Finite Group Databasebeta

← #24919705

Subgroups — #24919705

Order 10624.

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

normal subgroups only

This group has 460 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
1#19128416C211yesmaximal, characteristic, complemented
2#19128047C211yesmaximal, characteristic, complemented
3#19128022C212yesmaximal, complemented
4#19127991C211yesmaximal, characteristic, complemented
5#19127952C212yesmaximal, complemented
6#19127893C211yesmaximal, characteristic, complemented
7#19127835C212yesmaximal, complemented
8C4×#13050575C211yesmaximal, characteristic, complemented
9#19127779C211yesmaximal, characteristic, complemented
10#19127706C212yesmaximal, complemented
11D42×C83C211yesmaximal, characteristic, complemented
12C2×#13050575C2211yescharacteristic
13C2×#13050575C2212yescomplemented
14C2×#13050574C2212yescomplemented
15#14581792C2211yescharacteristic, complemented
16#14581778—22no—
17#14581766C2212yescomplemented
18#14581763—21no—
19C4×D332C2211yescharacteristic, complemented
20#14581751C2212yescomplemented
21#14581750—21no—
22#14581725C2211yescharacteristic, complemented
23#14581723C2212yescomplemented
24C4⋊(C83⋊C8)C2211yescharacteristic, complemented
25#14581713C2212yes—
26#14581713C2211yescharacteristic
27#14581713C2212yescomplemented
28#14581713C2212yescomplemented
29#14581713C2212yescomplemented
30#14581712C2211yescharacteristic, complemented
31C22⋊(C83⋊C8)C2212yescomplemented
32C4×C83⋊C8C2211yescharacteristic, complemented
33C22×D4×C83—21no—
34C22×D4×C83C2212yes—
35#127×C83C2211yescharacteristic
36#121×C83—21no—
37#121×C83C2212yes—
38#120×C83—22no—
39C4×D4×C83C2211yescharacteristic
40C4×D4×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×D332C2311yescharacteristic
43C2×C22⋊D83—22no—
44C2×C4×D83—21no—
45#13050575—22no—
46#13050575—42no—
47#13050574—42no—
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2312yes—
50C2×C83⋊C8C2311yescharacteristic
51C4⋊Dic83—21no—
52C4⋊Dic83C2312yes—
53C4⋊Dic83C2311yescharacteristic
54C22⋊Dic83—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—21no—
57C4×Dic83—22no—
58C24×C83—22no—
59C2×D4×C83—24no—
60C2×D4×C83—22no—
61C2×D4×C83—22no—
62C2×D4×C83—21no—
63C2×D4×C83—21no—
64C2×D4×C83—42no—
65C2×D4×C83—44no—
66C2×D4×C83D411yescharacteristic
67C2×D4×C83C2312yes—
68C2×D4×C83C2311yescharacteristic
69C22×C4×C83—21no—
70C22×C4×C83C2312yes—
71C4⋊C4×C83D411yescharacteristic
72C4⋊C4×C83C2311yescharacteristic
73C22⋊C4×C83—21no—
74C22⋊C4×C83—22no—
75C22⋊C4×C83D412yes—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83—22no—
79C23×C83—44no—
80C23×C83—44no—
81C23×C83—82no—
82C23×C83C2×D412yes—
83C22×D83—21no—
84D4×C83—22no—
85D4×C83—22no—
86D4×C83—21no—
87D4×C83—22no—
88D4×C83—44no—
89D4×C83—44no—
90D4×C83—42no—
91D4×C83—81no—
92D4×C83#4112yes—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—42no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D412yes—
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—82no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104D332—21no—
105D332—21no—
106C4×D83—42no—
107Dic166—81no—
108C83⋊C8—21no—
109C83⋊C8—21no—
110C83⋊C8—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83—22no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—88no—
121C22×C83C22×D411yescharacteristic
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—41no—
125C4×C83—22no—
126C4×C83—21no—
127C4×C83—41no—
128C4×C83AGL(1,ℤ/8)12yes—
129C4×C83C2×#4111yescharacteristic
130C4×C83H(5,𝔽2)11yescharacteristic
131Dic83—41no—
132Dic83—42no—
133C2×C83—42no—
134C2×C83—42no—
135C2×C83—82no—
136C2×C83#54611yescharacteristic
137C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C83#57611yescharacteristic
139D83—81no—
140#3325—831nomaximal, Sylow
141C83#332511yescharacteristic, complemented, Sylow
142#555—831no—
143D42D8311yescharacteristic, complemented
144#486—831no—
145#482—832no—
146#476—831no—
147#469—832no—
148#459—831no—
149#447—832no—
150C4×D8—831no—
151#436—831no—
152#418—832no—
153C22×D4—21no—
154C22×D4C2×D8312yes—
155C2×D8—832no—
156C2×D8—831no—
157C2×M16—832no—
158C4⋊Q8—831no—
159#127C2×D8311yescharacteristic
160#124—1662no—
161#121—21no—
162#121—832no—
163#121—1661no—
164#121C2×D8312yes—
165#120—22no—
166C4×D4—831no—
167C4×D4C2×D8311yescharacteristic
168C4×D4C2×D8311yescharacteristic
169#117—832no—
170#117—1661no—
171#107—831no—
172#106—832no—
173C4⋊C8—831no—
174#102—831no—
175#102—832no—
176#102—832no—
177#102—832no—
178#102—832no—
179#102—831no—
180C22⋊C8—832no—
181C4×C8—831no—
182C24—22no—
183C2×Q8—1661no—
184C2×D4—24no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—831no—
192C2×D4—1662no—
193C2×D4C22⋊D8311yescharacteristic
194C2×D4C22×D8312yes—
195C2×D4C22×D8311yescharacteristic
196C22×C4—21no—
197C22×C4—1661no—
198C22×C4C22×D8312yes—
199D8—3322no—
200D8—1662no—
201M16—3322no—
202C2×C8—831no—
203C2×C8—831no—
204C2×C8—832no—
205C4⋊C4—832no—
206C4⋊C4—831no—
207C4⋊C4—1661no—
208C4⋊C4C22⋊D8311yescharacteristic
209C4⋊C4C22×D8311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—22no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C22⋊C4—1661no—
215C22⋊C4C22⋊D8312yes—
216C42—1662no—
217C42C22×D8311yescharacteristic
218C23—21no—
219C23—22no—
220C23—44no—
221C23—44no—
222C23—82no—
223C23—1661no—
224C23C2×C22⋊D8312yes—
225Q8—6641no—
226D4—22no—
227D4—22no—
228D4—21no—
229D4—22no—
230D4—44no—
231D4—44no—
232D4—42no—
233D4—81no—
234D4—6642no—
235D4—1661no—
236D4—1661no—
237D4#1305058912yes—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—3322no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4C2×C22⋊D8312yes—
246C2×C4C2×C22⋊D8311yescharacteristic
247C2×C4C2×C22⋊D8311yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—1661no—
250C8—1661no—
251C8—1662no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—88no—
262C22—3321no—
263C22—3321no—
264C22—3321no—
265C22C22×C22⋊D8311yescharacteristic
266C4—22no—
267C4—21no—
268C4—41no—
269C4—3321no—
270C4—3322no—
271C4#1458181812yes—
272C4C2×#1305058911yescharacteristic
273C4#1458183711yescharacteristic
274C2—42no—
275C2—42no—
276C2—82no—
277C2—6641no—
278C2#1912834511yescharacteristic
279C2C2×#1458181811yescharacteristic
280C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingD42×C83C2
socleC22×C83C22×D4
hypercenterD42D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #19128416C2semidirect
2. #19128047C2semidirect
3. #19128022C2semidirect
4. #19127991C2semidirect
5. #19127952C2semidirect
6. #19127893C2semidirect
7. #19127835C2semidirect
8. C4×#13050575C2semidirect
9. #19127779C2semidirect
10. #19127706C2semidirect
11. D42×C83C2semidirect
13. C2×#13050575C22semidirect
14. C2×#13050574C22semidirect
15. #14581792C22semidirect
17. #14581766C22semidirect
19. C4×D332C22semidirect
20. #14581751C22semidirect
22. #14581725C22semidirect
23. #14581723C22semidirect
24. C4⋊(C83⋊C8)C22semidirect
27. #14581713C22semidirect
28. #14581713C22semidirect
29. #14581713C22semidirect
30. #14581712C22semidirect
31. C22⋊(C83⋊C8)C22semidirect
32. C4×C83⋊C8C22semidirect
141. C83#3325semidirect
143. D42D83semidirect

Maximal subgroups

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

subgroupindex
#191284162
#191280472
#191280222
#191279912
#191279522
#191278932
#191278352
C4×#130505752
#191277792
#191277062
D42×C832
#332583