Finite Group Databasebeta

← #480×D83 #24917331

Subgroups — #480×D83 #24917331

Order 10624.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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#121×D83C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19128016C212yesmaximal, complemented
4#19128015C211yesmaximal, characteristic, complemented
5#19128014C211yesmaximal, characteristic, complemented
6#19128013C212yesmaximal, complemented
7#19128012C211yesmaximal, characteristic, complemented
8#107×D83C212yesmaximal, complemented
9#102×D83C212yesmaximal, complemented
10C22⋊C8×D83C211yesmaximal, characteristic, complemented
11#480×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—21no—
13C2×D4×D83C2211yescharacteristic
14C22×C4×D83—21no—
15C22×C4×D83C2211yescharacteristic
16C2×C8×D83C2212yes—
17#14581767C2211yescharacteristic
18#14581766C2211yescharacteristic
19#14581765—21no—
20#14581764—21no—
21#14581763C2211yescharacteristic
22C2×C4⋊Dic83C2211yescharacteristic, complemented
23C2×Dic83⋊C4—21no—
24C4⋊C4×D83—21no—
25C4⋊C4×D83—21no—
26C4⋊C4×D83C2212yes—
27C4⋊C4×D83C2211yescharacteristic
28C4⋊C4×D83C2212yescomplemented
29C22⋊C4×D83—21no—
30#14581726C2212yescomplemented
31#14581725C2212yescomplemented
32#14581725C2212yescomplemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic, complemented
37(C2×D83)⋊C8C2212yescomplemented
38#121×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic, complemented
40#107×C83C2212yescomplemented
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44C23×D83D411yescharacteristic
45C2×D332—21no—
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C2×C22⋊D83—21no—
49C2×C22⋊D83—21no—
50D4×D83—24no—
51D4×D83—44no—
52C22×Dic83—21no—
53C22×Dic83D411yescharacteristic
54C2×C4×D83—24no—
55C2×C4×D83—22no—
56C2×C4×D83—22no—
57C2×C4×D83—21no—
58C2×C4×D83—44no—
59C2×C4×D83D411yescharacteristic
60C2×C4×D83D411yescharacteristic
61C2×C4×D83C2311yescharacteristic
62C2×C4×D83C2311yescharacteristic
63C2×C83⋊C8C2312yes—
64C8×D83—24no—
65C4⋊Dic83—21no—
66C4⋊Dic83C2311yescharacteristic
67C4⋊Dic83C2312yes—
68Dic83⋊C4—21no—
69Dic83⋊C4—22no—
70Dic83⋊C4—22no—
71C22⋊Dic83—21no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—21no—
74C2×D4×C83—21no—
75C2×D4×C83C2311yescharacteristic
76C22×C4×C83—21no—
77C22×C4×C83C2311yescharacteristic
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83C2311yescharacteristic
81C4⋊C4×C83C2312yes—
82C22⋊C4×C83—21no—
83C23×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—21no—
88C22×D83—42no—
89C22×D83—42no—
90C22×D83—42no—
91C22×D83D814yes—
92C22×D83C2×D411yescharacteristic
93C22×D83C2×D411yescharacteristic
94D4×C83—22no—
95D4×C83—42no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—42no—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2411yescharacteristic
102C22⋊D83—42no—
103C22⋊D83—42no—
104C22⋊D83—42no—
105C2×Dic83—22no—
106C2×Dic83—21no—
107C2×Dic83—22no—
108C2×Dic83—42no—
109C2×Dic83C2×D411yescharacteristic
110C2×Dic83C2×D411yescharacteristic
111D332—22no—
112D332—42no—
113C4×D83—22no—
114C4×D83—22no—
115C4×D83—22no—
116C4×D83—44no—
117C4×D83—44no—
118C4×D83—42no—
119C4×D83#4114yes—
120C8×C83—22no—
121C83⋊C8—22no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83—41no—
127C22×C83C2×D812yes—
128C22×C83C22×D411yescharacteristic
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—28no—
132C2×D83—41no—
133C2×D83—41no—
134C2×D83—41no—
135C2×D83—84no—
136C2×D83#12312yes—
137C2×D83C2×D812yes—
138C2×D83C2×D812yes—
139C2×D83#13712yes—
140C4×C83—21no—
141C4×C83—42no—
142C4×C83—41no—
143C4×C83C2×#4112yes—
144Dic83—21no—
145Dic83—42no—
146Dic83—41no—
147Dic83#12312yes—
148C2×C83—22no—
149C2×C83—81no—
150C2×C83C2×#12311yescharacteristic
151C2×C83C22×D811yescharacteristic
152C2×C83C2×#13711yescharacteristic
153D83—22no—
154D83—81no—
155D83#48014yesdirect product factor
156C2×#480—831nomaximal, Sylow
157C83C2×#48011yescharacteristic, complemented, Sylow
158C2×#121—831no—
159C22×C4⋊C4—831no—
160#480—831no—
161#480—831no—
162#480—832no—
163#480—832no—
164#480—831no—
165#480D8311yescharacteristic, direct product factor
166C2×#107—832no—
167C2×#102—832no—
168C2×C22⋊C8—831no—
169C22×D4—831no—
170C22×D4—1661no—
171C23×C4—831no—
172C23×C4—1661no—
173C22×C8—832no—
174#121—831no—
175#121—831no—
176#121—831no—
177#121—1661no—
178#121—1661no—
179#121C2×D8311yescharacteristic
180C2×C4⋊C4—831no—
181C2×C4⋊C4—832no—
182C2×C4⋊C4—832no—
183C2×C4⋊C4—831no—
184C2×C4⋊C4—1661no—
185C2×C4⋊C4—1661no—
186C2×C4⋊C4—1661no—
187C2×C4⋊C4C2×D8311yescharacteristic, complemented
188C2×C22⋊C4—1661no—
189#107—832no—
190#107—832no—
191#107—832no—
192#107C2×D8312yescomplemented
193#102—832no—
194#102—832no—
195#102—832no—
196#102C2×D8312yescomplemented
197C22⋊C8—831no—
198C22⋊C8—832no—
199C22⋊C8C2×D8311yescharacteristic, complemented
200C24—831no—
201C24—1661no—
202C2×D4—21no—
203C2×D4—3324no—
204C2×D4—831no—
205C2×D4—1661no—
206C2×D4—1664no—
207C2×D4—1661no—
208C2×D4—1661no—
209C2×D4—1661no—
210C2×D4C22×D8311yescharacteristic
211C22×C4—21no—
212C22×C4—3324no—
213C22×C4—831no—
214C22×C4—831no—
215C22×C4—831no—
216C22×C4—831no—
217C22×C4—831no—
218C22×C4—1661no—
219C22×C4—1662no—
220C22×C4—1662no—
221C22×C4—1664no—
222C22×C4—1661no—
223C22×C4C22×D8311yescharacteristic
224C2×C8—832no—
225C2×C8—1664no—
226C2×C8C22×D8312yes—
227C4⋊C4—21no—
228C4⋊C4—832no—
229C4⋊C4—831no—
230C4⋊C4—1662no—
231C4⋊C4—1662no—
232C4⋊C4—1661no—
233C4⋊C4—1661no—
234C4⋊C4C22×D8312yes—
235C4⋊C4C22×D8311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1661no—
238C22⋊C4—1661no—
239C22⋊C4—1661no—
240C23—21no—
241C23—3322no—
242C23—3322no—
243C23—3322no—
244C23—831no—
245C23—834no—
246C23—831no—
247C23—1662no—
248C23—1662no—
249C23—1661no—
250C23D4×D8311yescharacteristic
251D4—22no—
252D4—42no—
253D4—3322no—
254D4—3322no—
255D4—3322no—
256D4—3322no—
257D4—1662no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—3324no—
263C2×C4—3324no—
264C2×C4—3322no—
265C2×C4—3322no—
266C2×C4—831no—
267C2×C4—834no—
268C2×C4—831no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1661no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4—1662no—
275C2×C4D4×D8311yescharacteristic
276C2×C4C23×D8311yescharacteristic
277C8—22no—
278C8—1662no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—3321no—
285C22—3321no—
286C22—3321no—
287C22—832no—
288C22—832no—
289C22—832no—
290C22—832no—
291C22—6644no—
292C22—1668no—
293C22—1661no—
294C22—1661no—
295C22D8×D8312yes—
296C22C2×D4×D8311yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—3322no—
301C4—3321no—
302C4—832no—
303C4—1661no—
304C4#41×D8312yes—
305C2—22no—
306C2—81no—
307C2—834no—
308C2—6641no—
309C2—1662no—
310C2#123×D8311yescharacteristic
311C2C2×D8×D8311yescharacteristic
312C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#480×C83C2
socleC22×C83C22×D4
prime radical#121×D83C2
hypercenter#480D83
hypocenterC83C2×#480
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D83C2semidirect
3. #19128016C2semidirect
4. #19128015C2semidirect
5. #19128014C2semidirect
6. #19128013C2semidirect
7. #19128012C2semidirect
8. #107×D83C2semidirect
9. #102×D83C2semidirect
10. C22⋊C8×D83C2semidirect
11. #480×C83C2semidirect
22. C2×C4⋊Dic83C22semidirect
28. C4⋊C4×D83C22semidirect
30. #14581726C22semidirect
31. #14581725C22semidirect
32. #14581725C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
36. C22⋊(C83⋊C8)C22semidirect
37. (C2×D83)⋊C8C22semidirect
39. C2×C4⋊C4×C83C22semidirect
40. #107×C83C22semidirect
41. #102×C83C22semidirect
42. C22⋊C8×C83C22semidirect
155. D83#480direct (N × H)
157. C83C2×#480semidirect
165. #480D83direct (N × H)
187. C2×C4⋊C4C2×D83semidirect
192. #107C2×D83semidirect
196. #102C2×D83semidirect
199. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#121×D832
C2×C4⋊C4×D832
#191280162
#191280152
#191280142
#191280132
#191280122
#107×D832
#102×D832
C22⋊C8×D832
#480×C832
C2×#48083