Finite Group Databasebeta

← #24917564

Subgroups — #24917564

Order 10624.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128256C211yesmaximal, characteristic, complemented
2C4×D4×D83C211yesmaximal, characteristic
3#19128038C212yesmaximal, complemented
4#19127978C211yesmaximal, characteristic, complemented
5#19127907C211yesmaximal, characteristic, complemented
6#19127906C211yesmaximal, characteristic, complemented
7#19127853C212yesmaximal
8#19127800C212yesmaximal, complemented
9#19127798C212yesmaximal, complemented
10#19127757C211yesmaximal, characteristic, complemented
11#474×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83C2211yescharacteristic
13C22×C4×D83—21no—
14C2×#13050579C2212yes—
15C2×#13050578C2212yes—
16#14581792C2211yescharacteristic, complemented
17#14581783—21no—
18#14581772C2212yes—
19#14581768—21no—
20Q8×Dic83—21no—
21#14581757C2212yescomplemented
22C4×D332C2211yescharacteristic
23D4×Dic83C2211yescharacteristic
24#14581754C2211yescharacteristic
25C4×C22⋊D83—21no—
26(C22⋊D83)⋊C4—21no—
27#14581750—21no—
28C4⋊C4×D83C2211yescharacteristic
29C22⋊C4×D83—21no—
30C42×D83C2211yescharacteristic
31#14581725C2212yescomplemented
32C4⋊(C83⋊C8)C2211yescharacteristic, complemented
33#14581717C2212yescomplemented
34#14581715C2212yescomplemented
35#14581712C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581702C2212yescomplemented
38C4⋊Q8×C83C2211yescharacteristic, complemented
39C4×D4×C83C2211yescharacteristic
40#107×C83C2212yescomplemented
41C4⋊C8×C83C2211yescharacteristic, complemented
42#102×C83C2212yes—
43C23×D83—21no—
44C2×Dic166C2312yes—
45C2×D332C2311yescharacteristic
46C2×C22⋊D83—21no—
47D4×D83—22no—
48D4×D83—22no—
49C22×Dic83—21no—
50C2×C4×D83—21no—
51C2×C4×D83—21no—
52C2×C4×D83—21no—
53C2×C4×D83—44no—
54C2×C4×D83D411yescharacteristic
55C2×C4×D83D411yescharacteristic
56C2×C4×D83C2311yescharacteristic
57#13050579—24no—
58#13050578—24no—
59C2×C83⋊C8C2312yes—
60C4⋊Dic83—21no—
61C4⋊Dic83—21no—
62C4⋊Dic83—22no—
63C4⋊Dic83C2311yescharacteristic
64Dic83⋊C4—22no—
65Dic83⋊C4—21no—
66C22⋊Dic83—21no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70C4×Dic83—22no—
71C4×Dic83D411yescharacteristic
72C4×Dic83D411yescharacteristic
73C4×Dic83C2311yescharacteristic
74C2×Q8×C83—21no—
75C2×D4×C83C2311yescharacteristic
76C22×C4×C83—21no—
77C2×C8×C83C2312yes—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2312yes—
80C4⋊C4×C83C2311yescharacteristic
81C22⋊C4×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C22×D83—24no—
85C22×D83—21no—
86C22×D83—42no—
87C22×D83—42no—
88C22×D83C2×D411yescharacteristic
89Q8×C83—42no—
90D4×C83—21no—
91D4×C83#4112yes—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—42no—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—24no—
99C2×Dic83—22no—
100C2×Dic83—21no—
101C2×Dic83—42no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105D332—21no—
106D332#4112yes—
107C4×D83—22no—
108C4×D83—22no—
109C4×D83—21no—
110C4×D83—21no—
111C4×D83—42no—
112C4×D83—42no—
113C4×D83—42no—
114Dic166—22no—
115Dic166—22no—
116C8×C83—22no—
117C83⋊C8—22no—
118C22×C83—22no—
119C22×C83—41no—
120C22×C83—41no—
121C22×C83C22×D411yescharacteristic
122C2×D83—21no—
123C2×D83—21no—
124C2×D83—22no—
125C2×D83—48no—
126C2×D83—41no—
127C2×D83—41no—
128C2×D83#13712yes—
129C4×C83—21no—
130C4×C83—41no—
131C4×C83—42no—
132C4×C83AGL(1,ℤ/8)12yes—
133C4×C83C2×#4111yescharacteristic
134C4×C83C2×#4111yescharacteristic
135Dic83—21no—
136Dic83—21no—
137Dic83—21no—
138Dic83—42no—
139Dic83—41no—
140C2×C83—42no—
141C2×C83#52511yescharacteristic
142C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C83C2×#13711yescharacteristic
144D83—22no—
145D83—42no—
146#3110—831nomaximal, Sylow
147C83#311011yescharacteristic, complemented, Sylow
148#533—831no—
149C2×C4×D4—831no—
150#484—832no—
151#474—831no—
152#474D8311yescharacteristic, complemented
153#461—831no—
154#461—831no—
155#450—832no—
156#440—832no—
157#440—832no—
158#432—831no—
159C22×D4—831no—
160C23×C4—1661no—
161C2×SD16—832no—
162C2×SD16—832no—
163C4⋊Q8—831no—
164C4⋊Q8C2×D8311yescharacteristic, complemented
165#125—1661no—
166C22⋊Q8—832no—
167C22⋊Q8—1661no—
168C4×Q8—832no—
169C4×Q8—1661no—
170C4×D4—831no—
171C4×D4—831no—
172C4×D4—831no—
173C4×D4—1661no—
174C4×D4—1661no—
175C4×D4C2×D8311yescharacteristic
176#117—1661no—
177C2×C4⋊C4—831no—
178C2×C22⋊C4—1661no—
179C2×C42—831no—
180#107—832no—
181#107C2×D8312yescomplemented
182C4⋊C8—831no—
183C4⋊C8C2×D8311yescharacteristic, complemented
184#103—832no—
185#103—832no—
186#102—832no—
187#102C2×D8312yes—
188C22⋊C8—832no—
189#97—832no—
190C24—1661no—
191C2×Q8—21no—
192C2×Q8—832no—
193C2×D4—831no—
194C2×D4—1661no—
195C2×D4—1662no—
196C2×D4—1662no—
197C2×D4C22×D8311yescharacteristic
198C22×C4—21no—
199C22×C4—3324no—
200C22×C4—831no—
201C22×C4—831no—
202C22×C4—831no—
203C22×C4—1661no—
204C22×C4—1661no—
205C22×C4—1661no—
206C22×C4—1661no—
207SD16—1664no—
208SD16—1664no—
209C2×C8—832no—
210C2×C8C22×D8312yes—
211C4⋊C4—21no—
212C4⋊C4—831no—
213C4⋊C4—1661no—
214C4⋊C4—1662no—
215C4⋊C4—1662no—
216C4⋊C4—1661no—
217C4⋊C4—1661no—
218C4⋊C4C22×D8312yes—
219C4⋊C4C22×D8311yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—1661no—
222C22⋊C4—1662no—
223C22⋊C4—1661no—
224C22⋊C4—1661no—
225C42—831no—
226C42—831no—
227C42—831no—
228C42—1662no—
229C42C22×D8311yescharacteristic
230C23—21no—
231C23—3322no—
232C23—3322no—
233C23—831no—
234C23—1664no—
235C23—1661no—
236Q8—42no—
237Q8—1662no—
238Q8—1662no—
239D4—21no—
240D4—832no—
241D4—1664no—
242D4—1661no—
243D4#1305059112yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—3322no—
248C2×C4—3322no—
249C2×C4—3322no—
250C2×C4—3322no—
251C2×C4—831no—
252C2×C4—831no—
253C2×C4—831no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4—1661no—
257C2×C4—1662no—
258C2×C4—1661no—
259C2×C4—1661no—
260C2×C4D4×D8311yescharacteristic
261C2×C4D4×D8311yescharacteristic
262C2×C4C23×D8311yescharacteristic
263C8—22no—
264C8—1662no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—3328no—
269C22—3321no—
270C22—3321no—
271C22—832no—
272C22—1662no—
273C22—1661no—
274C22—1661no—
275C22C2×D4×D8311yescharacteristic
276C4—21no—
277C4—41no—
278C4—42no—
279C4—3322no—
280C4—3321no—
281C4—1661no—
282C4—1661no—
283C4—1661no—
284C4#1458181612yes—
285C4#41×D8311yescharacteristic
286C4C2×#1305059111yescharacteristic
287C2—42no—
288C2—3322no—
289C2—1662no—
290C2#1912821411yescharacteristic
291C2C2×#1458181611yescharacteristic
292C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
hypercenter#474D83
hypocenterC83#3110
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128038C2semidirect
4. #19127978C2semidirect
5. #19127907C2semidirect
6. #19127906C2semidirect
8. #19127800C2semidirect
9. #19127798C2semidirect
10. #19127757C2semidirect
11. #474×C83C2semidirect
16. #14581792C22semidirect
21. #14581757C22semidirect
31. #14581725C22semidirect
32. C4⋊(C83⋊C8)C22semidirect
33. #14581717C22semidirect
34. #14581715C22semidirect
37. #14581702C22semidirect
38. C4⋊Q8×C83C22semidirect
40. #107×C83C22semidirect
41. C4⋊C8×C83C22semidirect
147. C83#3110semidirect
152. #474D83semidirect
164. C4⋊Q8C2×D83semidirect
181. #107C2×D83semidirect
183. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
C4×D4×D832
#191280382
#191279782
#191279072
#191279062
#191278532
#191278002
#191277982
#191277572
#474×C832
#311083