Finite Group Databasebeta

← #24919477

Subgroups — #24919477

Order 10624.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128385C211yesmaximal, characteristic, complemented
2D4×C22⋊D83C211yesmaximal, characteristic
3#19128064C211yesmaximal, characteristic, complemented
4#19128026C211yesmaximal, characteristic, complemented
5#19128013C211yesmaximal, characteristic, complemented
6#19128003C211yesmaximal, characteristic, complemented
7#19127949C211yesmaximal, characteristic, complemented
8#19127927C211yesmaximal, characteristic, complemented
9#19127899C211yesmaximal, characteristic, complemented
10#19127850C211yesmaximal, characteristic, complemented
11#19127802C211yesmaximal, characteristic, complemented
12#19127775C211yesmaximal, characteristic, complemented
13#19127706C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#450×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83—21no—
17C2×D4×D83C2211yescharacteristic
18C2×#13050577C2211yescharacteristic, complemented
19C2×#13050574C2211yescharacteristic, complemented
20C2×C8×D83C2211yescharacteristic, complemented
21#14581788C2211yescharacteristic
22#14581785—21no—
23#14581783C2211yescharacteristic, complemented
24#14581782—21no—
25#14581774—21no—
26#14581768C2211yescharacteristic, complemented
27#14581766C2211yescharacteristic
28#14581764—21no—
29#14581763C2211yescharacteristic
30#14581761—21no—
31#14581760—21no—
32Q8×Dic83C2211yescharacteristic, complemented
33D4×Dic83C2211yescharacteristic
34C4×C22⋊D83C2211yescharacteristic
35#14581751C2211yescharacteristic, complemented
36#14581750—21no—
37C4⋊C4×D83C2211yescharacteristic, complemented
38#14581726C2211yescharacteristic, complemented
39#14581725C2211yescharacteristic, complemented
40#14581724C2211yescharacteristic, complemented
41Dic83⋊C8C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic, complemented
45#14581713C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581713C2211yescharacteristic, complemented
48#14581713C2211yescharacteristic, complemented
49#14581712C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51#14581702C2211yescharacteristic, complemented
52C22×D4×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic
54C2×SD16×C83C2211yescharacteristic, complemented
55C22⋊Q8×C83C2211yescharacteristic, complemented
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C2×D332C2311yescharacteristic
61C2×C22⋊D83—21no—
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C2×C22⋊D83—21no—
65C2×C22⋊D83—21no—
66C2×C22⋊D83—44no—
67C2×C22⋊D83D411yescharacteristic
68D4×D83—22no—
69D4×D83—41no—
70C22×Dic83—21no—
71C2×C4×D83—21no—
72C2×C4×D83C2311yescharacteristic
73#13050577—41no—
74#13050574—41no—
75C2×C83⋊C8C2311yescharacteristic
76C2×C83⋊C8C2311yescharacteristic
77C8×D83—41no—
78C4⋊Dic83—21no—
79C4⋊Dic83—21no—
80C4⋊Dic83C2311yescharacteristic
81C4⋊Dic83C2311yescharacteristic
82C4⋊Dic83C2311yescharacteristic
83Dic83⋊C4—21no—
84Dic83⋊C4—21no—
85Dic83⋊C4—21no—
86Dic83⋊C4D411yescharacteristic
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83D411yescharacteristic
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4D411yescharacteristic
94C4×Dic83—21no—
95C4×Dic83—21no—
96C4×Dic83C2311yescharacteristic
97C24×C83—21no—
98C2×Q8×C83C2311yescharacteristic
99C2×D4×C83—22no—
100C2×D4×C83—21no—
101C2×D4×C83—41no—
102C2×D4×C83C2311yescharacteristic
103C2×D4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105SD16×C83—22no—
106SD16×C83—41no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83—21no—
110C4⋊C4×C83C2311yescharacteristic
111C22⋊C4×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—42no—
115C23×C83—42no—
116C23×C83—81no—
117C23×C83C2×D411yescharacteristic
118C22×D83—21no—
119C22×D83—42no—
120C22×D83—42no—
121C22×D83—81no—
122C22×D83C2×D411yescharacteristic
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—42no—
129D4×C83#4112yes—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2411yescharacteristic
136C22⋊D83—22no—
137C22⋊D83—42no—
138C22⋊D83—42no—
139C22⋊D83—42no—
140C22⋊D83—42no—
141C22⋊D83—81no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—42no—
146C2×Dic83C2×D411yescharacteristic
147C2×Dic83C2×D411yescharacteristic
148C2×Dic83C2×D411yescharacteristic
149D332—21no—
150D332—21no—
151C4×D83—21no—
152C4×D83—21no—
153C4×D83—42no—
154C8×C83—21no—
155C8×C83—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C22×C83—22no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—84no—
168C22×C83C22×D411yescharacteristic
169C2×D83—21no—
170C2×D83—21no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—84no—
175C2×D83AGL(1,ℤ/8)12yes—
176C4×C83—21no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83C2×#4111yescharacteristic
180C4×C83H(5,𝔽2)11yescharacteristic
181Dic83—21no—
182Dic83—21no—
183Dic83—21no—
184Dic83—41no—
185Dic83—41no—
186Dic83—41no—
187C2×C83—42no—
188C2×C83—41no—
189C2×C83—81no—
190C2×C83#54611yescharacteristic
191C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C83#57711yescharacteristic
193D83—22no—
194D83—81no—
195#3310—831nomaximal, Sylow
196C83#331011yescharacteristic, complemented, Sylow
197#552—831no—
198D42—831no—
199#490—831no—
200#482—831no—
201#480—831no—
202#478—831no—
203#468—831no—
204#465—831no—
205#460—831no—
206#450—831no—
207#450D8311yescharacteristic, complemented
208#440—831no—
209#435—831no—
210#418—831no—
211C2×#102—831no—
212C22×D4—831no—
213C22×D4—1661no—
214C22×D4C2×D8311yescharacteristic
215C2×SD16—831no—
216C2×SD16C2×D8311yescharacteristic
217C2×SD16C2×D8311yescharacteristic, complemented
218C2×M16—831no—
219C22×C8—831no—
220#127—831no—
221#126—1661no—
222#125—831no—
223#125—1661no—
224#123—1661no—
225C22⋊Q8—831no—
226C22⋊Q8C2×D8311yescharacteristic, complemented
227#121—831no—
228#121—831no—
229#121—1661no—
230#120—1661no—
231#120—1661no—
232C4×Q8—831no—
233C4×D4—831no—
234C4×D4—831no—
235#117—831no—
236#117—1661no—
237C2×C4⋊C4—831no—
238#107—831no—
239#107—831no—
240#106—831no—
241C4⋊C8—831no—
242#103—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102C2×D8311yescharacteristic
251#102C2×D8311yescharacteristic, complemented
252C22⋊C8—831no—
253C22⋊C8C2×D8311yescharacteristic, complemented
254#97—831no—
255C24—21no—
256C24—1661no—
257C2×Q8C22×D8311yescharacteristic
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—3324no—
262C2×D4—3321no—
263C2×D4—831no—
264C2×D4—831no—
265C2×D4—1661no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1662no—
271C2×D4C22×D8311yescharacteristic
272C2×D4C22×D8311yescharacteristic
273C22×C4—831no—
274C22×C4—1661no—
275C22×C4—1661no—
276C22×C4C22×D8311yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—3321no—
280M16—3321no—
281C2×C8—3321no—
282C2×C8—831no—
283C2×C8—831no—
284C2×C8C22×D8311yescharacteristic
285C2×C8C22×D8311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—831no—
288C4⋊C4—831no—
289C4⋊C4—831no—
290C4⋊C4—831no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
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—
305C42—831no—
306C42—1661no—
307C42—1661no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—3322no—
314C23—3322no—
315C23—831no—
316C23—6641no—
317C23—1661no—
318C23D4×D8311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—3322no—
326D4—3322no—
327D4—3322no—
328D4—3322no—
329D4—6641no—
330D4—1662no—
331D4—1661no—
332D4—1661no—
333D4#1305058912yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3322no—
339C2×C4—3322no—
340C2×C4—831no—
341C2×C4—831no—
342C2×C4—831no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4D4×D8311yescharacteristic
349C2×C4C23×D8311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1661no—
353C8—1661no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—832no—
368C22—6644no—
369C22—1661no—
370C22—1661no—
371C22C2×D4×D8311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3321no—
376C4—3321no—
377C4—3321no—
378C4—1661no—
379C4—1661no—
380C4—1661no—
381C4C2×#1305058911yescharacteristic
382C4#1458183711yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—6641no—
387C2—1662no—
388C2#1912833811yescharacteristic
389C2AGL(1,ℤ/8)×D8311yescharacteristic
390C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
prime radicalD4×C22⋊D83C2
hypercenter#450D83
hypocenterC83#3310
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. #19128385C2semidirect
3. #19128064C2semidirect
4. #19128026C2semidirect
5. #19128013C2semidirect
6. #19128003C2semidirect
7. #19127949C2semidirect
8. #19127927C2semidirect
9. #19127899C2semidirect
10. #19127850C2semidirect
11. #19127802C2semidirect
12. #19127775C2semidirect
13. #19127706C2semidirect
14. #102×D83C2semidirect
15. #450×C83C2semidirect
18. C2×#13050577C22semidirect
19. C2×#13050574C22semidirect
20. C2×C8×D83C22semidirect
23. #14581783C22semidirect
26. #14581768C22semidirect
32. Q8×Dic83C22semidirect
35. #14581751C22semidirect
37. C4⋊C4×D83C22semidirect
38. #14581726C22semidirect
39. #14581725C22semidirect
40. #14581724C22semidirect
41. Dic83⋊C8C22semidirect
42. #14581716C22semidirect
43. #14581714C22semidirect
44. #14581714C22semidirect
47. #14581713C22semidirect
48. #14581713C22semidirect
49. #14581712C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. #14581702C22semidirect
54. C2×SD16×C83C22semidirect
55. C22⋊Q8×C83C22semidirect
57. #102×C83C22semidirect
58. C22⋊C8×C83C22semidirect
196. C83#3310semidirect
207. #450D83semidirect
217. C2×SD16C2×D83semidirect
226. C22⋊Q8C2×D83semidirect
251. #102C2×D83semidirect
253. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283852
D4×C22⋊D832
#191280642
#191280262
#191280132
#191280032
#191279492
#191279272
#191278992
#191278502
#191278022
#191277752
#191277062
#102×D832
#450×C832
#331083