Finite Group Databasebeta

← #24919086

Subgroups — #24919086

Order 10624.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581816C212yesmaximal, complemented
2#19128371C211yesmaximal, characteristic, complemented
3D4×D332C211yesmaximal, characteristic, complemented
4#19128077C211yesmaximal, characteristic, complemented
5#19127953C212yesmaximal, complemented
6#19127902C211yesmaximal, characteristic, complemented
7#19127847C212yesmaximal, complemented
8#19127833C212yesmaximal, complemented
9C4×#13050578C211yesmaximal, characteristic, complemented
10#19127778C211yesmaximal, characteristic, complemented
11#459×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yescomplemented
13C22×D332—21no—
14C2×D4×D83C2212yes—
15#14581816—28no—
16C2×#13050579C2212yescomplemented
17C2×#13050578C2211yescharacteristic
18C2×#13050575C2212yescomplemented
19C2×#13050575C2212yescomplemented
20C2×#13050573C2212yescomplemented
21#14581789C2211yescharacteristic
22#14581788—21no—
23#14581767C2212yes—
24#14581766—21no—
25#14581764—22no—
26#14581763C2212yescomplemented
27#14581762—22no—
28C4×Dic166C2211yescharacteristic, complemented
29C4×D332C2211yescharacteristic
30D4×Dic83—21no—
31#14581723C2211yescharacteristic, complemented
32#14581715C2211yescharacteristic, complemented
33#14581714C2212yescomplemented
34#14581713C2211yescharacteristic
35(C2×D83)⋊C8C2212yescomplemented
36C4×C83⋊C8C2211yescharacteristic, complemented
37C2×D8×C83C2212yes—
38#127×C83C2211yescharacteristic, complemented
39C4×D4×C83C2211yescharacteristic
40C4⋊C8×C83C2211yescharacteristic, complemented
41#102×C83C2212yescomplemented
42C23×D83—22no—
43#13050589—24no—
44#13050589—42no—
45C2×Dic166C2311yescharacteristic
46C2×D332—22no—
47C2×D332—21no—
48C2×D332—21no—
49C2×D332—44no—
50C2×D332D411yescharacteristic
51C2×D332C2312yes—
52C2×C22⋊D83—22no—
53C2×C22⋊D83—22no—
54C2×C22⋊D83—22no—
55D4×D83—24no—
56D4×D83—42no—
57C22×Dic83—22no—
58C2×C4×D83C2312yes—
59#13050579—24no—
60#13050578D414yescomplemented
61#13050575—24no—
62#13050575—42no—
63#13050573—24no—
64C2×C83⋊C8C2311yescharacteristic
65C2×C83⋊C8C2311yescharacteristic
66C4⋊Dic83D411yescharacteristic
67C4⋊Dic83C2311yescharacteristic
68Dic83⋊C4—21no—
69C22⋊Dic83—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4D412yes—
72C4×Dic83—21no—
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83C2312yes—
76C2×D4×C83C2311yescharacteristic
77C22×C4×C83—21no—
78D8×C83—24no—
79C2×C8×C83C2312yes—
80C4⋊C4×C83C2311yescharacteristic
81C22⋊C4×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C23×C83—22no—
85C22×D83—22no—
86C22×D83—44no—
87C22×D83—44no—
88C22×D83—82no—
89C22×D83C2×D412yes—
90D4×C83—22no—
91D4×C83—21no—
92D4×C83—22no—
93D4×C83—42no—
94D4×C83—81no—
95D4×C83C2×D412yes—
96C2×C4×C83—21no—
97C2×C4×C83—42no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—44no—
102C22⋊D83—44no—
103C22⋊D83—82no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—44no—
107C2×Dic83C2×D412yes—
108D332—22no—
109D332—22no—
110D332—22no—
111D332—44no—
112D332—42no—
113D332—81no—
114C4×D83—22no—
115C4×D83—22no—
116Dic166—21no—
117Dic166C2×D412yescomplemented
118C8×C83—22no—
119C83⋊C8—21no—
120C83⋊C8C2×D412yescomplemented
121C22×C83—22no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83C22×D411yescharacteristic
128C2×D83—22no—
129C2×D83—22no—
130C2×D83—22no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—88no—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83AGL(1,ℤ/8)12yescomplemented
138C4×C83C22×D411yescharacteristic
139C4×C83H(5,𝔽2)11yescharacteristic
140Dic83—22no—
141Dic83—22no—
142Dic83—41no—
143C2×C83—42no—
144C2×C83—82no—
145C2×C83D4211yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C83#57611yescharacteristic
148D83—42no—
149D83—82no—
150#3274—831nomaximal, Sylow
151C83#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—832no—
153#550—831no—
154D42—831no—
155#493—831no—
156#469—832no—
157#460—831no—
158#459D8311yescharacteristic, complemented
159#449—832no—
160#447—832no—
161C4×SD16—831no—
162#436—831no—
163C2×#41—832no—
164C22×D4—832no—
165C22×D4—1661no—
166AGL(1,ℤ/8)—1668no—
167C2×SD16—831no—
168C2×SD16—832no—
169C2×D8—832no—
170C2×D8—832no—
171C2×D8C2×D8312yes—
172C2×M16—832no—
173#127—831no—
174#127—1661no—
175#127C2×D8311yescharacteristic, complemented
176#121—832no—
177#121—832no—
178#121—1662no—
179#121—1661no—
180#120—1662no—
181C4×Q8—831no—
182C4×D4—831no—
183C4×D4—1661no—
184C4×D4C2×D8311yescharacteristic
185#106—831no—
186C4⋊C8C2×D8311yescharacteristic, complemented
187#103—831no—
188#102—831no—
189#102—832no—
190#102C2×D8312yescomplemented
191C22⋊C8—832no—
192C4×C8—831no—
193C24—1662no—
194#41—3322no—
195#41—1664no—
196C2×Q8—831no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—3324no—
200C2×D4—3322no—
201C2×D4—831no—
202C2×D4—832no—
203C2×D4—1662no—
204C2×D4—1662no—
205C2×D4—1662no—
206C2×D4—1664no—
207C2×D4—1662no—
208C2×D4—1661no—
209C2×D4—1661no—
210C2×D4C22×D8312yes—
211C2×D4C22×D8311yescharacteristic
212C22×C4—21no—
213C22×C4—832no—
214C22×C4—1662no—
215SD16—834no—
216SD16—1664no—
217D8—24no—
218D8—3322no—
219D8—1664no—
220M16—1664no—
221C2×C8—831no—
222C2×C8—831no—
223C2×C8C22×D8312yes—
224C4⋊C4—831no—
225C4⋊C4—831no—
226C4⋊C4—1661no—
227C4⋊C4C22×D8311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—832no—
230C22⋊C4—1662no—
231C22⋊C4—1662no—
232C42—1661no—
233C42C22×D8311yescharacteristic
234C23—21no—
235C23—22no—
236C23—3324no—
237C23—3324no—
238C23—832no—
239C23—6642no—
240C23—1662no—
241Q8—832no—
242Q8—1661no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—3324no—
249D4—3324no—
250D4—3324no—
251D4—3322no—
252D4—6641no—
253D4—6642no—
254D4—1662no—
255D4—1662no—
256D4—1662no—
257D4D4×D8312yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—3324no—
261C2×C4—832no—
262C2×C4—1661no—
263C2×C4—1661no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4D4×D8311yescharacteristic
267C2×C4D4×D8311yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—22no—
270C8—832no—
271C8—1661no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—6648no—
282C22—1662no—
283C22—1662no—
284C22—1662no—
285C22C2×D4×D8311yescharacteristic
286C4—21no—
287C4—41no—
288C4—3321no—
289C4—1662no—
290C4—1662no—
291C4#1458181612yescomplemented
292C4C2×D4×D8311yescharacteristic
293C4#1458183811yescharacteristic
294C2—42no—
295C2—82no—
296C2—3322no—
297C2—6642no—
298C2#1912833511yescharacteristic
299C2C2×#1458181611yescharacteristic
300C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#459×C83C2
socleC22×C83C22×D4
hypercenter#459D83
hypocenterC83#3274
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14581816C2semidirect
2. #19128371C2semidirect
3. D4×D332C2semidirect
4. #19128077C2semidirect
5. #19127953C2semidirect
6. #19127902C2semidirect
7. #19127847C2semidirect
8. #19127833C2semidirect
9. C4×#13050578C2semidirect
10. #19127778C2semidirect
11. #459×C83C2semidirect
12. C2×#13050589C22semidirect
16. C2×#13050579C22semidirect
18. C2×#13050575C22semidirect
19. C2×#13050575C22semidirect
20. C2×#13050573C22semidirect
26. #14581763C22semidirect
28. C4×Dic166C22semidirect
31. #14581723C22semidirect
32. #14581715C22semidirect
33. #14581714C22semidirect
35. (C2×D83)⋊C8C22semidirect
36. C4×C83⋊C8C22semidirect
38. #127×C83C22semidirect
40. C4⋊C8×C83C22semidirect
41. #102×C83C22semidirect
60. #13050578D4semidirect
117. Dic166C2×D4semidirect
120. C83⋊C8C2×D4semidirect
137. C4×C83AGL(1,ℤ/8)semidirect
151. C83#3274semidirect
158. #459D83semidirect
175. #127C2×D83semidirect
186. C4⋊C8C2×D83semidirect
190. #102C2×D83semidirect
291. C4#14581816semidirect

Maximal subgroups

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

subgroupindex
C2×#145818162
#191283712
D4×D3322
#191280772
#191279532
#191279022
#191278472
#191278332
C4×#130505782
#191277782
#459×C832
#327483