Finite Group Databasebeta

← #24919151

Subgroups — #24919151

Order 10624.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14581814C211yesmaximal, characteristic, complemented
2C2×D8×D83C211yesmaximal, characteristic, complemented
3#19128348C212yesmaximal, complemented
4#19128084C211yesmaximal, characteristic, complemented
5#19127943C211yesmaximal, characteristic, complemented
6#19127915C212yesmaximal, complemented
7#19127853C212yesmaximal, complemented
8#19127844C212yesmaximal, complemented
9#19127783C211yesmaximal, characteristic, complemented
10C8×C22⋊D83C211yesmaximal, characteristic, complemented
11#466×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2212yescomplemented
13C2×D4×D83C2212yes—
14#14581814—24no—
15C2×Dic332C2211yescharacteristic, complemented
16C2×#13050579C2212yescomplemented
17C2×#13050578C2212yescomplemented
18C2×D664C2211yescharacteristic
19C2×#13050575C2212yes—
20D8×D83—24no—
21C2×C8×D83C2211yescharacteristic
22#14581779C2212yescomplemented
23#14581774—22no—
24#14581772C2212yes—
25#14581763—22no—
26#14581760—22no—
27#14581754C2212yescomplemented
28C4×C22⋊D83C2211yescharacteristic
29C22⋊C4×D83—22no—
30Dic83⋊C8C2211yescharacteristic, complemented
31#14581715C2212yescomplemented
32#14581712C2212yescomplemented
33C22⋊(C83⋊C8)C2211yescharacteristic, complemented
34(C2×D83)⋊C8C2211yescharacteristic
35C8×Dic83C2211yescharacteristic, complemented
36C2×D8×C83C2211yescharacteristic
37C22×C8×C83C2211yescharacteristic, complemented
38#121×C83C2212yescomplemented
39#107×C83C2211yescharacteristic, complemented
40#102×C83C2212yescomplemented
41C23×D83—22no—
42#13050590—24no—
43#13050590—42no—
44C2×Dic166C2312yes—
45C2×D332C2312yes—
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48C2×C22⋊D83D411yescharacteristic
49D4×D83—24no—
50D4×D83—42no—
51C2×C4×D83—22no—
52C2×C4×D83—22no—
53C2×C4×D83C2311yescharacteristic
54Dic332—22no—
55#13050579—24no—
56#13050578—42no—
57D664D414yescomplemented
58#13050575—24no—
59C2×C83⋊C8C2311yescharacteristic
60C2×C83⋊C8C2311yescharacteristic
61C8×D83—22no—
62C4⋊Dic83—22no—
63Dic83⋊C4D411yescharacteristic
64C22⋊Dic83—22no—
65C22⋊Dic83—22no—
66C22⋊Dic83D411yescharacteristic
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C2×D4×C83—22no—
72C2×D4×C83C2312yes—
73C22×C4×C83C2311yescharacteristic
74D8×C83—22no—
75C2×C8×C83—22no—
76C2×C8×C83C2311yescharacteristic
77C2×C8×C83C2311yescharacteristic
78C4⋊C4×C83C2312yes—
79C22⋊C4×C83—22no—
80C23×C83—22no—
81C23×C83C2×D411yescharacteristic
82C22×D83—22no—
83C22×D83—44no—
84C22×D83—44no—
85C22×D83—82no—
86C22×D83C2×D411yescharacteristic
87D4×C83—22no—
88D4×C83—22no—
89D4×C83—82no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—22no—
96C22⋊D83—44no—
97C22⋊D83—44no—
98C2×Dic83—22no—
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102D332—22no—
103D332C2×D414yes—
104C4×D83—21no—
105C4×D83—21no—
106C4×D83—44no—
107C4×D83—82no—
108Dic166—22no—
109Dic166—22no—
110C8×C83—21no—
111C8×C83C2×D412yes—
112C83⋊C8—21no—
113C83⋊C8—21no—
114C22×C83—21no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83C22×D411yescharacteristic
121C2×D83—21no—
122C2×D83—21no—
123C2×D83—24no—
124C2×D83—42no—
125C2×D83—42no—
126C2×D83—88no—
127C2×D83#13512yescomplemented
128C4×C83—21no—
129C4×C83—42no—
130C4×C83C22×D411yescharacteristic
131C4×C83H(5,𝔽2)11yescharacteristic
132Dic83—21no—
133Dic83—21no—
134Dic83—21no—
135Dic83—42no—
136C2×C83—41no—
137C2×C83—82no—
138C2×C83D4211yescharacteristic
139C2×C83C2×#13511yescharacteristic
140C2×C83#57611yescharacteristic
141D83—22no—
142D83—44no—
143#3280—831nomaximal, Sylow
144C83#328011yescharacteristic, complemented, Sylow
145C2×#135—831no—
146C22×D8—831no—
147#546—832no—
148#495—831no—
149#467—831no—
150#466D8311yescharacteristic, complemented
151#463—832no—
152#450—832no—
153#449—832no—
154C4×D8—831no—
155C8×D4—831no—
156C2×#41—832no—
157C22×D4—832no—
158#135—1664no—
159C2×Q16—831no—
160C2×SD16—832no—
161C2×SD16—832no—
162C2×D8—832no—
163C2×D8—831no—
164C2×D8—1664no—
165C2×D8C2×D8311yescharacteristic
166C22×C8—831no—
167C22×C8C2×D8311yescharacteristic, complemented
168#124—832no—
169#123—1662no—
170C22⋊Q8—832no—
171#121—1662no—
172#121C2×D8312yescomplemented
173#120—1662no—
174C4×D4—831no—
175C4×D4—832no—
176C2×C22⋊C4—1662no—
177#107C2×D8311yescharacteristic, complemented
178C4⋊C8—831no—
179#103—832no—
180#102—832no—
181#102C2×D8312yescomplemented
182C22⋊C8—831no—
183C22⋊C8—831no—
184C4×C8—831no—
185C24—1662no—
186#41—3322no—
187#41—1664no—
188C2×Q8—832no—
189C2×D4—22no—
190C2×D4—3322no—
191C2×D4—831no—
192C2×D4—832no—
193C2×D4—1662no—
194C2×D4—1664no—
195C2×D4—1662no—
196C2×D4C22×D8312yes—
197C22×C4—831no—
198C22×C4—1662no—
199C22×C4—1662no—
200C22×C4C22×D8311yescharacteristic
201Q16—1662no—
202SD16—3322no—
203SD16—1664no—
204D8—22no—
205D8—834no—
206D8—1664no—
207C2×C8—22no—
208C2×C8—831no—
209C2×C8—831no—
210C2×C8—1662no—
211C2×C8C22×D8311yescharacteristic
212C2×C8C22×D8311yescharacteristic
213C4⋊C4—831no—
214C4⋊C4—1662no—
215C4⋊C4C22×D8312yes—
216C22⋊C4—22no—
217C22⋊C4—831no—
218C22⋊C4—831no—
219C22⋊C4—1662no—
220C22⋊C4—1662no—
221C22⋊C4—1662no—
222C22⋊C4—1662no—
223C42—831no—
224C23—22no—
225C23—3324no—
226C23—3324no—
227C23—831no—
228C23—6642no—
229C23—1662no—
230C23D4×D8311yescharacteristic
231Q8—1662no—
232Q8—1662no—
233D4—22no—
234D4—22no—
235D4—82no—
236D4—3324no—
237D4—3324no—
238D4—834no—
239D4—1662no—
240D4—1662no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—3324no—
245C2×C4—831no—
246C2×C4—831no—
247C2×C4—831no—
248C2×C4—6642no—
249C2×C4—1662no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4D4×D8311yescharacteristic
253C2×C4C23×D8311yescharacteristic
254C8—21no—
255C8—1661no—
256C8—1661no—
257C8D4×D8312yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—3322no—
265C22—3322no—
266C22—832no—
267C22—6648no—
268C22—1664no—
269C22—1661no—
270C22—1661no—
271C22C2×D4×D8311yescharacteristic
272C4—21no—
273C4—42no—
274C4—3322no—
275C4—1661no—
276C4—1661no—
277C4—1661no—
278C4C2×D4×D8311yescharacteristic
279C4#1458183811yescharacteristic
280C2—41no—
281C2—82no—
282C2—3324no—
283C2—1662no—
284C2#1912833511yescharacteristic
285C2#135×D8311yescharacteristic
286C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#466×C83C2
socleC22×C83C22×D4
hypercenter#466D83
hypocenterC83#3280
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#14581814C2semidirect
2. C2×D8×D83C2semidirect
3. #19128348C2semidirect
4. #19128084C2semidirect
5. #19127943C2semidirect
6. #19127915C2semidirect
7. #19127853C2semidirect
8. #19127844C2semidirect
9. #19127783C2semidirect
10. C8×C22⋊D83C2semidirect
11. #466×C83C2semidirect
12. C2×#13050590C22semidirect
15. C2×Dic332C22semidirect
16. C2×#13050579C22semidirect
17. C2×#13050578C22semidirect
22. #14581779C22semidirect
27. #14581754C22semidirect
30. Dic83⋊C8C22semidirect
31. #14581715C22semidirect
32. #14581712C22semidirect
33. C22⋊(C83⋊C8)C22semidirect
35. C8×Dic83C22semidirect
37. C22×C8×C83C22semidirect
38. #121×C83C22semidirect
39. #107×C83C22semidirect
40. #102×C83C22semidirect
57. D664D4semidirect
127. C2×D83#135semidirect
144. C83#3280semidirect
150. #466D83semidirect
167. C22×C8C2×D83semidirect
172. #121C2×D83semidirect
177. #107C2×D83semidirect
181. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818142
C2×D8×D832
#191283482
#191280842
#191279432
#191279152
#191278532
#191278442
#191277832
C8×C22⋊D832
#466×C832
#328083