Finite Group Databasebeta

← #24919083

Subgroups — #24919083

Order 10624.

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

normal subgroups only

This group has 556 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×#14581817C212yesmaximal, complemented
2D4×D332C211yesmaximal, characteristic, complemented
3#19128077C211yesmaximal, characteristic, complemented
4#19127952C212yesmaximal, complemented
5#19127901C211yesmaximal, characteristic, complemented
6#19127893C211yesmaximal, characteristic, complemented
7#19127846C212yesmaximal, complemented
8#19127834C212yesmaximal, complemented
9C4×#13050577C211yesmaximal, characteristic, complemented
10#19127779C211yesmaximal, characteristic, complemented
11#550×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C2×D4×D83—21no—
14#14581817—28no—
15C2×#13050577C2211yescharacteristic
16C2×#13050577C2212yescomplemented
17C2×#13050575C2212yes—
18C2×#13050575C2212yescomplemented
19C2×#13050575C2212yescomplemented
20C2×#13050574C2212yescomplemented
21#14581789C2211yescharacteristic
22#14581767—21no—
23#14581766C2212yes—
24#14581762—22no—
25C4×D332C2211yescharacteristic
26#14581723C2211yescharacteristic, complemented
27C4⋊(C83⋊C8)C2211yescharacteristic, complemented
28#14581716C2211yescharacteristic, complemented
29#14581713C2212yescomplemented
30#14581712C2211yescharacteristic
31#14581712C2212yescomplemented
32C22⋊(C83⋊C8)C2212yescomplemented
33C4×C83⋊C8C2211yescharacteristic, complemented
34C2×#41×C83C2212yescomplemented
35#127×C83—21no—
36#127×C83C2211yescharacteristic, complemented
37#121×C83—22no—
38#121×C83C2212yescomplemented
39C4×Q8×C83C2211yescharacteristic, complemented
40C4×D4×C83—21no—
41C4×D4×C83C2211yescharacteristic
42C23×D83—22no—
43C2×D332—24no—
44C2×D332—22no—
45C2×D332—21no—
46C2×D332—21no—
47C2×D332—42no—
48C2×D332C2312yes—
49C2×D332C2311yescharacteristic
50C2×C22⋊D83—22no—
51D4×D83—44no—
52C2×C4×D83—21no—
53#13050577—24no—
54#13050577D414yescomplemented
55#13050575—24no—
56#13050575—24no—
57#13050575—42no—
58#13050574—24no—
59C2×C83⋊C8C2312yes—
60C2×C83⋊C8C2311yescharacteristic
61C2×C83⋊C8C2311yescharacteristic
62C4⋊Dic83C2311yescharacteristic
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—22no—
65#41×C83—24no—
66#41×C83—42no—
67C2×Q8×C83C2311yescharacteristic
68C2×D4×C83—22no—
69C2×D4×C83—22no—
70C2×D4×C83—21no—
71C2×D4×C83—21no—
72C2×D4×C83D411yescharacteristic
73C2×D4×C83C2312yes—
74C22×C4×C83—22no—
75C22×C4×C83C2312yes—
76C4⋊C4×C83—21no—
77C4⋊C4×C83D411yescharacteristic
78C4⋊C4×C83C2311yescharacteristic
79C22⋊C4×C83—22no—
80C22⋊C4×C83D412yes—
81C42×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—22no—
84C23×C83C2×D412yes—
85C22×D83—21no—
86C22×D83—22no—
87C22×D83—44no—
88C22×D83—44no—
89C22×D83—82no—
90Q8×C83—21no—
91Q8×C83C2×D412yescomplemented
92D4×C83—22no—
93D4×C83—22no—
94D4×C83—22no—
95D4×C83—44no—
96D4×C83—42no—
97D4×C83—81no—
98D4×C83—82no—
99C2×C4×C83—21no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—21no—
103C2×C4×C83—44no—
104C2×C4×C83C2×D411yescharacteristic
105C2×C4×C83C2×D412yes—
106C2×C4×C83C2×D411yescharacteristic
107C2×C4×C83C2411yescharacteristic
108C22⋊D83—44no—
109C2×Dic83—21no—
110D332—22no—
111D332—21no—
112D332—22no—
113D332—44no—
114D332—42no—
115D332—81no—
116D332C2×D412yes—
117C4×D83—42no—
118C83⋊C8—22no—
119C83⋊C8—21no—
120C83⋊C8C2×D412yescomplemented
121C22×C83—22no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83C22×D411yescharacteristic
128C2×D83—22no—
129C2×D83—41no—
130C2×D83—41no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—88no—
135C4×C83—22no—
136C4×C83—22no—
137C4×C83—21no—
138C4×C83—41no—
139C4×C83AGL(1,ℤ/8)12yescomplemented
140C4×C83C22×D411yescharacteristic
141C4×C83H(5,𝔽2)11yescharacteristic
142Dic83—41no—
143C2×C83—42no—
144C2×C83—82no—
145C2×C83D4211yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C83#57611yescharacteristic
148D83—42no—
149D83—82no—
150#3274—831nomaximal, Sylow
151C83#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—832no—
153#550D8311yescharacteristic, complemented
154D42—831no—
155#493—831no—
156#469—832no—
157#460—831no—
158#459—831no—
159#449—832no—
160#447—832no—
161C4×SD16—831no—
162#436—831no—
163C2×#41C2×D8312yescomplemented
164C22×D4—832no—
165C22×D4—1661no—
166AGL(1,ℤ/8)—1668no—
167C2×SD16—831no—
168C2×SD16—832no—
169C2×D8—832no—
170C2×D8—832no—
171C2×D8—832no—
172C2×M16—832no—
173#127—21no—
174#127—831no—
175#127C2×D8311yescharacteristic, complemented
176#121—22no—
177#121—832no—
178#121—1661no—
179#121C2×D8312yescomplemented
180#120—1662no—
181C4×Q8C2×D8311yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—831no—
184C4×D4C2×D8311yescharacteristic
185#106—831no—
186C4⋊C8—831no—
187#103—831no—
188#102—832no—
189#102—831no—
190#102—832no—
191C22⋊C8—832no—
192C4×C8—831no—
193C24—1662no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D8311yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3322no—
202C2×D4—3324no—
203C2×D4—831no—
204C2×D4—832no—
205C2×D4—1664no—
206C2×D4—1662no—
207C2×D4—1662no—
208C2×D4—1661no—
209C2×D4—1661no—
210C2×D4C22⋊D8311yescharacteristic
211C2×D4C22×D8312yes—
212C22×C4—22no—
213C22×C4—1661no—
214C22×C4C22×D8312yes—
215SD16—834no—
216SD16—1664no—
217D8—3322no—
218D8—1664no—
219D8—1664no—
220M16—1664no—
221C2×C8—831no—
222C2×C8—831no—
223C2×C8—832no—
224C4⋊C4—21no—
225C4⋊C4—831no—
226C4⋊C4C22⋊D8311yescharacteristic
227C4⋊C4C22×D8311yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—1661no—
230C22⋊C4—1662no—
231C22⋊C4C22⋊D8312yes—
232C42—21no—
233C42C22×D8311yescharacteristic
234C23—22no—
235C23—3324no—
236C23—3324no—
237C23—6642no—
238C23—1662no—
239C23—1661no—
240C23C2×C22⋊D8312yes—
241Q8—21no—
242Q8D4×D8312yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—3324no—
251D4—3324no—
252D4—3322no—
253D4—832no—
254D4—6641no—
255D4—1662no—
256D4—1662no—
257D4—1661no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—3322no—
264C2×C4—1661no—
265C2×C4C2×C22⋊D8311yescharacteristic
266C2×C4C2×C22⋊D8312yes—
267C2×C4C2×C22⋊D8311yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—832no—
270C8—1662no—
271C8—1661no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3321no—
279C22—3321no—
280C22—3322no—
281C22—3322no—
282C22—3322no—
283C22—6648no—
284C22—1662no—
285C22C22×C22⋊D8311yescharacteristic
286C4—22no—
287C4—22no—
288C4—21no—
289C4—41no—
290C4—3321no—
291C4#1458181712yescomplemented
292C4C2×D4×D8311yescharacteristic
293C4#1458183711yescharacteristic
294C2—42no—
295C2—82no—
296C2—3322no—
297C2—6642no—
298C2D4×C22⋊D8311yescharacteristic
299C2C2×#1458181711yescharacteristic
300C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#550×C83C2
socleC22×C83C22×D4
hypercenter#550D83
hypocenterC83#3274
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#14581817C2semidirect
2. D4×D332C2semidirect
3. #19128077C2semidirect
4. #19127952C2semidirect
5. #19127901C2semidirect
6. #19127893C2semidirect
7. #19127846C2semidirect
8. #19127834C2semidirect
9. C4×#13050577C2semidirect
10. #19127779C2semidirect
11. #550×C83C2semidirect
16. C2×#13050577C22semidirect
18. C2×#13050575C22semidirect
19. C2×#13050575C22semidirect
20. C2×#13050574C22semidirect
26. #14581723C22semidirect
27. C4⋊(C83⋊C8)C22semidirect
28. #14581716C22semidirect
29. #14581713C22semidirect
31. #14581712C22semidirect
32. C22⋊(C83⋊C8)C22semidirect
33. C4×C83⋊C8C22semidirect
34. C2×#41×C83C22semidirect
36. #127×C83C22semidirect
38. #121×C83C22semidirect
39. C4×Q8×C83C22semidirect
54. #13050577D4semidirect
91. Q8×C83C2×D4semidirect
120. C83⋊C8C2×D4semidirect
139. C4×C83AGL(1,ℤ/8)semidirect
151. C83#3274semidirect
153. #550D83semidirect
163. C2×#41C2×D83semidirect
175. #127C2×D83semidirect
179. #121C2×D83semidirect
181. C4×Q8C2×D83semidirect
242. Q8D4×D83semidirect
291. C4#14581817semidirect

Maximal subgroups

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

subgroupindex
C2×#145818172
D4×D3322
#191280772
#191279522
#191279012
#191278932
#191278462
#191278342
C4×#130505772
#191277792
#550×C832
#327483