Finite Group Databasebeta

← #24918445

Subgroups — #24918445

Order 10624.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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#19128249C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic, complemented
3#19128088C211yesmaximal, characteristic, complemented
4#19128061C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19127918C212yesmaximal, complemented
7#19127914C212yesmaximal, complemented
8#19127838C212yesmaximal, complemented
9#19127833C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic, complemented
11#496×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C2×Q8×D83C2211yescharacteristic
14C2×D4×D83C2211yescharacteristic
15C2×#13050578C2212yescomplemented
16C2×#13050575C2212yes—
17#14581792C2211yescharacteristic, complemented
18#14581781C2211yescharacteristic
19#14581780C2211yescharacteristic
20#14581779—21no—
21#14581779C2211yescharacteristic
22#14581778—21no—
23#14581778—21no—
24#14581763—21no—
25#14581763C2212yes—
26D4×Dic83—21no—
27D4×Dic83C2212yescomplemented
28C22⋊C4×D83—21no—
29C22⋊C4×D83—21no—
30C42×D83C2211yescharacteristic
31Dic83⋊C8C2212yescomplemented
32#14581717C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yes—
35#14581713C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic, complemented
38C2×SD16×C83C2212yescomplemented
39C2×D8×C83C2212yes—
40#127×C83C2211yescharacteristic, complemented
41#124×C83C2211yescharacteristic
42#97×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44#13050589—44no—
45C2×Dic166—21no—
46C2×Dic166—21no—
47C2×Dic166C2311yescharacteristic
48Q8×D83—42no—
49C2×D332C2311yescharacteristic
50C2×C22⋊D83—22no—
51C2×C22⋊D83—21no—
52D4×D83—42no—
53C22×Dic83—22no—
54C2×C4×D83—21no—
55C2×C4×D83D411yescharacteristic
56C2×C4×D83D411yescharacteristic
57C2×C4×D83C2311yescharacteristic
58#13050578—42no—
59#13050575—42no—
60C2×C83⋊C8C2312yes—
61C4⋊Dic83—21no—
62C4⋊Dic83C2312yes—
63C22⋊Dic83—22no—
64C22⋊Dic83—22no—
65C22⋊Dic83—21no—
66C22⋊Dic83—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71C4×Dic83D411yescharacteristic
72C4×Dic83D411yescharacteristic
73C4×Dic83C2311yescharacteristic
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83—21no—
76C2×D4×C83—21no—
77C2×D4×C83C2312yes—
78C2×D4×C83C2311yescharacteristic
79SD16×C83—42no—
80D8×C83—42no—
81C2×C8×C83C2312yes—
82C22⋊C4×C83—21no—
83C22⋊C4×C83—21no—
84C42×C83C2311yescharacteristic
85C23×C83—21no—
86C23×C83—22no—
87C22×D83—21no—
88C22×D83—42no—
89C22×D83—42no—
90C22×D83—42no—
91C22×D83C2×D411yescharacteristic
92Q8×C83—22no—
93D4×C83—22no—
94D4×C83—22no—
95D4×C83—22no—
96D4×C83—42no—
97D4×C83—42no—
98C2×C4×C83—21no—
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2411yescharacteristic
102C22⋊D83—42no—
103C22⋊D83—44no—
104C2×Dic83—21no—
105C2×Dic83—22no—
106C2×Dic83—44no—
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110D332—22no—
111C4×D83—22no—
112C4×D83—22no—
113C4×D83—41no—
114C4×D83—42no—
115Dic166—22no—
116Dic166—42no—
117Dic166—42no—
118C8×C83—22no—
119C83⋊C8—22no—
120C22×C83—41no—
121C22×C83—42no—
122C22×C83—42no—
123C22×C83—41no—
124C22×C83—42no—
125C22×C83—41no—
126C22×C83C22×D411yescharacteristic
127C2×D83—21no—
128C2×D83—21no—
129C2×D83—41no—
130C2×D83—41no—
131C2×D83—41no—
132C2×D83—84no—
133C2×D83AGL(1,ℤ/8)12yes—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83H(5,𝔽2)12yes—
138Dic83—21no—
139Dic83—21no—
140Dic83—21no—
141Dic83—41no—
142Dic83—42no—
143C2×C83—81no—
144C2×C83—82no—
145C2×C83#53511yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C83#57511yescharacteristic
148D83—22no—
149D83—81no—
150#3213—831nomaximal, Sylow
151C83#321311yescharacteristic, complemented, Sylow
152#532—831no—
153C2×#124—831no—
154#496—831no—
155#496D8311yescharacteristic, complemented
156#489—831no—
157#489—831no—
158#463—832no—
159#463—832no—
160#448—832no—
161#447—832no—
162#432—831no—
163C2×#41—1661no—
164C22×Q8—831no—
165C22×D4—831no—
166C2×SD16—832no—
167C2×SD16C2×D8312yescomplemented
168C2×D8—832no—
169C2×D8C2×D8312yes—
170C4⋊Q8—831no—
171#127C2×D8311yescharacteristic, complemented
172#124—831no—
173#124—831no—
174#124—831no—
175#124—1661no—
176#124—1661no—
177#124—1661no—
178#124C2×D8311yescharacteristic
179#121—832no—
180#121—1661no—
181C4×D4—832no—
182C4×D4—1661no—
183C2×C22⋊C4—1661no—
184C2×C22⋊C4—1661no—
185C2×C42—831no—
186C4⋊C8—832no—
187#103—832no—
188#103—832no—
189#102—832no—
190#102—832no—
191C22⋊C8—832no—
192#97—831no—
193#97C2×D8311yescharacteristic, complemented
194C24—1661no—
195#41—3324no—
196C2×Q8—3322no—
197C2×Q8—831no—
198C2×Q8—1661no—
199C2×Q8—1661no—
200C2×Q8C22×D8311yescharacteristic
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—3322no—
204C2×D4—831no—
205C2×D4—1661no—
206C2×D4—1662no—
207C2×D4C22×D8311yescharacteristic
208C2×D4C22×D8312yes—
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—831no—
212C22×C4—1661no—
213C22×C4—1662no—
214SD16—42no—
215SD16—3322no—
216D8—42no—
217D8—3322no—
218C2×C8—832no—
219C2×C8C22×D8312yes—
220C4⋊C4—832no—
221C4⋊C4—1661no—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—1661no—
225C22⋊C4—1661no—
226C22⋊C4—1661no—
227C22⋊C4—1661no—
228C22⋊C4—1662no—
229C22⋊C4—1662no—
230C22⋊C4—1661no—
231C22⋊C4—1661no—
232C42—831no—
233C42—831no—
234C42—831no—
235C42C22×D8311yescharacteristic
236C23—21no—
237C23—22no—
238C23—3322no—
239C23—3322no—
240C23—3322no—
241C23—831no—
242C23—1661no—
243Q8—22no—
244Q8—3322no—
245Q8—3322no—
246Q8—1662no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—3324no—
253D4—3322no—
254D4—1662no—
255C2×C4—21no—
256C2×C4—3321no—
257C2×C4—3322no—
258C2×C4—3324no—
259C2×C4—831no—
260C2×C4—831no—
261C2×C4—831no—
262C2×C4—1662no—
263C2×C4—1661no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4D4×D8311yescharacteristic
267C2×C4D4×D8311yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—22no—
270C8—1662no—
271C22—42no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—3321no—
278C22—3321no—
279C22—3321no—
280C22—832no—
281C22—6644no—
282C22—1661no—
283C22—1661no—
284C22C2×D4×D8311yescharacteristic
285C4—21no—
286C4—21no—
287C4—41no—
288C4—3322no—
289C4—3321no—
290C4—1661no—
291C4—1661no—
292C4—1661no—
293C4#1458183712yes—
294C2—81no—
295C2—82no—
296C2—6641no—
297C2—1662no—
298C2#1912826411yescharacteristic
299C2AGL(1,ℤ/8)×D8311yescharacteristic
300C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#496×C83C2
socleC22×C83C22×D4
hypercenter#496D83
hypocenterC83#3213
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #19128249C2semidirect
2. #124×D83C2semidirect
3. #19128088C2semidirect
4. #19128061C2semidirect
5. #19128060C2semidirect
6. #19127918C2semidirect
7. #19127914C2semidirect
8. #19127838C2semidirect
9. #19127833C2semidirect
10. #19127756C2semidirect
11. #496×C83C2semidirect
15. C2×#13050578C22semidirect
17. #14581792C22semidirect
27. D4×Dic83C22semidirect
31. Dic83⋊C8C22semidirect
32. #14581717C22semidirect
33. #14581716C22semidirect
37. #14581703C22semidirect
38. C2×SD16×C83C22semidirect
40. #127×C83C22semidirect
42. #97×C83C22semidirect
151. C83#3213semidirect
155. #496D83semidirect
167. C2×SD16C2×D83semidirect
171. #127C2×D83semidirect
193. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282492
#124×D832
#191280882
#191280612
#191280602
#191279182
#191279142
#191278382
#191278332
#191277562
#496×C832
#321383