Finite Group Databasebeta

← #24919142

Subgroups — #24919142

Order 10624.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×D8×D83C211yesmaximal, characteristic, complemented
2#19128339C212yesmaximal, complemented
3#19128086C211yesmaximal, characteristic, complemented
4#19127939C211yesmaximal, characteristic, complemented
5#19127937C211yesmaximal, characteristic, complemented
6#19127918C212yesmaximal, complemented
7#19127850C212yesmaximal, complemented
8#19127846C212yesmaximal, complemented
9D8×Dic83C211yesmaximal, characteristic, complemented
10C8×C22⋊D83C211yesmaximal, characteristic, complemented
11C2×#135×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83C2212yes—
13C2×#13050577C2212yescomplemented
14C2×D664C2211yescharacteristic
15C2×#13050575C2212yes—
16D8×D83—24no—
17C2×C8×D83C2211yescharacteristic
18#14581781C2212yescomplemented
19#14581775—22no—
20#14581768C2212yes—
21#14581766C2212yescomplemented
22#14581764—22no—
23#14581760—22no—
24D4×Dic83C2212yescomplemented
25C4×C22⋊D83C2211yescharacteristic
26C2×(C2×D83)⋊C4—22no—
27#14581726C2211yescharacteristic, complemented
28Dic83⋊C8C2211yescharacteristic, complemented
29#14581716C2212yescomplemented
30#14581714C2212yescomplemented
31#14581713C2212yescomplemented
32C22⋊(C83⋊C8)C2211yescharacteristic, complemented
33(C2×D83)⋊C8C2211yescharacteristic
34C8×Dic83C2211yescharacteristic, complemented
35C2×#41×C83C2212yescomplemented
36#135×C83—24no—
37C2×Q16×C83C2211yescharacteristic, complemented
38C2×SD16×C83C2212yescomplemented
39C2×D8×C83C2211yescharacteristic
40C22×C8×C83C2211yescharacteristic, complemented
41C23×D83—22no—
42C2×D332C2312yes—
43C2×C22⋊D83—22no—
44C2×C22⋊D83—22no—
45C2×C22⋊D83D411yescharacteristic
46D4×D83—24no—
47D4×D83—42no—
48C22×Dic83—22no—
49C2×C4×D83C2311yescharacteristic
50#13050577—42no—
51D664—22no—
52#13050575—24no—
53C2×C83⋊C8C2311yescharacteristic
54C2×C83⋊C8C2311yescharacteristic
55C8×D83—22no—
56C4⋊Dic83C2312yes—
57Dic83⋊C4—22no—
58Dic83⋊C4D411yescharacteristic
59C22⋊Dic83—22no—
60C22⋊Dic83D411yescharacteristic
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4D411yescharacteristic
66C4×Dic83C2311yescharacteristic
67#41×C83—24no—
68#41×C83—42no—
69C2×Q8×C83C2312yes—
70C2×D4×C83—22no—
71C2×D4×C83C2312yes—
72C22×C4×C83—22no—
73C22×C4×C83C2311yescharacteristic
74Q16×C83—22no—
75SD16×C83—24no—
76D8×C83D414yescomplemented
77C2×C8×C83—22no—
78C2×C8×C83C2311yescharacteristic
79C2×C8×C83C2311yescharacteristic
80C23×C83—22no—
81C23×C83C2×D411yescharacteristic
82C22×D83—22no—
83C22×D83—44no—
84C22×D83—44no—
85C22×D83—82no—
86C22×D83C2×D411yescharacteristic
87Q8×C83—22no—
88Q8×C83—22no—
89D4×C83—22no—
90D4×C83—44no—
91D4×C83C2×D414yes—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—44no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—22no—
99C22⋊D83—44no—
100C22⋊D83—82no—
101C2×Dic83—22no—
102C2×Dic83—82no—
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105C2×Dic83C2×D411yescharacteristic
106D332—22no—
107D332—22no—
108C4×D83—21no—
109C4×D83—21no—
110C8×C83—21no—
111C8×C83C2×D412yes—
112C83⋊C8—21no—
113C83⋊C8—21no—
114C22×C83—24no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83C22×D411yescharacteristic
121C2×D83—21no—
122C2×D83—21no—
123C2×D83—42no—
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—44no—
137C2×C83—41no—
138C2×C83D4211yescharacteristic
139C2×C83C2×#13511yescharacteristic
140C2×C83#57611yescharacteristic
141D83—22no—
142D83—82no—
143#3280—831nomaximal, Sylow
144C83#328011yescharacteristic, complemented, Sylow
145C2×#135D8311yescharacteristic, complemented
146C22×D8—831no—
147#546—832no—
148#495—831no—
149#467—831no—
150#466—831no—
151#463—832no—
152#450—832no—
153#449—832no—
154C4×D8—831no—
155C8×D4—831no—
156C2×#41C2×D8312yescomplemented
157C22×D4—832no—
158#135—24no—
159C2×Q16C2×D8311yescharacteristic, complemented
160C2×SD16—832no—
161C2×SD16C2×D8312yescomplemented
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—832no—
172#121—1662no—
173#120—1662no—
174C4×D4—831no—
175C4×D4—832no—
176C2×C22⋊C4—1662no—
177#107—831no—
178C4⋊C8—831no—
179#103—832no—
180#102—832no—
181#102—832no—
182C22⋊C8—831no—
183C22⋊C8—831no—
184C4×C8—831no—
185C24—1662no—
186#41—24no—
187#41—42no—
188C2×Q8C22×D8312yes—
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—22no—
198C22×C4—831no—
199C22×C4—1662no—
200C22×C4C22×D8311yescharacteristic
201Q16—22no—
202SD16—24no—
203SD16—3322no—
204D8—1664no—
205D8—1662no—
206D8C22⋊D8314yescomplemented
207C2×C8—22no—
208C2×C8—831no—
209C2×C8—831no—
210C2×C8—1662no—
211C2×C8C22×D8311yescharacteristic
212C2×C8C22×D8311yescharacteristic
213C4⋊C4—831no—
214C4⋊C4—832no—
215C4⋊C4—1662no—
216C22⋊C4—831no—
217C22⋊C4—831no—
218C22⋊C4—1662no—
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—22no—
232Q8—22no—
233D4—22no—
234D4—44no—
235D4—3324no—
236D4—6642no—
237D4—1662no—
238D4—1662no—
239D4—1662no—
240D4C2×C22⋊D8314yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
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—
257C8C2×C22⋊D8312yes—
258C22—21no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—3322no—
265C22—3322no—
266C22—3322no—
267C22—832no—
268C22—6648no—
269C22—1661no—
270C22—1661no—
271C22C2×D4×D8311yescharacteristic
272C4—21no—
273C4—42no—
274C4—3322no—
275C4—1661no—
276C4—1661no—
277C4—1661no—
278C4C22×C22⋊D8311yescharacteristic
279C4#1458183711yescharacteristic
280C2—41no—
281C2—44no—
282C2—6642no—
283C2—1662no—
284C2D4×C22⋊D8311yescharacteristic
285C2#135×D8311yescharacteristic
286C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#135×C83C2
socleC22×C83C22×D4
hypercenterC2×#135D83
hypocenterC83#3280
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×D8×D83C2semidirect
2. #19128339C2semidirect
3. #19128086C2semidirect
4. #19127939C2semidirect
5. #19127937C2semidirect
6. #19127918C2semidirect
7. #19127850C2semidirect
8. #19127846C2semidirect
9. D8×Dic83C2semidirect
10. C8×C22⋊D83C2semidirect
11. C2×#135×C83C2semidirect
13. C2×#13050577C22semidirect
18. #14581781C22semidirect
21. #14581766C22semidirect
24. D4×Dic83C22semidirect
27. #14581726C22semidirect
28. Dic83⋊C8C22semidirect
29. #14581716C22semidirect
30. #14581714C22semidirect
31. #14581713C22semidirect
32. C22⋊(C83⋊C8)C22semidirect
34. C8×Dic83C22semidirect
35. C2×#41×C83C22semidirect
37. C2×Q16×C83C22semidirect
38. C2×SD16×C83C22semidirect
40. C22×C8×C83C22semidirect
76. D8×C83D4semidirect
127. C2×D83#135semidirect
144. C83#3280semidirect
145. C2×#135D83semidirect
156. C2×#41C2×D83semidirect
159. C2×Q16C2×D83semidirect
161. C2×SD16C2×D83semidirect
167. C22×C8C2×D83semidirect
206. D8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D832
#191283392
#191280862
#191279392
#191279372
#191279182
#191278502
#191278462
D8×Dic832
C8×C22⋊D832
C2×#135×C832
#328083