Finite Group Databasebeta

← #24919704

Subgroups — #24919704

Order 10624.

460 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128417C211yesmaximal, characteristic, complemented
2#19128336C211yesmaximal, characteristic, complemented
3#19128023C212yesmaximal, complemented
4#19127988C211yesmaximal, characteristic, complemented
5#19127953C212yesmaximal, complemented
6#19127893C211yesmaximal, characteristic, complemented
7#19127836C212yesmaximal, complemented
8C4×D664C211yesmaximal, characteristic, complemented
9#19127780C211yesmaximal, characteristic, complemented
10#19127704C212yesmaximal, complemented
11#486×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—21no—
13C2×D4×D83C2212yes—
14C2×D664C2211yescharacteristic
15C2×#13050575C2212yescomplemented
16C2×#13050573C2212yescomplemented
17#14581781—22no—
18#14581767—21no—
19#14581767C2212yescomplemented
20#14581763—21no—
21#14581763C2212yes—
22#14581760—22no—
23C4×D332C2211yescharacteristic
24C4×D332C2211yescharacteristic
25C4×D332C2211yescharacteristic, complemented
26#14581750—21no—
27#14581750C2212yescomplemented
28#14581726C2211yescharacteristic, complemented
29#14581723C2212yescomplemented
30C4⋊(C83⋊C8)C2211yescharacteristic, complemented
31#14581714C2211yescharacteristic
32#14581714C2211yescharacteristic, complemented
33#14581713C2212yescomplemented
34#14581712C2212yes—
35(C2×D83)⋊C8C2212yescomplemented
36C4⋊Q8×C83C2211yescharacteristic, complemented
37#127×C83C2211yescharacteristic
38#102×C83C2212yescomplemented
39#102×C83C2212yescomplemented
40C4×C8×C83C2211yescharacteristic, complemented
41C23×D83—22no—
42C2×D332—22no—
43C2×D332D411yescharacteristic
44C2×D332C2311yescharacteristic
45C2×D332C2311yescharacteristic
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48D4×D83—24no—
49D4×D83—42no—
50D4×D83—44no—
51C2×C4×D83—21no—
52C2×C4×D83—21no—
53C2×C4×D83C2312yes—
54D664—22no—
55#13050575—42no—
56#13050573—42no—
57C2×C83⋊C8C2312yes—
58C4⋊Dic83D411yescharacteristic
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2311yescharacteristic
61C22⋊Dic83—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4D412yes—
67C4×Dic83—22no—
68C2×Q8×C83—21no—
69C2×D4×C83—21no—
70C2×D4×C83—21no—
71C2×D4×C83C2312yes—
72C2×C8×C83C2311yescharacteristic
73C2×C8×C83C2311yescharacteristic
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C42×C83C2311yescharacteristic
77C23×C83—22no—
78C22×D83—21no—
79C22×D83—21no—
80C22×D83—44no—
81C22×D83—44no—
82C22×D83—82no—
83C22×D83C2×D412yes—
84Q8×C83—81no—
85D4×C83—22no—
86D4×C83—22no—
87D4×C83—42no—
88D4×C83—81no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—44no—
94C22⋊D83—44no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83C2×D412yes—
98D332—22no—
99D332—21no—
100D332—21no—
101D332—21no—
102D332—82no—
103D332#4112yes—
104C4×D83—22no—
105C4×D83—22no—
106C4×D83—42no—
107C4×D83—42no—
108C8×C83—21no—
109C8×C83—21no—
110C83⋊C8—22no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83C22×D411yescharacteristic
115C2×D83—22no—
116C2×D83—22no—
117C2×D83—22no—
118C2×D83—22no—
119C2×D83—41no—
120C2×D83—41no—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—88no—
125C4×C83—21no—
126C4×C83—42no—
127C4×C83AGL(1,ℤ/8)12yes—
128C4×C83C2×#4111yescharacteristic
129C4×C83H(5,𝔽2)11yescharacteristic
130Dic83—22no—
131Dic83—41no—
132Dic83—41no—
133C2×C83—82no—
134C2×C83#54611yescharacteristic
135C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C83#57611yescharacteristic
137D83—42no—
138D83—42no—
139D83—81no—
140#3325—831nomaximal, Sylow
141C83#332511yescharacteristic, complemented, Sylow
142#555—831no—
143D42—831no—
144#486D8311yescharacteristic, complemented
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—832no—
154C22×D4—1661no—
155C2×D8—832no—
156C2×D8—831no—
157C2×M16—832no—
158C4⋊Q8C2×D8311yescharacteristic, complemented
159#127C2×D8311yescharacteristic
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×D4—831no—
169#117—832no—
170#117—1661no—
171#107—831no—
172#106—832no—
173C4⋊C8—831no—
174#102—832no—
175#102—831no—
176#102—831no—
177#102—832no—
178#102C2×D8312yescomplemented
179#102C2×D8312yescomplemented
180C22⋊C8—832no—
181C4×C8C2×D8311yescharacteristic, complemented
182C24—1662no—
183C2×Q8—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—3324no—
187C2×D4—3322no—
188C2×D4—831no—
189C2×D4—831no—
190C2×D4—831no—
191C2×D4—1662no—
192C2×D4—1662no—
193C2×D4—1664no—
194C2×D4—1662no—
195C2×D4C22×D8312yes—
196C22×C4—832no—
197C22×C4—1661no—
198C22×C4—1661no—
199D8—3322no—
200D8—1662no—
201M16—3322no—
202C2×C8—832no—
203C2×C8C22×D8311yescharacteristic
204C2×C8C22×D8311yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—831no—
207C4⋊C4—831no—
208C4⋊C4—831no—
209C4⋊C4C22×D8312yes—
210C22⋊C4—832no—
211C22⋊C4—1662no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C22⋊C4—1661no—
215C22⋊C4—1661no—
216C42—1662no—
217C42C22×D8311yescharacteristic
218C23—22no—
219C23—3324no—
220C23—3324no—
221C23—832no—
222C23—6642no—
223C23—1661no—
224C23—1661no—
225Q8—81no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—81no—
230D4—3324no—
231D4—3324no—
232D4—832no—
233D4—6642no—
234D4—1662no—
235D4—1661no—
236D4—1661no—
237D4—1661no—
238C2×C4—22no—
239C2×C4—3322no—
240C2×C4—3322no—
241C2×C4—832no—
242C2×C4—1661no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4D4×D8311yescharacteristic
247C2×C4D4×D8311yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—21no—
250C8—21no—
251C8—1662no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—3321no—
256C22—3321no—
257C22—3321no—
258C22—3321no—
259C22—3321no—
260C22—6648no—
261C22—1662no—
262C22—1662no—
263C22—1662no—
264C22—1662no—
265C22C2×D4×D8311yescharacteristic
266C4—21no—
267C4—42no—
268C4—3321no—
269C4—3321no—
270C4—1662no—
271C4#1458181912yes—
272C4#41×D8311yescharacteristic
273C4#1458183811yescharacteristic
274C2—82no—
275C2—3322no—
276C2—3322no—
277C2—6641no—
278C2#1912834411yescharacteristic
279C2C2×#1458181911yescharacteristic
280C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#486×C83C2
socleC22×C83C22×D4
hypercenter#486D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #19128417C2semidirect
2. #19128336C2semidirect
3. #19128023C2semidirect
4. #19127988C2semidirect
5. #19127953C2semidirect
6. #19127893C2semidirect
7. #19127836C2semidirect
8. C4×D664C2semidirect
9. #19127780C2semidirect
10. #19127704C2semidirect
11. #486×C83C2semidirect
15. C2×#13050575C22semidirect
16. C2×#13050573C22semidirect
19. #14581767C22semidirect
25. C4×D332C22semidirect
27. #14581750C22semidirect
28. #14581726C22semidirect
29. #14581723C22semidirect
30. C4⋊(C83⋊C8)C22semidirect
32. #14581714C22semidirect
33. #14581713C22semidirect
35. (C2×D83)⋊C8C22semidirect
36. C4⋊Q8×C83C22semidirect
38. #102×C83C22semidirect
39. #102×C83C22semidirect
40. C4×C8×C83C22semidirect
141. C83#3325semidirect
144. #486D83semidirect
158. C4⋊Q8C2×D83semidirect
178. #102C2×D83semidirect
179. #102C2×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
#191284172
#191283362
#191280232
#191279882
#191279532
#191278932
#191278362
C4×D6642
#191277802
#191277042
#486×C832
#332583