Finite Group Databasebeta

← #24919064

Subgroups — #24919064

Order 10624.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14581816C212yesmaximal, complemented
2#19128355C211yesmaximal, characteristic, complemented
3D4×D332C211yesmaximal, characteristic, complemented
4#19128074C211yesmaximal, characteristic, complemented
5#19127948C212yesmaximal, complemented
6#19127906C211yesmaximal, characteristic, complemented
7#19127896C211yesmaximal, characteristic, complemented
8#19127853C212yesmaximal, complemented
9#19127847C212yesmaximal, complemented
10#19127780C211yesmaximal, characteristic, complemented
11C4×D8×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yescomplemented
13C22×D332—21no—
14C2×D4×D83C2212yes—
15#14581816—28no—
16C2×#13050579C2212yescomplemented
17C2×#13050579C2212yescomplemented
18C2×#13050578C2212yes—
19C2×#13050575C2212yescomplemented
20C2×#13050573C2212yescomplemented
21#14581792C2211yescharacteristic, complemented
22#14581789C2211yescharacteristic
23#14581776—22no—
24#14581772C2212yescomplemented
25#14581767C2212yes—
26#14581766—21no—
27#14581766—21no—
28#14581762—22no—
29C4×D332C2211yescharacteristic
30C2×C4⋊Dic83—21no—
31C4⋊(C83⋊C8)C2211yescharacteristic, complemented
32#14581715C2212yescomplemented
33#14581712C2212yescomplemented
34(C2×D83)⋊C8C2212yescomplemented
35C2×D8×C83C2211yescharacteristic
36C4×D4×C83C2211yescharacteristic
37C4×D4×C83C2211yescharacteristic, complemented
38#107×C83C2211yescharacteristic, complemented
39#102×C83C2211yescharacteristic
40#102×C83C2211yescharacteristic, complemented
41C4×C8×C83C2211yescharacteristic, complemented
42C23×D83—22no—
43#13050589—24no—
44#13050589—42no—
45C2×Dic166C2312yes—
46C2×D332—22no—
47C2×D332—21no—
48C2×D332—21no—
49C2×D332—44no—
50C2×D332D411yescharacteristic
51C2×D332C2312yes—
52C2×C22⋊D83—22no—
53C2×C22⋊D83—22no—
54D4×D83—24no—
55D4×D83—42no—
56C22×Dic83—22no—
57C2×C4×D83C2312yes—
58#13050579—24no—
59#13050579—42no—
60#13050578—24no—
61#13050575—24no—
62#13050573—24no—
63C2×C83⋊C8C2312yes—
64C4⋊Dic83—21no—
65C4⋊Dic83—21no—
66C4⋊Dic83—22no—
67C4⋊Dic83D411yescharacteristic
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4D412yes—
71C2×D4×C83C2311yescharacteristic
72C2×D4×C83C2311yescharacteristic
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75D8×C83D414yescomplemented
76C2×C8×C83C2311yescharacteristic
77C2×C8×C83C2311yescharacteristic
78C4⋊C4×C83C2311yescharacteristic
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C22⋊C4×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C23×C83—21no—
85C22×D83—22no—
86C22×D83—44no—
87C22×D83—44no—
88C22×D83—82no—
89C22×D83C2×D412yes—
90D4×C83—21no—
91D4×C83—21no—
92D4×C83C2×D412yes—
93D4×C83C2×D412yescomplemented
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—42no—
97C2×C4×C83—42no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—44no—
102C22⋊D83—44no—
103C2×Dic83—22no—
104C2×Dic83—44no—
105C2×Dic83C2×D412yes—
106D332—22no—
107D332—22no—
108D332—22no—
109D332—44no—
110D332—42no—
111D332—81no—
112C4×D83—22no—
113C4×D83—22no—
114Dic166—22no—
115Dic166—22no—
116C8×C83—21no—
117C8×C83C2×D412yescomplemented
118C83⋊C8—22no—
119C22×C83—22no—
120C22×C83—22no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83C22×D411yescharacteristic
126C2×D83—22no—
127C2×D83—22no—
128C2×D83—22no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—88no—
133C4×C83—21no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83AGL(1,ℤ/8)12yescomplemented
137C4×C83C22×D411yescharacteristic
138C4×C83H(5,𝔽2)11yescharacteristic
139Dic83—22no—
140Dic83—42no—
141C2×C83—42no—
142C2×C83—42no—
143C2×C83D4211yescharacteristic
144C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C83#57711yescharacteristic
146D83—42no—
147D83—82no—
148#3272—831nomaximal, Sylow
149C83#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—832no—
151#547—831no—
152D42—831no—
153#492—831no—
154#468—832no—
155#461—831no—
156#459—831no—
157#450—832no—
158#449—832no—
159C4×D8D8311yescharacteristic, complemented
160#436—831no—
161C2×#41—832no—
162C22×D4—832no—
163C22×D4—1661no—
164AGL(1,ℤ/8)—1668no—
165C2×SD16—832no—
166C2×SD16—832no—
167C2×SD16—832no—
168C2×D8—832no—
169C2×D8C2×D8311yescharacteristic
170C2×M16—832no—
171C4⋊Q8—831no—
172#127—831no—
173#123—1662no—
174C22⋊Q8—832no—
175#121—832no—
176#121—1661no—
177#121—1661no—
178#120—1662no—
179C4×D4—831no—
180C4×D4C2×D8311yescharacteristic
181C4×D4C2×D8311yescharacteristic, complemented
182C2×C4⋊C4—1661no—
183#107C2×D8311yescharacteristic, complemented
184C4⋊C8—831no—
185#103—832no—
186#102—832no—
187#102C2×D8311yescharacteristic
188#102C2×D8311yescharacteristic, complemented
189C22⋊C8—832no—
190C4×C8C2×D8311yescharacteristic, complemented
191C24—1662no—
192#41—3322no—
193#41—1664no—
194C2×Q8—832no—
195C2×D4—3324no—
196C2×D4—3322no—
197C2×D4—831no—
198C2×D4—832no—
199C2×D4—1662no—
200C2×D4—1662no—
201C2×D4—1662no—
202C2×D4—1664no—
203C2×D4—1661no—
204C2×D4—1661no—
205C2×D4C22×D8311yescharacteristic
206C2×D4C22×D8311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—832no—
210C22×C4—1662no—
211SD16—3322no—
212SD16—1664no—
213SD16—1664no—
214D8—1664no—
215D8D33214yescomplemented
216M16—1664no—
217C2×C8—832no—
218C2×C8C22×D8311yescharacteristic
219C2×C8C22×D8311yescharacteristic
220C4⋊C4—831no—
221C4⋊C4—1662no—
222C4⋊C4—1661no—
223C4⋊C4—1661no—
224C4⋊C4C22×D8311yescharacteristic
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—832no—
229C22⋊C4—1662no—
230C22⋊C4—1662no—
231C42C22×D8311yescharacteristic
232C23—21no—
233C23—21no—
234C23—3324no—
235C23—3324no—
236C23—832no—
237C23—6642no—
238C23—1662no—
239Q8—1662no—
240Q8—1662no—
241D4—21no—
242D4—21no—
243D4—3324no—
244D4—3324no—
245D4—3324no—
246D4—3322no—
247D4—6641no—
248D4—1662no—
249D4—1662no—
250D4—1662no—
251D4C2×D33212yes—
252D4C2×D33212yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3324no—
258C2×C4—832no—
259C2×C4—1662no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4D4×D8311yescharacteristic
263C2×C4D4×D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—21no—
266C8—1662no—
267C8C2×D33212yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3322no—
275C22—3322no—
276C22—3322no—
277C22—6648no—
278C22—1662no—
279C22—1662no—
280C22—1662no—
281C22C2×D4×D8311yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—3322no—
286C4—1662no—
287C4#1458181612yescomplemented
288C4C22×D33211yescharacteristic
289C4#1458183811yescharacteristic
290C2—42no—
291C2—42no—
292C2—3322no—
293C2—6642no—
294C2D4×D33211yescharacteristic
295C2C2×#1458181611yescharacteristic
296C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
hypercenterC4×D8D83
hypocenterC83#3272
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#14581816C2semidirect
2. #19128355C2semidirect
3. D4×D332C2semidirect
4. #19128074C2semidirect
5. #19127948C2semidirect
6. #19127906C2semidirect
7. #19127896C2semidirect
8. #19127853C2semidirect
9. #19127847C2semidirect
10. #19127780C2semidirect
11. C4×D8×C83C2semidirect
12. C2×#13050589C22semidirect
16. C2×#13050579C22semidirect
17. C2×#13050579C22semidirect
19. C2×#13050575C22semidirect
20. C2×#13050573C22semidirect
21. #14581792C22semidirect
24. #14581772C22semidirect
31. C4⋊(C83⋊C8)C22semidirect
32. #14581715C22semidirect
33. #14581712C22semidirect
34. (C2×D83)⋊C8C22semidirect
37. C4×D4×C83C22semidirect
38. #107×C83C22semidirect
40. #102×C83C22semidirect
41. C4×C8×C83C22semidirect
75. D8×C83D4semidirect
93. D4×C83C2×D4semidirect
117. C8×C83C2×D4semidirect
136. C4×C83AGL(1,ℤ/8)semidirect
149. C83#3272semidirect
159. C4×D8D83semidirect
181. C4×D4C2×D83semidirect
183. #107C2×D83semidirect
188. #102C2×D83semidirect
190. C4×C8C2×D83semidirect
215. D8D332semidirect
252. D4C2×D332semidirect
267. C8C2×D332semidirect
287. C4#14581816semidirect

Maximal subgroups

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

subgroupindex
C2×#145818162
#191283552
D4×D3322
#191280742
#191279482
#191279062
#191278962
#191278532
#191278472
#191277802
C4×D8×C832
#327283