Finite Group Databasebeta

← #24917413

Subgroups — #24917413

Order 10624.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14581772C211yesmaximal, characteristic
2#117×D83C211yesmaximal, characteristic
3#19128045C212yesmaximal, complemented
4#19128044C212yesmaximal, complemented
5#19128038C212yesmaximal, complemented
6#19128033C212yesmaximal, complemented
7#19127741C211yesmaximal, characteristic, complemented
8#19127711C211yesmaximal, characteristic, complemented
9#19127709C211yesmaximal, characteristic, complemented
10#19127664C211yesmaximal, characteristic
11#428×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C2×#13050574C2211yescharacteristic
15#14581772—21no—
16#14581772—21no—
17#14581772C2212yes—
18#14581772C2212yes—
19#14581751C2211yescharacteristic
20#14581750C2211yescharacteristic
21#14581750C2211yescharacteristic
22#14581749—21no—
23#14581748—21no—
24C2×C4⋊Dic83—21no—
25C4⋊C4×D83C2211yescharacteristic
26C4⋊C4×D83C2211yescharacteristic
27C2×(C2×D83)⋊C4—21no—
28C22⋊C4×D83—21no—
29C42×D83—21no—
30#14581725C2212yescomplemented
31#14581723C2212yescomplemented
32#14581717C2212yescomplemented
33#14581717C2212yescomplemented
34#14581715C2212yescomplemented
35#14581715C2212yescomplemented
36(C2×D83)⋊C8C2212yes—
37(C2×D83)⋊C8C2212yes—
38C2×M16×C83C2211yescharacteristic
39#117×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#107×C83C2212yescomplemented
42#106×C83C2212yescomplemented
43C23×D83D411yescharacteristic
44C2×Dic166—22no—
45C2×Dic166—22no—
46C2×Dic166—21no—
47C2×Dic166C2312yes—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C2×C4×D83—22no—
51C2×C4×D83—22no—
52C2×C4×D83—21no—
53C2×C4×D83—21no—
54C2×C4×D83D411yescharacteristic
55C2×C4×D83D411yescharacteristic
56C2×C4×D83C2311yescharacteristic
57C2×C4×D83C2311yescharacteristic
58#13050574—22no—
59C2×C83⋊C8C2312yes—
60C4⋊Dic83—22no—
61C4⋊Dic83—22no—
62C4⋊Dic83C2311yescharacteristic
63C4⋊Dic83C2311yescharacteristic
64Dic83⋊C4—21no—
65Dic83⋊C4—21no—
66C22⋊Dic83—21no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71C4×Dic83—21no—
72C4×Dic83—21no—
73C4×Dic83—21no—
74C22×C4×C83—21no—
75C22×C4×C83C2311yescharacteristic
76M16×C83—22no—
77C2×C8×C83C2312yes—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2312yes—
80C4⋊C4×C83C2311yescharacteristic
81C4⋊C4×C83C2311yescharacteristic
82C22⋊C4×C83—21no—
83C42×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—41no—
88C22×D83C2×D411yescharacteristic
89C22×D83C2×D411yescharacteristic
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—42no—
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83#4112yes—
96C2×C4×C83#4112yes—
97C2×C4×C83C2411yescharacteristic
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—42no—
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105C4×D83—22no—
106C4×D83—22no—
107C4×D83—21no—
108C4×D83—21no—
109C4×D83—21no—
110C4×D83—21no—
111C4×D83—42no—
112C4×D83—42no—
113Dic166—22no—
114Dic166—22no—
115Dic166—42no—
116C8×C83—22no—
117C83⋊C8—22no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83#12312yes—
121C22×C83C22×D411yescharacteristic
122C2×D83—21no—
123C2×D83—21no—
124C2×D83—21no—
125C2×D83—21no—
126C2×D83—21no—
127C2×D83—44no—
128C2×D83#13712yes—
129C4×C83—41no—
130C4×C83—41no—
131C4×C83—42no—
132C4×C83#12312yes—
133C4×C83C2×#4111yescharacteristic
134C4×C83C2×#4111yescharacteristic
135Dic83—21no—
136Dic83—21no—
137Dic83—42no—
138Dic83—41no—
139Dic83—41no—
140C2×C83—22no—
141C2×C83C2×#12311yescharacteristic
142C2×C83C2×#13711yescharacteristic
143C2×C83#57511yescharacteristic
144D83—22no—
145D83—41no—
146#3095—831nomaximal, Sylow
147C83#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—831no—
149C2×#117—831no—
150#485—832no—
151#485—832no—
152#484—832no—
153#483—832no—
154#428—831no—
155#428D8311yescharacteristic, complemented
156#419—831no—
157#419—831no—
158#407—831no—
159C22×Q8—831no—
160C23×C4—831no—
161C2×M16—831no—
162C2×M16C2×D8311yescharacteristic
163C22⋊Q8—832no—
164C22⋊Q8—832no—
165C22⋊Q8—1661no—
166C22⋊Q8—1661no—
167#117—831no—
168#117—831no—
169#117—831no—
170#117—1661no—
171#117—1661no—
172#117C2×D8311yescharacteristic
173C2×C4⋊C4—831no—
174C2×C4⋊C4—831no—
175C2×C4⋊C4—1661no—
176C2×C4⋊C4C2×D8311yescharacteristic
177C2×C22⋊C4—1661no—
178C2×C22⋊C4—1661no—
179C2×C42—1661no—
180#107—832no—
181#107C2×D8312yescomplemented
182#106—832no—
183#106C2×D8312yescomplemented
184#103—832no—
185#103—832no—
186#103—832no—
187#103—832no—
188C22⋊C8—832no—
189C22⋊C8—832no—
190C24—831no—
191C2×Q8—832no—
192C2×Q8—1662no—
193C2×Q8—1662no—
194C2×Q8—1661no—
195C22×C4—21no—
196C22×C4—831no—
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—831no—
201C22×C4—1661no—
202C22×C4—1661no—
203C22×C4—1661no—
204C22×C4—1662no—
205C22×C4—1662no—
206C22×C4C22×D8311yescharacteristic
207M16—22no—
208M16—1662no—
209C2×C8—832no—
210C2×C8C22×D8312yes—
211C4⋊C4—21no—
212C4⋊C4—831no—
213C4⋊C4—831no—
214C4⋊C4—1661no—
215C4⋊C4—1661no—
216C4⋊C4—1662no—
217C4⋊C4—1662no—
218C4⋊C4C22×D8311yescharacteristic
219C4⋊C4C22×D8311yescharacteristic
220C4⋊C4C22×D8312yes—
221C22⋊C4—21no—
222C22⋊C4—1661no—
223C22⋊C4—1661no—
224C22⋊C4—1662no—
225C22⋊C4—1662no—
226C22⋊C4—1661no—
227C42—21no—
228C42—1661no—
229C42—1661no—
230C42—1661no—
231C23—3321no—
232C23—831no—
233C23—831no—
234C23—1662no—
235C23—1662no—
236C23D4×D8311yescharacteristic
237Q8—3322no—
238Q8—1662no—
239Q8—1662no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—3322no—
245C2×C4—3322no—
246C2×C4—3322no—
247C2×C4—831no—
248C2×C4—831no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1661no—
256C2×C4—1661no—
257C2×C4—1661no—
258C2×C4—1661no—
259C2×C4D4×D8311yescharacteristic
260C2×C4#1305059012yes—
261C2×C4#1305059112yes—
262C2×C4C23×D8311yescharacteristic
263C8—22no—
264C8—1662no—
265C22—21no—
266C22—21no—
267C22—3324no—
268C22—832no—
269C22—1661no—
270C22—1661no—
271C22—1661no—
272C22—1661no—
273C22—1661no—
274C22#1458177712yes—
275C22C2×D4×D8311yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3322no—
280C4—3321no—
281C4—3321no—
282C4—1661no—
283C4—1661no—
284C4#1458177712yes—
285C4C2×#1305059011yescharacteristic
286C4C2×#1305059111yescharacteristic
287C2—22no—
288C2—3321no—
289C2—1662no—
290C2C2×#1458177711yescharacteristic
291C2#137×D8311yescharacteristic
292C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#428×C83C2
socleC22×C83C22×D4
prime radicalC23×D83D4
hypercenter#428D83
hypocenterC83#3095
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #19128045C2semidirect
4. #19128044C2semidirect
5. #19128038C2semidirect
6. #19128033C2semidirect
7. #19127741C2semidirect
8. #19127711C2semidirect
9. #19127709C2semidirect
11. #428×C83C2semidirect
30. #14581725C22semidirect
31. #14581723C22semidirect
32. #14581717C22semidirect
33. #14581717C22semidirect
34. #14581715C22semidirect
35. #14581715C22semidirect
41. #107×C83C22semidirect
42. #106×C83C22semidirect
147. C83#3095semidirect
155. #428D83semidirect
181. #107C2×D83semidirect
183. #106C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817722
#117×D832
#191280452
#191280442
#191280382
#191280332
#191277412
#191277112
#191277092
#191276642
#428×C832
#309583