Finite Group Databasebeta

← #24917549

Subgroups — #24917549

Order 10624.

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

normal subgroups only

This group has 488 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#19128246C211yesmaximal, characteristic, complemented
2C2×D4×Dic83C211yesmaximal, characteristic, complemented
3#19128020C211yesmaximal, characteristic, complemented
4#19128017C211yesmaximal, characteristic, complemented
5#19127989C212yesmaximal, complemented
6#19127894C212yesmaximal, complemented
7#19127835C211yesmaximal, characteristic, complemented
8#19127813C212yesmaximal, complemented
9#19127810C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#447×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C23×Dic83—21no—
14C2×#13050575C2212yes—
15#14581791C2211yescharacteristic, complemented
16#14581788C2211yescharacteristic, complemented
17#14581778—21no—
18#14581766C2211yescharacteristic
19#14581764—21no—
20D4×Dic83—21no—
21D4×Dic83—21no—
22D4×Dic83C2212yes—
23D4×Dic83C2212yes—
24D4×Dic83C2211yescharacteristic, complemented
25#14581754C2211yescharacteristic, complemented
26(C22⋊D83)⋊C4—21no—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83—21no—
29C2×C4×Dic83C2211yescharacteristic
30#14581724C2212yescomplemented
31#14581723C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581714C2212yescomplemented
34#14581713C2212yes—
35#14581713C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×D8×C83C2212yes—
40#121×C83C2211yescharacteristic
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43#13050589—44no—
44C2×Dic166—21no—
45C2×D332C2311yescharacteristic
46C2×C22⋊D83—21no—
47C2×C22⋊D83—21no—
48C2×C22⋊D83—21no—
49C22×Dic83—21no—
50C22×Dic83—22no—
51C22×Dic83—22no—
52C22×Dic83—21no—
53C22×Dic83—44no—
54C22×Dic83D411yescharacteristic
55C22×Dic83D411yescharacteristic
56C2×C4×D83—21no—
57#13050575—24no—
58C2×C83⋊C8C2312yes—
59C4⋊Dic83—21no—
60C4⋊Dic83C2311yescharacteristic
61C4⋊Dic83C2312yes—
62Dic83⋊C4—21no—
63C22⋊Dic83—21no—
64C22⋊Dic83—22no—
65C22⋊Dic83—22no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C4×Dic83C2311yescharacteristic
72C24×C83—21no—
73C2×D4×C83—24no—
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83C2311yescharacteristic
77C2×D4×C83C2312yes—
78C22×C4×C83C2311yescharacteristic
79D8×C83—24no—
80C2×C8×C83C2312yes—
81C4⋊C4×C83C2311yescharacteristic
82C22⋊C4×C83—21no—
83C23×C83—24no—
84C23×C83—22no—
85C23×C83—21no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83C2×D411yescharacteristic
89C22×D83—21no—
90D4×C83—24no—
91D4×C83—22no—
92D4×C83—22no—
93D4×C83—42no—
94D4×C83#4114yescomplemented
95C2×C4×C83—22no—
96C2×C4×C83—21no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C22⋊D83—42no—
101C22⋊D83—42no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—21no—
105C2×Dic83—22no—
106C2×Dic83—44no—
107C2×Dic83—42no—
108C2×Dic83—42no—
109C2×Dic83—44no—
110C2×Dic83C2×D411yescharacteristic
111C2×Dic83C2×D411yescharacteristic
112C2×Dic83C2×D411yescharacteristic
113C2×Dic83C2×D411yescharacteristic
114D332—22no—
115C4×D83—42no—
116Dic166—42no—
117C8×C83—22no—
118C83⋊C8—22no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—24no—
122C22×C83—48no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—41no—
126C22×C83—41no—
127C22×C83—41no—
128C22×C83AGL(1,ℤ/8)12yes—
129C22×C83C22×D411yescharacteristic
130C2×D83—41no—
131C2×D83—41no—
132C2×D83—41no—
133C4×C83—21no—
134C4×C83—41no—
135C4×C83C2×#4112yes—
136Dic83—21no—
137Dic83—21no—
138Dic83—21no—
139Dic83—41no—
140Dic83—42no—
141Dic83AGL(1,ℤ/8)12yes—
142C2×C83—22no—
143C2×C83—44no—
144C2×C83—81no—
145C2×C83#52511yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
148D83—81no—
149#3109—831nomaximal, Sylow
150C83#310911yescharacteristic, complemented, Sylow
151#532—831no—
152C2×C4×D4—831no—
153#481—831no—
154#481—831no—
155#476—832no—
156#459—832no—
157#447—831no—
158#447D8311yescharacteristic, complemented
159#442—832no—
160#442—832no—
161#432—831no—
162C2×#41—1661no—
163C22×D4C2×D8311yescharacteristic
164C23×C4—1661no—
165C2×D8—832no—
166C2×D8C2×D8312yes—
167C4⋊Q8—831no—
168#127—831no—
169#124—1661no—
170#121—831no—
171#121—1661no—
172#121C2×D8311yescharacteristic
173C4×D4—832no—
174C4×D4—832no—
175C4×D4—831no—
176C4×D4—831no—
177C4×D4—1661no—
178C4×D4—1661no—
179C4×D4—1661no—
180C2×C4⋊C4—831no—
181C2×C22⋊C4—1661no—
182C2×C42—831no—
183#106—832no—
184#106—832no—
185C4⋊C8—832no—
186#102—832no—
187#102—832no—
188#102—832no—
189#102C2×D8312yes—
190C22⋊C8—831no—
191C22⋊C8C2×D8311yescharacteristic
192#97—832no—
193C24—21no—
194#41—3324no—
195C2×Q8—1661no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—831no—
200C2×D4—1661no—
201C2×D4—1661no—
202C2×D4—1661no—
203C2×D4C22×D8311yescharacteristic
204C2×D4C22×D8312yes—
205C22×C4—3324no—
206C22×C4—831no—
207C22×C4—831no—
208C22×C4—1661no—
209C22×C4—1662no—
210C22×C4—1662no—
211C22×C4—1661no—
212C22×C4—1661no—
213C22×C4C22×D8311yescharacteristic
214D8—24no—
215D8—1664no—
216C2×C8—832no—
217C2×C8C22×D8312yes—
218C4⋊C4—832no—
219C4⋊C4—831no—
220C4⋊C4—1661no—
221C4⋊C4—1661no—
222C4⋊C4C22×D8311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1661no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C22⋊C4—1661no—
228C22⋊C4—1661no—
229C42—831no—
230C42—831no—
231C42—831no—
232C42—831no—
233C23—24no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—1661no—
239C23D4×D8311yescharacteristic
240Q8—3322no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—3322no—
246D4—3322no—
247D4—3322no—
248D4—1662no—
249D4#1305058914yescomplemented
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—3322no—
253C2×C4—3322no—
254C2×C4—3324no—
255C2×C4—3324no—
256C2×C4—3322no—
257C2×C4—831no—
258C2×C4—831no—
259C2×C4—831no—
260C2×C4—831no—
261C2×C4—1662no—
262C2×C4—1661no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4D4×D8311yescharacteristic
266C2×C4C23×D8311yescharacteristic
267C8—22no—
268C8—1662no—
269C22—21no—
270C22—21no—
271C22—24no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—3321no—
279C22—3321no—
280C22—3321no—
281C22#1458181612yes—
282C22C2×D4×D8311yescharacteristic
283C4—21no—
284C4—41no—
285C4—3322no—
286C4—3321no—
287C4—832no—
288C4—1661no—
289C4—1661no—
290C4—1661no—
291C4C2×#1305058912yes—
292C2—22no—
293C2—44no—
294C2—81no—
295C2—6641no—
296C2#1912820711yescharacteristic
297C2AGL(1,ℤ/8)×D8311yescharacteristic
298C2C2×#1458181611yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#447×C83C2
socleC22×C83C22×D4
hypercenter#447D83
hypocenterC83#3109
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #19128246C2semidirect
2. C2×D4×Dic83C2semidirect
3. #19128020C2semidirect
4. #19128017C2semidirect
5. #19127989C2semidirect
6. #19127894C2semidirect
7. #19127835C2semidirect
8. #19127813C2semidirect
9. #19127810C2semidirect
10. #19127755C2semidirect
11. #447×C83C2semidirect
15. #14581791C22semidirect
16. #14581788C22semidirect
24. D4×Dic83C22semidirect
25. #14581754C22semidirect
30. #14581724C22semidirect
31. #14581723C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581714C22semidirect
35. #14581713C22semidirect
37. #14581702C22semidirect
94. D4×C83#41semidirect
150. C83#3109semidirect
158. #447D83semidirect
249. D4#13050589semidirect

Maximal subgroups

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

subgroupindex
#191282462
C2×D4×Dic832
#191280202
#191280172
#191279892
#191278942
#191278352
#191278132
#191278102
#191277552
#447×C832
#310983