Finite Group Databasebeta

← #24919706

Subgroups — #24919706

Order 10624.

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

normal subgroups only

This group has 460 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
1#19128416C211yesmaximal, characteristic, complemented
2D4×D332C211yesmaximal, characteristic, complemented
3#19128048C211yesmaximal, characteristic, complemented
4#19128026C212yesmaximal, complemented
5#19127992C211yesmaximal, characteristic, complemented
6#19127954C212yesmaximal, complemented
7#19127897C211yesmaximal, characteristic, complemented
8#19127837C212yesmaximal, complemented
9C4×D664C211yesmaximal, characteristic, complemented
10#19127707C212yesmaximal, complemented
11#436×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C2×D4×D83—21no—
14C2×D664C2211yescharacteristic
15C2×D664C2212yescomplemented
16#14581792C2211yescharacteristic, complemented
17#14581789C2211yescharacteristic
18#14581778—22no—
19#14581767—21no—
20#14581766C2212yes—
21#14581766C2212yescomplemented
22#14581763—21no—
23#14581762—22no—
24C4×D332C2211yescharacteristic
25C4×D332C2211yescharacteristic, complemented
26#14581751C2212yescomplemented
27#14581750—21no—
28#14581726C2211yescharacteristic, complemented
29#14581724C2212yescomplemented
30#14581714C2212yes—
31#14581714C2211yescharacteristic
32#14581714C2212yescomplemented
33#14581714C2212yescomplemented
34#14581714C2211yescharacteristic, complemented
35#14581714C2212yescomplemented
36C2×M16×C83C2212yescomplemented
37C4×D4×C83C2211yescharacteristic
38C4⋊C8×C83C2211yescharacteristic, complemented
39C22⋊C8×C83C2212yescomplemented
40C4×C8×C83C2211yescharacteristic, complemented
41C23×D83—22no—
42C2×Dic166—21no—
43C2×D332—24no—
44C2×D332—22no—
45C2×D332—21no—
46C2×D332—21no—
47C2×D332—42no—
48C2×D332C2312yes—
49C2×D332C2311yescharacteristic
50C2×D332C2311yescharacteristic
51C2×C22⋊D83—22no—
52C2×C22⋊D83—22no—
53D4×D83—44no—
54C2×C4×D83—21no—
55C2×C4×D83—21no—
56D664—22no—
57D664—42no—
58C4⋊Dic83—21no—
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2312yes—
61C4⋊Dic83C2311yescharacteristic
62C22⋊Dic83—22no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4—21no—
67C4×Dic83—22no—
68C2×D4×C83D411yescharacteristic
69C22×C4×C83C2312yes—
70M16×C83—42no—
71C2×C8×C83C2311yescharacteristic
72C2×C8×C83C2312yes—
73C2×C8×C83C2311yescharacteristic
74C4⋊C4×C83D411yescharacteristic
75C22⋊C4×C83D412yes—
76C42×C83C2311yescharacteristic
77C23×C83C2×D412yes—
78C22×D83—21no—
79C22×D83—21no—
80C22×D83—22no—
81C22×D83—44no—
82C22×D83—44no—
83C22×D83—82no—
84D4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C22⋊D83—44no—
92C22⋊D83—82no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—21no—
96D332—21no—
97D332—22no—
98D332—21no—
99D332—21no—
100D332—22no—
101D332—44no—
102D332—42no—
103D332—81no—
104D332#4112yes—
105C4×D83—42no—
106C4×D83—42no—
107Dic166—81no—
108C8×C83—21no—
109C8×C83—21no—
110C8×C83—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83C22×D411yescharacteristic
115C2×D83—22no—
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—42no—
119C2×D83—41no—
120C2×D83—41no—
121C2×D83—42no—
122C2×D83—42no—
123C2×D83—41no—
124C2×D83—88no—
125C4×C83—22no—
126C4×C83—21no—
127C4×C83AGL(1,ℤ/8)12yes—
128C4×C83C2×#4111yescharacteristic
129C4×C83H(5,𝔽2)11yescharacteristic
130Dic83—41no—
131Dic83—41no—
132Dic83—42no—
133C2×C83—42no—
134C2×C83#54611yescharacteristic
135C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C83#57611yescharacteristic
137D83—42no—
138D83—81no—
139D83—82no—
140#3325—831nomaximal, Sylow
141C83#332511yescharacteristic, complemented, Sylow
142#555—831no—
143D42—831no—
144#486—831no—
145#482—832no—
146#476—831no—
147#469—832no—
148#459—831no—
149#447—832no—
150C4×D8—831no—
151#436D8311yescharacteristic, complemented
152#418—832no—
153C22×D4—832no—
154C22×D4—1661no—
155C2×D8—832no—
156C2×D8—831no—
157C2×M16C2×D8312yescomplemented
158C4⋊Q8—831no—
159#127—831no—
160#124—1662no—
161#121—832no—
162#121—832no—
163#121—1661no—
164#121—1661no—
165#120—1662no—
166C4×D4—831no—
167C4×D4—831no—
168C4×D4C2×D8311yescharacteristic
169#117—832no—
170#117—1661no—
171#107—831no—
172#106—832no—
173C4⋊C8C2×D8311yescharacteristic, complemented
174#102—831no—
175#102—832no—
176#102—832no—
177#102—832no—
178#102—831no—
179#102—832no—
180C22⋊C8C2×D8312yescomplemented
181C4×C8C2×D8311yescharacteristic, complemented
182C24—1662no—
183C2×Q8—1661no—
184C2×D4—3322no—
185C2×D4—3324no—
186C2×D4—831no—
187C2×D4—832no—
188C2×D4—831no—
189C2×D4—1664no—
190C2×D4—1662no—
191C2×D4—1662no—
192C2×D4—1662no—
193C2×D4—1661no—
194C2×D4—1661no—
195C2×D4D33211yescharacteristic
196C22×C4—1661no—
197C22×C4—1661no—
198C22×C4C22×D8312yes—
199D8—3322no—
200D8—1662no—
201M16—42no—
202C2×C8C22×D8311yescharacteristic
203C2×C8C22×D8311yescharacteristic
204C2×C8C22×D8312yes—
205C4⋊C4—832no—
206C4⋊C4—831no—
207C4⋊C4—831no—
208C4⋊C4—1661no—
209C4⋊C4D33211yescharacteristic
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C22⋊C4—1661no—
213C22⋊C4—1661no—
214C22⋊C4—1662no—
215C22⋊C4D33212yes—
216C42—1662no—
217C42C22×D8311yescharacteristic
218C23—3324no—
219C23—3324no—
220C23—6642no—
221C23—1661no—
222C23—1662no—
223C23—1661no—
224C23C2×D33212yes—
225Q8—6641no—
226D4—22no—
227D4—3324no—
228D4—3324no—
229D4—3322no—
230D4—832no—
231D4—6641no—
232D4—6642no—
233D4—1661no—
234D4—1662no—
235D4—1661no—
236D4—1662no—
237D4—1661no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—3322no—
241C2×C4—3322no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4—1661no—
245C2×C4C2×D33212yes—
246C2×C4C2×D33211yescharacteristic
247C2×C4C2×D33211yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—21no—
250C8—21no—
251C8—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—3321no—
256C22—3321no—
257C22—3322no—
258C22—3321no—
259C22—3321no—
260C22—3322no—
261C22—3321no—
262C22—3322no—
263C22—6648no—
264C22—1662no—
265C22C22×D33211yescharacteristic
266C4—22no—
267C4—21no—
268C4—3321no—
269C4—3321no—
270C4—3322no—
271C4#1458182012yes—
272C4#41×D8311yescharacteristic
273C4#1458183711yescharacteristic
274C2—42no—
275C2—3322no—
276C2—6642no—
277C2—6641no—
278C2#1912834611yescharacteristic
279C2C2×#1458182011yescharacteristic
280C2#1912850611yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#436×C83C2
socleC22×C83C22×D4
hypercenter#436D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #19128416C2semidirect
2. D4×D332C2semidirect
3. #19128048C2semidirect
4. #19128026C2semidirect
5. #19127992C2semidirect
6. #19127954C2semidirect
7. #19127897C2semidirect
8. #19127837C2semidirect
9. C4×D664C2semidirect
10. #19127707C2semidirect
11. #436×C83C2semidirect
15. C2×D664C22semidirect
16. #14581792C22semidirect
21. #14581766C22semidirect
25. C4×D332C22semidirect
26. #14581751C22semidirect
28. #14581726C22semidirect
29. #14581724C22semidirect
32. #14581714C22semidirect
33. #14581714C22semidirect
34. #14581714C22semidirect
35. #14581714C22semidirect
36. C2×M16×C83C22semidirect
38. C4⋊C8×C83C22semidirect
39. C22⋊C8×C83C22semidirect
40. C4×C8×C83C22semidirect
141. C83#3325semidirect
151. #436D83semidirect
157. C2×M16C2×D83semidirect
173. C4⋊C8C2×D83semidirect
180. C22⋊C8C2×D83semidirect
181. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284162
D4×D3322
#191280482
#191280262
#191279922
#191279542
#191278972
#191278372
C4×D6642
#191277072
#436×C832
#332583