Finite Group Databasebeta

← #24919702

Subgroups — #24919702

Order 10624.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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
2D4×D332C211yesmaximal, characteristic, complemented
3#19128049C211yesmaximal, characteristic, complemented
4#19128027C212yesmaximal, complemented
5#19127955C212yesmaximal, complemented
6#19127897C211yesmaximal, characteristic, complemented
7#19127836C212yesmaximal, complemented
8C4×#13050575C211yesmaximal, characteristic, complemented
9#19127778C211yesmaximal, characteristic, complemented
10#19127704C212yesmaximal, complemented
11#476×C83C211yesmaximal, characteristic, complemented
12C22×D332—21no—
13C2×D4×D83C2212yes—
14C2×D664C2212yescomplemented
15C2×#13050575C2211yescharacteristic
16C2×#13050573C2212yescomplemented
17#14581789C2211yescharacteristic
18#14581781—22no—
19#14581767—21no—
20#14581767C2212yes—
21#14581767C2212yescomplemented
22#14581766—21no—
23#14581762—22no—
24C4×D332C2211yescharacteristic
25C4×D332C2211yescharacteristic, complemented
26#14581750—21no—
27#14581750C2212yescomplemented
28#14581725C2211yescharacteristic, complemented
29#14581714C2212yescomplemented
30#14581713C2211yescharacteristic
31#14581712C2212yescomplemented
32#14581712C2211yescharacteristic, complemented
33#14581712C2212yescomplemented
34(C2×D83)⋊C8C2212yescomplemented
35C4×C83⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C83C2211yescharacteristic, complemented
37C4×D4×C83C2211yescharacteristic
38#106×C83C2212yescomplemented
39C4⋊C8×C83C2211yescharacteristic, complemented
40#102×C83C2212yes—
41C23×D83—22no—
42C2×D332—22no—
43C2×D332—21no—
44C2×D332—22no—
45C2×D332—21no—
46C2×D332—44no—
47C2×D332D411yescharacteristic
48C2×D332C2312yes—
49C2×D332C2311yescharacteristic
50C2×C22⋊D83—22no—
51D4×D83—24no—
52D4×D83—42no—
53C2×C4×D83—21no—
54C2×C4×D83C2312yes—
55D664—42no—
56#13050575—22no—
57#13050573—42no—
58C2×C83⋊C8C2311yescharacteristic
59C2×C83⋊C8C2311yescharacteristic
60C4⋊Dic83D411yescharacteristic
61C4⋊Dic83C2311yescharacteristic
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4D412yes—
67C4×Dic83—22no—
68C2×Q8×C83—21no—
69C2×D4×C83C2311yescharacteristic
70C22×C4×C83—21no—
71C2×C8×C83C2312yes—
72C4⋊C4×C83—21no—
73C4⋊C4×C83C2311yescharacteristic
74C4⋊C4×C83C2312yes—
75C22⋊C4×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C22×D83—21no—
79C22×D83—22no—
80C22×D83—44no—
81C22×D83—44no—
82C22×D83—82no—
83C22×D83C2×D412yes—
84Q8×C83—81no—
85D4×C83—21no—
86D4×C83#4112yes—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—42no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—44no—
94C2×Dic83—21no—
95C2×Dic83C2×D412yes—
96D332—22no—
97D332—21no—
98D332—22no—
99D332—21no—
100D332—22no—
101D332—44no—
102D332—42no—
103D332—81no—
104D332—82no—
105C4×D83—22no—
106C4×D83—22no—
107C4×D83—42no—
108C8×C83—22no—
109C83⋊C8—21no—
110C83⋊C8—21no—
111C22×C83—22no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83C22×D411yescharacteristic
115C2×D83—22no—
116C2×D83—22no—
117C2×D83—22no—
118C2×D83—42no—
119C2×D83—41no—
120C2×D83—41no—
121C2×D83—42no—
122C2×D83—41no—
123C2×D83—42no—
124C2×D83—88no—
125C4×C83—21no—
126C4×C83—41no—
127C4×C83—42no—
128C4×C83AGL(1,ℤ/8)12yes—
129C4×C83C2×#4111yescharacteristic
130C4×C83H(5,𝔽2)11yescharacteristic
131Dic83—22no—
132Dic83—41no—
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#476D8311yescharacteristic, complemented
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—831no—
156C2×D8—832no—
157C2×M16—832no—
158C4⋊Q8C2×D8311yescharacteristic, complemented
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#106C2×D8312yescomplemented
173C4⋊C8C2×D8311yescharacteristic, complemented
174#102—831no—
175#102—832no—
176#102—832no—
177#102—831no—
178#102—832no—
179#102C2×D8312yes—
180C22⋊C8—832no—
181C4×C8—831no—
182C24—1662no—
183C2×Q8—21no—
184C2×D4—3324no—
185C2×D4—3322no—
186C2×D4—831no—
187C2×D4—832no—
188C2×D4—831no—
189C2×D4—1662no—
190C2×D4—1662no—
191C2×D4—1662no—
192C2×D4—1664no—
193C2×D4—1661no—
194C2×D4—1661no—
195C2×D4C22×D8311yescharacteristic
196C22×C4—21no—
197C22×C4—832no—
198C22×C4—1661no—
199D8—3322no—
200D8—1662no—
201M16—3322no—
202C2×C8—831no—
203C2×C8—831no—
204C2×C8C22×D8312yes—
205C4⋊C4—21no—
206C4⋊C4—831no—
207C4⋊C4—831no—
208C4⋊C4C22×D8312yes—
209C4⋊C4C22×D8311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—832no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C22⋊C4—1661no—
216C42—1662no—
217C42C22×D8311yescharacteristic
218C23—21no—
219C23—3324no—
220C23—3324no—
221C23—832no—
222C23—6642no—
223C23—1661no—
224C23—1662no—
225Q8—81no—
226D4—21no—
227D4—3324no—
228D4—3324no—
229D4—3322no—
230D4—6641no—
231D4—6642no—
232D4—1662no—
233D4—1661no—
234D4—1662no—
235D4—1661no—
236D4—1662no—
237D4#1305059112yes—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—3322no—
242C2×C4—832no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4D4×D8311yescharacteristic
247C2×C4D4×D8311yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—22no—
250C8—1661no—
251C8—1661no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—3322no—
256C22—3321no—
257C22—3321no—
258C22—3322no—
259C22—3321no—
260C22—3322no—
261C22—6648no—
262C22—1662no—
263C22—1662no—
264C22—1662no—
265C22C2×D4×D8311yescharacteristic
266C4—21no—
267C4—41no—
268C4—42no—
269C4—3321no—
270C4—1662no—
271C4#1458181912yes—
272C4C2×#1305059111yescharacteristic
273C4#1458183811yescharacteristic
274C2—42no—
275C2—3322no—
276C2—6642no—
277C2—6641no—
278C2#1912834211yescharacteristic
279C2C2×#1458181911yescharacteristic
280C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#476×C83C2
socleC22×C83C22×D4
hypercenter#476D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #19128417C2semidirect
2. D4×D332C2semidirect
3. #19128049C2semidirect
4. #19128027C2semidirect
5. #19127955C2semidirect
6. #19127897C2semidirect
7. #19127836C2semidirect
8. C4×#13050575C2semidirect
9. #19127778C2semidirect
10. #19127704C2semidirect
11. #476×C83C2semidirect
14. C2×D664C22semidirect
16. C2×#13050573C22semidirect
21. #14581767C22semidirect
25. C4×D332C22semidirect
27. #14581750C22semidirect
28. #14581725C22semidirect
29. #14581714C22semidirect
31. #14581712C22semidirect
32. #14581712C22semidirect
33. #14581712C22semidirect
34. (C2×D83)⋊C8C22semidirect
35. C4×C83⋊C8C22semidirect
36. C4⋊Q8×C83C22semidirect
38. #106×C83C22semidirect
39. C4⋊C8×C83C22semidirect
141. C83#3325semidirect
146. #476D83semidirect
158. C4⋊Q8C2×D83semidirect
172. #106C2×D83semidirect
173. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284172
D4×D3322
#191280492
#191280272
#191279552
#191278972
#191278362
C4×#130505752
#191277782
#191277042
#476×C832
#332583