191C2×C83#53511yescharacteristic
192C2×C83#57611yescharacteristic
193C2×C83#57711yescharacteristic
194D83—41no—
195D83—81no—
196#3224—831nomaximal, Sylow
197C83#322411yescharacteristic, complemented, Sylow
198#542—831no—
199#535—831no—
200#496D8311yescharacteristic, complemented
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×SD16C2×D8311yescharacteristic
219C2×SD16C2×D8311yescharacteristic, complemented
220C2×D8—831no—
221C2×D8C2×D8311yescharacteristic
222C2×D8C2×D8311yescharacteristic, complemented
223C4⋊Q8—831no—
224#127—831no—
225#127C2×D8311yescharacteristic
226#126—1661no—
227#124—831no—
228#124—831no—
229#124—1661no—
230#124C2×D8311yescharacteristic, complemented
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×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#102—831no—
251#102—831no—
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97C2×D8311yescharacteristic, complemented
255C4×C8—831no—
256C24—1661no—
257#41—3321no—
258#41—3321no—
259C2×Q8—831no—
260C2×Q8—1661no—
261C2×Q8C22×D8311yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—3321no—
265C2×D4—3321no—
266C2×D4—831no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4—1661no—
272C2×D4C22×D8311yescharacteristic
273C2×D4C22×D8311yescharacteristic
274C2×D4C22×D8311yescharacteristic
275C22×C4—831no—
276C22×C4—1661no—
277C22×C4—1661no—
278Q16—3321no—
279SD16—41no—
280SD16—41no—
281SD16—3321no—
282SD16—3321no—
283D8—41no—
284D8—41no—
285D8—3321no—
286C2×C8—831no—
287C2×C8—831no—
288C2×C8C22×D8311yescharacteristic
289C2×C8C22×D8311yescharacteristic
290C4⋊C4—831no—
291C4⋊C4—831no—
292C4⋊C4—831no—
293C4⋊C4—831no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4—1661no—
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—831no—
309C42—1661no—
310C42C22×D8311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—3322no—
315C23—831no—
316C23—6641no—
317C23—6641no—
318C23—1661no—
319Q8—21no—
320Q8—21no—
321Q8—6641no—
322Q8—1661no—
323Q8—1661no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—81no—
331D4—81no—
332D4—3322no—
333D4—3322no—
334D4—6641no—
335D4—6641no—
336D4—6641no—
337D4—1661no—
338D4—1661no—
339C2×C4—21no—
340C2×C4—3322no—
341C2×C4—831no—
342C2×C4—831no—
343C2×C4—831no—
344C2×C4—6641no—
345C2×C4—1661no—
346C2×C4—1661no—
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—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—6644no—
370C22—1661no—
371C22—1661no—
372C22—1661no—
373C22C2×D4×D8311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—3321no—
378C4—3321no—
379C4—3321no—
380C4—1661no—
381C4—1661no—
382C4—1661no—
383C4#1458183711yescharacteristic
384C4#1458183711yescharacteristic
385C2—81no—
386C2—81no—
387C2—81no—
388C2—3321no—
389C2—6641no—
390C2#1912826411yescharacteristic
391C2#1912850511yescharacteristic
392C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#496×C83C2
socleC22×C83C22×D4
hypercenter#496D83
hypocenterC83#3224
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #19128320C2semidirect
2. #19128264C2semidirect
3. #19128083C2semidirect
4. #19128061C2semidirect
5. #19128047C2semidirect
6. #19127921C2semidirect
7. #19127914C2semidirect
8. #19127899C2semidirect
9. #19127894C2semidirect
10. #19127860C2semidirect
11. #19127850C2semidirect
12. #19127843C2semidirect
13. #19127833C2semidirect
14. #19127763C2semidirect
15. #496×C83C2semidirect
18. C2×#13050580C22semidirect
19. C2×#13050578C22semidirect
22. #14581792C22semidirect
36. Q8×Dic83C22semidirect
37. D4×Dic83C22semidirect
39. Dic83⋊C8C22semidirect
40. Dic83⋊C8C22semidirect
41. #14581717C22semidirect
42. #14581717C22semidirect
47. #14581713C22semidirect
48. #14581713C22semidirect
51. C4×C83⋊C8C22semidirect
53. C2×SD16×C83C22semidirect
55. C2×D8×C83C22semidirect
57. #124×C83C22semidirect
58. #97×C83C22semidirect
197. C83#3224semidirect
200. #496D83semidirect
219. C2×SD16C2×D83semidirect
222. C2×D8C2×D83semidirect
230. #124C2×D83semidirect
254. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283202
#191282642
#191280832
#191280612
#191280472
#191279212
#191279142
#191278992
#191278942
#191278602
#191278502
#191278432
#191278332
#191277632
#496×C832
#322483