Finite Group Databasebeta

← #24917389

Subgroups — #24917389

Order 10624.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#19128213C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19128038C212yesmaximal, complemented
4#19128033C212yesmaximal, complemented
5#19128021C212yesmaximal, complemented
6#19128016C212yesmaximal, complemented
7#19127744C211yesmaximal, characteristic, complemented
8#19127703C211yesmaximal, characteristic, complemented
9#19127700C211yesmaximal, characteristic, complemented
10#19127664C211yesmaximal, characteristic, complemented
11#428×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×C4×D83—21no—
14C22×C4×D83C2211yescharacteristic
15C2×#13050574C2211yescharacteristic
16#14581776—21no—
17#14581772C2211yescharacteristic
18#14581772C2211yescharacteristic
19#14581767C2211yescharacteristic
20#14581767C2211yescharacteristic
21#14581762—21no—
22C4×D332—21no—
23C4×D332—21no—
24C2×C4⋊Dic83C2211yescharacteristic
25C2×Dic83⋊C4—21no—
26C4⋊C4×D83—21no—
27C4⋊C4×D83—21no—
28C4⋊C4×D83C2212yes—
29C4⋊C4×D83C2212yes—
30#14581725C2212yescomplemented
31#14581723C2212yescomplemented
32#14581717C2212yescomplemented
33#14581715C2212yescomplemented
34#14581714C2212yescomplemented
35#14581712C2212yescomplemented
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
44#13050590—22no—
45#13050590—22no—
46C2×Dic166C2311yescharacteristic
47C2×D332—21no—
48C2×D332—21no—
49C2×D332C2311yescharacteristic
50C2×C22⋊D83—21no—
51C22×Dic83—21no—
52C22×Dic83D411yescharacteristic
53C2×C4×D83—22no—
54C2×C4×D83—22no—
55C2×C4×D83—22no—
56C2×C4×D83—22no—
57C2×C4×D83—21no—
58C2×C4×D83—44no—
59C2×C4×D83D411yescharacteristic
60C2×C4×D83D411yescharacteristic
61C2×C4×D83C2311yescharacteristic
62C2×C4×D83C2311yescharacteristic
63#13050574—22no—
64C2×C83⋊C8C2312yes—
65C4⋊Dic83—21no—
66C4⋊Dic83—21no—
67C4⋊Dic83—21no—
68C4⋊Dic83C2312yes—
69Dic83⋊C4—22no—
70Dic83⋊C4—22no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—21no—
74(C2×D83)⋊C4—21no—
75C22×C4×C83—21no—
76C22×C4×C83C2311yescharacteristic
77M16×C83—22no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83C2311yescharacteristic
81C4⋊C4×C83C2311yescharacteristic
82C4⋊C4×C83C2312yes—
83C22⋊C4×C83—21no—
84C42×C83—21no—
85C23×C83C2×D411yescharacteristic
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—21no—
90C22×D83C2×D411yescharacteristic
91C22×D83C2×D411yescharacteristic
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—42no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83#4112yes—
98C2×C4×C83#4112yes—
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—42no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104C2×Dic83—42no—
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107D332—21no—
108D332—21no—
109D332—42no—
110D332—42no—
111C4×D83—22no—
112C4×D83—22no—
113C4×D83—21no—
114C4×D83—21no—
115C4×D83—21no—
116C4×D83—21no—
117C4×D83—44no—
118C4×D83—44no—
119C4×D83—42no—
120Dic166—21no—
121Dic166—21no—
122C8×C83—22no—
123C83⋊C8—22no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83#12312yes—
127C22×C83C22×D411yescharacteristic
128C2×D83—28no—
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—21no—
132C2×D83—21no—
133C2×D83—41no—
134C2×D83—41no—
135C2×D83—41no—
136C2×D83AGL(1,ℤ/8)12yes—
137C2×D83#13712yes—
138C4×C83—42no—
139C4×C83—41no—
140C4×C83—41no—
141C4×C83#12312yes—
142C4×C83C2×#4111yescharacteristic
143C4×C83C2×#4111yescharacteristic
144Dic83—21no—
145Dic83—21no—
146Dic83—41no—
147Dic83—42no—
148C2×C83—22no—
149C2×C83C2×#12311yescharacteristic
150C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C83C2×#13711yescharacteristic
152D83—22no—
153D83—22no—
154D83—81no—
155#3092—831nomaximal, Sylow
156C83#309211yescharacteristic, complemented, Sylow
157#525—831no—
158C22×C4⋊C4—831no—
159#484—832no—
160#483—832no—
161#481—832no—
162#480—832no—
163#428—831no—
164#428D8311yescharacteristic, complemented
165#417—831no—
166#417—831no—
167#407—831no—
168C2×#41—831no—
169C23×C4—831no—
170C23×C4—1661no—
171C2×M16—831no—
172C2×M16C2×D8311yescharacteristic
173#123—1661no—
174C22⋊Q8—831no—
175C22⋊Q8—831no—
176#121—831no—
177#121—831no—
178#120—1661no—
179C4×D4—1661no—
180C4×D4—1661no—
181#117C2×D8311yescharacteristic
182C2×C4⋊C4—831no—
183C2×C4⋊C4—832no—
184C2×C4⋊C4—832no—
185C2×C4⋊C4—1661no—
186C2×C4⋊C4—1661no—
187C2×C4⋊C4—1661no—
188C2×C4⋊C4C2×D8311yescharacteristic
189#107—832no—
190#107C2×D8312yescomplemented
191#106—832no—
192#106C2×D8312yescomplemented
193#103—832no—
194#103—832no—
195#102—832no—
196#102—832no—
197C22⋊C8—832no—
198C22⋊C8—832no—
199C24—831no—
200#41—1662no—
201#41—1662no—
202C2×Q8—831no—
203C2×D4—831no—
204C2×D4—1661no—
205C2×D4—1661no—
206C2×D4—1661no—
207C22×C4—21no—
208C22×C4—3324no—
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—831no—
212C22×C4—831no—
213C22×C4—831no—
214C22×C4—1662no—
215C22×C4—1662no—
216C22×C4—1661no—
217C22×C4—1661no—
218C22×C4—1662no—
219C22×C4—1662no—
220C22×C4C22×D8311yescharacteristic
221M16—22no—
222M16—1662no—
223C2×C8—832no—
224C2×C8C22×D8312yes—
225C4⋊C4—21no—
226C4⋊C4—832no—
227C4⋊C4—1662no—
228C4⋊C4—1662no—
229C4⋊C4—1661no—
230C4⋊C4—1661no—
231C4⋊C4—1661no—
232C4⋊C4C22×D8312yes—
233C4⋊C4C22×D8311yescharacteristic
234C4⋊C4C22×D8311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1661no—
237C22⋊C4—1661no—
238C22⋊C4—1661no—
239C22⋊C4—1661no—
240C42—21no—
241C23—831no—
242C23—831no—
243C23—1662no—
244C23—1662no—
245C23—1661no—
246C23—1662no—
247C23D4×D8311yescharacteristic
248Q8—1661no—
249Q8—1661no—
250D4—3322no—
251D4—3322no—
252D4—3322no—
253D4—1661no—
254D4—1661no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—3324no—
260C2×C4—3324no—
261C2×C4—3322no—
262C2×C4—3322no—
263C2×C4—831no—
264C2×C4—831no—
265C2×C4—1661no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1661no—
271C2×C4—1661no—
272C2×C4—1661no—
273C2×C4—1661no—
274C2×C4D4×D8311yescharacteristic
275C2×C4#1305059012yes—
276C2×C4#1305059112yes—
277C2×C4C23×D8311yescharacteristic
278C8—22no—
279C8—1662no—
280C22—21no—
281C22—21no—
282C22—3321no—
283C22—3321no—
284C22—3321no—
285C22—832no—
286C22—832no—
287C22—1668no—
288C22—1661no—
289C22—1661no—
290C22—1661no—
291C22—1661no—
292C22#1458177712yes—
293C22C2×D4×D8311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3321no—
298C4—3322no—
299C4—1661no—
300C4—1661no—
301C4#1458177712yes—
302C4C2×#1305059011yescharacteristic
303C4C2×#1305059111yescharacteristic
304C2—22no—
305C2—6641no—
306C2—1662no—
307C2—1662no—
308C2C2×#1458177711yescharacteristic
309C2AGL(1,ℤ/8)×D8311yescharacteristic
310C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#428×C83C2
socleC22×C83C22×D4
prime radical#19128213C2
hypercenter#428D83
hypocenterC83#3092
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
2. C2×C4⋊C4×D83C2semidirect
3. #19128038C2semidirect
4. #19128033C2semidirect
5. #19128021C2semidirect
6. #19128016C2semidirect
7. #19127744C2semidirect
8. #19127703C2semidirect
9. #19127700C2semidirect
10. #19127664C2semidirect
11. #428×C83C2semidirect
30. #14581725C22semidirect
31. #14581723C22semidirect
32. #14581717C22semidirect
33. #14581715C22semidirect
34. #14581714C22semidirect
35. #14581712C22semidirect
41. #107×C83C22semidirect
42. #106×C83C22semidirect
156. C83#3092semidirect
164. #428D83semidirect
190. #107C2×D83semidirect
192. #106C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282132
C2×C4⋊C4×D832
#191280382
#191280332
#191280212
#191280162
#191277442
#191277032
#191277002
#191276642
#428×C832
#309283