Finite Group Databasebeta

← #24919059

Subgroups — #24919059

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×#14581818C212yesmaximal, complemented
2#19128356C211yesmaximal, characteristic, complemented
3#19128073C211yesmaximal, characteristic, complemented
4#19127945C212yesmaximal, complemented
5#19127906C211yesmaximal, characteristic, complemented
6#19127893C211yesmaximal, characteristic, complemented
7#19127852C212yesmaximal, complemented
8#19127844C212yesmaximal, complemented
9C4×#13050575C211yesmaximal, characteristic, complemented
10#19127779C211yesmaximal, characteristic, complemented
11D42×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2212yescomplemented
13#14581818—28no—
14C2×#13050578C2212yes—
15C2×#13050578C2212yescomplemented
16C2×#13050578C2212yescomplemented
17C2×#13050575C2211yescharacteristic
18C2×#13050575C2212yescomplemented
19C2×#13050574C2212yescomplemented
20#14581792C2211yescharacteristic, complemented
21#14581774—22no—
22#14581771C2212yescomplemented
23#14581763—21no—
24C4×D332C2211yescharacteristic, complemented
25C4⋊C4×D83—21no—
26#14581725C2211yescharacteristic, complemented
27C4⋊(C83⋊C8)C2211yescharacteristic, complemented
28#14581715C2212yescomplemented
29#14581713C2211yescharacteristic
30#14581713C2212yescomplemented
31#14581712C2211yescharacteristic, complemented
32C22⋊(C83⋊C8)C2212yescomplemented
33C4×C83⋊C8C2211yescharacteristic, complemented
34C22×D4×C83—21no—
35C22×D4×C83C2212yes—
36#127×C83C2211yescharacteristic
37#121×C83—21no—
38#121×C83C2212yes—
39#120×C83—22no—
40C4×D4×C83C2211yescharacteristic
41C4×D4×C83C2211yescharacteristic
42#13050590—24no—
43#13050590—42no—
44C2×Dic166C2312yes—
45C2×D332C2311yescharacteristic
46C2×C22⋊D83—22no—
47C2×C4×D83—21no—
48C2×C4×D83—22no—
49#13050578—24no—
50#13050578—24no—
51#13050578—42no—
52#13050575—24no—
53#13050575D414yescomplemented
54#13050574—24no—
55C2×C83⋊C8C2312yes—
56C2×C83⋊C8C2311yescharacteristic
57C2×C83⋊C8C2311yescharacteristic
58C4⋊Dic83—21no—
59C4⋊Dic83—21no—
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—22no—
62C22⋊Dic83—22no—
63(C2×D83)⋊C4—21no—
64C24×C83—22no—
65C2×D4×C83—24no—
66C2×D4×C83—22no—
67C2×D4×C83—22no—
68C2×D4×C83—21no—
69C2×D4×C83—21no—
70C2×D4×C83—42no—
71C2×D4×C83—44no—
72C2×D4×C83D411yescharacteristic
73C2×D4×C83C2312yes—
74C2×D4×C83C2311yescharacteristic
75C22×C4×C83—21no—
76C22×C4×C83C2312yes—
77C4⋊C4×C83D411yescharacteristic
78C4⋊C4×C83C2311yescharacteristic
79C22⋊C4×C83—21no—
80C22⋊C4×C83—22no—
81C22⋊C4×C83D412yes—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C23×C83—22no—
85C23×C83—44no—
86C23×C83—44no—
87C23×C83—82no—
88C23×C83C2×D412yes—
89C22×D83—21no—
90D4×C83—22no—
91D4×C83—22no—
92D4×C83—21no—
93D4×C83—22no—
94D4×C83—44no—
95D4×C83—44no—
96D4×C83—42no—
97D4×C83—81no—
98D4×C83C2×D412yes—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—21no—
102C2×C4×C83—42no—
103C2×C4×C83C2×D411yescharacteristic
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2×D411yescharacteristic
106C2×C4×C83C2411yescharacteristic
107C22⋊D83—44no—
108C2×Dic83—21no—
109C2×Dic83—22no—
110D332—21no—
111D332C2×D412yescomplemented
112C4×D83—44no—
113C4×D83—42no—
114Dic166—22no—
115Dic166—22no—
116C83⋊C8—22no—
117C83⋊C8—21no—
118C83⋊C8C2×D412yescomplemented
119C22×C83—22no—
120C22×C83—22no—
121C22×C83—22no—
122C22×C83—22no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—88no—
129C22×C83C22×D411yescharacteristic
130C2×D83—22no—
131C2×D83—41no—
132C2×D83—41no—
133C4×C83—22no—
134C4×C83—21no—
135C4×C83—41no—
136C4×C83AGL(1,ℤ/8)12yescomplemented
137C4×C83C22×D411yescharacteristic
138C4×C83H(5,𝔽2)11yescharacteristic
139Dic83—42no—
140Dic83—41no—
141C2×C83—42no—
142C2×C83—42no—
143C2×C83—82no—
144C2×C83D4211yescharacteristic
145C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C83#57711yescharacteristic
147D83—42no—
148#3272—831nomaximal, Sylow
149C83#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—832no—
151#547—831no—
152D42D8311yescharacteristic, complemented
153#492—831no—
154#468—832no—
155#461—831no—
156#459—831no—
157#450—832no—
158#449—832no—
159C4×D8—831no—
160#436—831no—
161C2×#41—832no—
162C22×D4—21no—
163C22×D4C2×D8312yes—
164AGL(1,ℤ/8)—1668no—
165C2×SD16—832no—
166C2×SD16—832no—
167C2×SD16—832no—
168C2×D8—831no—
169C2×D8—832no—
170C2×M16—832no—
171C4⋊Q8—831no—
172#127C2×D8311yescharacteristic
173#123—1662no—
174C22⋊Q8—832no—
175#121—21no—
176#121—1661no—
177#121C2×D8312yes—
178#120—22no—
179C4×D4—831no—
180C4×D4C2×D8311yescharacteristic
181C4×D4C2×D8311yescharacteristic
182C2×C4⋊C4—1661no—
183#107—831no—
184C4⋊C8—831no—
185#103—832no—
186#102—831no—
187#102—832no—
188#102—831no—
189C22⋊C8—832no—
190C4×C8—831no—
191C24—22no—
192#41—3322no—
193#41—1664no—
194C2×Q8—832no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—831no—
203C2×D4—1662no—
204C2×D4C22⋊D8311yescharacteristic
205C2×D4C22×D8312yes—
206C2×D4C22×D8311yescharacteristic
207C22×C4—21no—
208C22×C4—1661no—
209C22×C4—1662no—
210C22×C4C22×D8312yes—
211SD16—3322no—
212SD16—1664no—
213SD16—1664no—
214D8—834no—
215D8—1664no—
216M16—1664no—
217C2×C8—831no—
218C2×C8—831no—
219C2×C8—832no—
220C4⋊C4—831no—
221C4⋊C4—1661no—
222C4⋊C4—1662no—
223C4⋊C4—1661no—
224C4⋊C4C22⋊D8311yescharacteristic
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—1662no—
229C22⋊C4—1661no—
230C22⋊C4C22⋊D8312yes—
231C42C22×D8311yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—1661no—
238C23C2×C22⋊D8312yes—
239Q8—1662no—
240Q8—1662no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—3324no—
250D4—832no—
251D4—1661no—
252D4D4×D8312yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3324no—
258C2×C4—3322no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4C2×C22⋊D8311yescharacteristic
262C2×C4C2×C22⋊D8312yes—
263C2×C4C2×C22⋊D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—832no—
266C8—1662no—
267C8—1661no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3321no—
279C22—3321no—
280C22—1662no—
281C22C22×C22⋊D8311yescharacteristic
282C4—22no—
283C4—21no—
284C4—41no—
285C4—3322no—
286C4—3321no—
287C4#1458181812yescomplemented
288C4C2×D4×D8311yescharacteristic
289C4#1458183711yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—3322no—
294C2D4×C22⋊D8311yescharacteristic
295C2C2×#1458181811yescharacteristic
296C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingD42×C83C2
socleC22×C83C22×D4
hypercenterD42D83
hypocenterC83#3272
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. #19128356C2semidirect
3. #19128073C2semidirect
4. #19127945C2semidirect
5. #19127906C2semidirect
6. #19127893C2semidirect
7. #19127852C2semidirect
8. #19127844C2semidirect
9. C4×#13050575C2semidirect
10. #19127779C2semidirect
11. D42×C83C2semidirect
12. C2×#13050590C22semidirect
15. C2×#13050578C22semidirect
16. C2×#13050578C22semidirect
18. C2×#13050575C22semidirect
19. C2×#13050574C22semidirect
20. #14581792C22semidirect
22. #14581771C22semidirect
24. C4×D332C22semidirect
26. #14581725C22semidirect
27. C4⋊(C83⋊C8)C22semidirect
28. #14581715C22semidirect
30. #14581713C22semidirect
31. #14581712C22semidirect
32. C22⋊(C83⋊C8)C22semidirect
33. C4×C83⋊C8C22semidirect
53. #13050575D4semidirect
111. D332C2×D4semidirect
118. C83⋊C8C2×D4semidirect
136. C4×C83AGL(1,ℤ/8)semidirect
149. C83#3272semidirect
152. D42D83semidirect
287. C4#14581818semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
#191283562
#191280732
#191279452
#191279062
#191278932
#191278522
#191278442
C4×#130505752
#191277792
D42×C832
#327283