Finite Group Databasebeta

← #24919063

Subgroups — #24919063

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×#14581817C212yesmaximal, complemented
2D4×D332C211yesmaximal, characteristic, complemented
3#19128075C211yesmaximal, characteristic, complemented
4#19127949C212yesmaximal, complemented
5#19127903C211yesmaximal, characteristic, complemented
6#19127896C211yesmaximal, characteristic, complemented
7#19127851C212yesmaximal, complemented
8#19127846C212yesmaximal, complemented
9C4×#13050575C211yesmaximal, characteristic, complemented
10#19127779C211yesmaximal, characteristic, complemented
11#547×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C2×D4×D83—21no—
14#14581817—28no—
15C2×#13050577C2212yes—
16C2×#13050577C2212yescomplemented
17C2×#13050577C2212yescomplemented
18C2×#13050575C2211yescharacteristic
19C2×#13050575C2212yescomplemented
20C2×#13050574C2212yescomplemented
21#14581789C2211yescharacteristic
22#14581767—21no—
23#14581766C2212yes—
24#14581762—22no—
25C4×D332C2211yescharacteristic
26#14581725C2211yescharacteristic, complemented
27C4⋊(C83⋊C8)C2211yescharacteristic, complemented
28#14581716C2212yescomplemented
29#14581713C2211yescharacteristic, complemented
30#14581712C2211yescharacteristic
31#14581712C2212yescomplemented
32C22⋊(C83⋊C8)C2212yescomplemented
33C4×C83⋊C8C2211yescharacteristic, complemented
34C2×#41×C83C2212yescomplemented
35C4⋊Q8×C83C2211yescharacteristic, complemented
36#123×C83—22no—
37C22⋊Q8×C83C2212yescomplemented
38#121×C83—21no—
39C4×D4×C83C2211yescharacteristic
40C4×D4×C83C2211yescharacteristic, complemented
41C2×C4⋊C4×C83—21no—
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#13050577—24no—
55#13050577—42no—
56#13050575—24no—
57#13050575D414yescomplemented
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×C83C2312yes—
68C2×D4×C83—22no—
69C2×D4×C83D411yescharacteristic
70C2×D4×C83C2311yescharacteristic
71C22×C4×C83—21no—
72C22×C4×C83—22no—
73C22×C4×C83C2312yes—
74C4⋊C4×C83—21no—
75C4⋊C4×C83—22no—
76C4⋊C4×C83—21no—
77C4⋊C4×C83D411yescharacteristic
78C4⋊C4×C83C2311yescharacteristic
79C22⋊C4×C83—22no—
80C22⋊C4×C83—21no—
81C22⋊C4×C83D412yes—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C23×C83C2×D412yes—
85C22×D83—21no—
86C22×D83—22no—
87C22×D83—44no—
88C22×D83—44no—
89C22×D83—82no—
90Q8×C83—22no—
91Q8×C83—22no—
92D4×C83—22no—
93D4×C83—21no—
94D4×C83—44no—
95D4×C83C2×D412yescomplemented
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—44no—
101C2×C4×C83—42no—
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2×D412yes—
104C2×C4×C83C2×D411yescharacteristic
105C2×C4×C83C2411yescharacteristic
106C22⋊D83—44no—
107C2×Dic83—21no—
108D332—22no—
109D332—21no—
110D332—22no—
111D332—44no—
112D332—42no—
113D332—81no—
114D332C2×D412yes—
115C4×D83—42no—
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×C83C22×D411yescharacteristic
126C2×D83—22no—
127C2×D83—41no—
128C2×D83—41no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—88no—
133C4×C83—22no—
134C4×C83—21no—
135C4×C83—42no—
136C4×C83—41no—
137C4×C83AGL(1,ℤ/8)12yescomplemented
138C4×C83C22×D411yescharacteristic
139C4×C83H(5,𝔽2)11yescharacteristic
140Dic83—41no—
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#547D8311yescharacteristic, complemented
152D42—831no—
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×#41C2×D8312yescomplemented
162C22×D4—832no—
163C22×D4—1661no—
164AGL(1,ℤ/8)—1668no—
165C2×SD16—832no—
166C2×SD16—832no—
167C2×SD16—832no—
168C2×D8—831no—
169C2×D8—832no—
170C2×M16—832no—
171C4⋊Q8C2×D8311yescharacteristic, complemented
172#127—831no—
173#123—22no—
174C22⋊Q8C2×D8312yescomplemented
175#121—21no—
176#121—832no—
177#121—1661no—
178#120—1662no—
179C4×D4—831no—
180C4×D4C2×D8311yescharacteristic
181C4×D4C2×D8311yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—831no—
184C4⋊C8—831no—
185#103—832no—
186#102—831no—
187#102—831no—
188#102—832no—
189C22⋊C8—832no—
190C4×C8—831no—
191C24—1662no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D8312yes—
195C2×D4—22no—
196C2×D4—3322no—
197C2×D4—3324no—
198C2×D4—831no—
199C2×D4—832no—
200C2×D4—1664no—
201C2×D4—1662no—
202C2×D4—1662no—
203C2×D4—1661no—
204C2×D4—1661no—
205C2×D4C22⋊D8311yescharacteristic
206C2×D4C22×D8311yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—1661no—
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—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—831no—
224C4⋊C4C22⋊D8311yescharacteristic
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—1661no—
229C22⋊C4—1662no—
230C22⋊C4C22⋊D8312yes—
231C42C22×D8311yescharacteristic
232C23—21no—
233C23—3324no—
234C23—3324no—
235C23—6642no—
236C23—1662no—
237C23—1661no—
238C23C2×C22⋊D8312yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—3324no—
245D4—3324no—
246D4—3322no—
247D4—832no—
248D4—6641no—
249D4—1662no—
250D4—1662no—
251D4—1661no—
252D4D4×D8312yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3322no—
260C2×C4—1661no—
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—3321no—
275C22—3321no—
276C22—3322no—
277C22—3322no—
278C22—3322no—
279C22—6648no—
280C22—1662no—
281C22C22×C22⋊D8311yescharacteristic
282C4—22no—
283C4—21no—
284C4—42no—
285C4—41no—
286C4—3321no—
287C4#1458181712yescomplemented
288C4C2×D4×D8311yescharacteristic
289C4#1458183711yescharacteristic
290C2—42no—
291C2—42no—
292C2—3322no—
293C2—6642no—
294C2D4×C22⋊D8311yescharacteristic
295C2C2×#1458181711yescharacteristic
296C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#547×C83C2
socleC22×C83C22×D4
hypercenter#547D83
hypocenterC83#3272
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#14581817C2semidirect
2. D4×D332C2semidirect
3. #19128075C2semidirect
4. #19127949C2semidirect
5. #19127903C2semidirect
6. #19127896C2semidirect
7. #19127851C2semidirect
8. #19127846C2semidirect
9. C4×#13050575C2semidirect
10. #19127779C2semidirect
11. #547×C83C2semidirect
16. C2×#13050577C22semidirect
17. C2×#13050577C22semidirect
19. C2×#13050575C22semidirect
20. C2×#13050574C22semidirect
26. #14581725C22semidirect
27. C4⋊(C83⋊C8)C22semidirect
28. #14581716C22semidirect
29. #14581713C22semidirect
31. #14581712C22semidirect
32. C22⋊(C83⋊C8)C22semidirect
33. C4×C83⋊C8C22semidirect
34. C2×#41×C83C22semidirect
35. C4⋊Q8×C83C22semidirect
37. C22⋊Q8×C83C22semidirect
40. C4×D4×C83C22semidirect
57. #13050575D4semidirect
95. D4×C83C2×D4semidirect
118. C83⋊C8C2×D4semidirect
137. C4×C83AGL(1,ℤ/8)semidirect
149. C83#3272semidirect
151. #547D83semidirect
161. C2×#41C2×D83semidirect
171. C4⋊Q8C2×D83semidirect
174. C22⋊Q8C2×D83semidirect
181. C4×D4C2×D83semidirect
252. D4D4×D83semidirect
287. 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
#191280752
#191279492
#191279032
#191278962
#191278512
#191278462
C4×#130505752
#191277792
#547×C832
#327283