Finite Group Databasebeta

← #24917939

Subgroups — #24917939

Order 10624.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581825C211yesmaximal, characteristic, complemented
2C2×#14581820C211yesmaximal, characteristic, complemented
3C2×#14581814C211yesmaximal, characteristic, complemented
4C22×#13050579C211yesmaximal, characteristic, complemented
5#19128277C211yesmaximal, characteristic, complemented
6#19128270C211yesmaximal, characteristic, complemented
7#19128095C212yesmaximal, complemented
8#19128090C212yesmaximal, complemented
9#19128085C212yesmaximal, complemented
10#19128074C212yesmaximal, complemented
11#405×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050590C2211yescharacteristic
14C22×Dic166C2211yescharacteristic
15C22×D332C2211yescharacteristic
16#14581825—24no—
17#14581820—24no—
18#14581814—24no—
19C2×Dic332C2212yescomplemented
20C2×Dic332C2211yescharacteristic, complemented
21C2×#13050579—24no—
22C2×#13050579C2212yes—
23C2×#13050579C2212yes—
24C2×#13050579C2212yes—
25C2×#13050579C2211yescharacteristic, complemented
26C2×#13050579C2212yescomplemented
27C2×#13050579C2211yescharacteristic, complemented
28C2×D664C2212yes—
29C2×D664C2211yescharacteristic, complemented
30#14581792C2212yescomplemented
31#14581789C2212yes—
32#14581780C2212yes—
33#14581780C2212yescomplemented
34#14581776—22no—
35#14581772—22no—
36#14581767—22no—
37#14581762—22no—
38C2×M16×C83C2211yescharacteristic
39C22×C8×C83C2211yescharacteristic
40#117×C83C2211yescharacteristic
41#97×C83C2212yescomplemented
42C4×C8×C83C2212yescomplemented
43C23×D83—21no—
44#13050590—22no—
45#13050590—22no—
46#13050590—22no—
47#13050590—22no—
48C2×Dic166—22no—
49C2×Dic166—22no—
50C2×Dic166—21no—
51C2×Dic166C2311yescharacteristic
52C2×Dic166C2312yes—
53C2×Dic166C2311yescharacteristic
54C2×D332—22no—
55C2×D332—22no—
56C2×D332—21no—
57C2×D332—24no—
58C2×D332C2312yes—
59C2×D332C2311yescharacteristic
60C2×D332C2311yescharacteristic
61C2×C22⋊D83—21no—
62C2×C22⋊D83—21no—
63C22×Dic83—21no—
64C2×C4×D83—21no—
65C2×C4×D83—21no—
66Dic332—22no—
67Dic332—24no—
68#13050579—24no—
69#13050579—24no—
70#13050579—24no—
71#13050579—22no—
72#13050579—22no—
73#13050579—24no—
74D664—24no—
75D664—22no—
76C4⋊Dic83—24no—
77(C2×D83)⋊C4—24no—
78(C2×D83)⋊C4—24no—
79C22×C4×C83C2311yescharacteristic
80M16×C83D414yescomplemented
81C2×C8×C83D414yescomplemented
82C2×C8×C83C2312yes—
83C2×C8×C83C2311yescharacteristic
84C2×C8×C83C2311yescharacteristic
85C4⋊C4×C83D412yes—
86C22⋊C4×C83D412yes—
87C42×C83C2312yes—
88C23×C83C2×D411yescharacteristic
89C22×D83—21no—
90C22×D83—22no—
91C22×D83—21no—
92C22×D83—42no—
93C22×D83—82no—
94C2×C4×C83C2×D414yes—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D412yes—
97C2×C4×C83C2×D412yes—
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C22⋊D83—42no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—21no—
104C2×Dic83—42no—
105D332—21no—
106D332—22no—
107D332—21no—
108D332—21no—
109D332—22no—
110D332—21no—
111D332—42no—
112D332—84no—
113C4×D83—42no—
114C4×D83—42no—
115Dic166—22no—
116Dic166—21no—
117Dic166—22no—
118Dic166—21no—
119Dic166—21no—
120Dic166—21no—
121Dic166—42no—
122C8×C83C2×D412yes—
123C8×C83C2×D412yes—
124C8×C83C2×D414yes—
125C22×C83—22no—
126C22×C83#12712yes—
127C22×C83C22×D411yescharacteristic
128C2×D83—41no—
129C2×D83—41no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—44no—
133C2×D83—42no—
134C2×D83—84no—
135C4×C83—24no—
136C4×C83#12712yes—
137C4×C83C22×D411yescharacteristic
138C4×C83C22×D411yescharacteristic
139Dic83—42no—
140Dic83—41no—
141Dic83—41no—
142C2×C83—22no—
143C2×C83C2×#12711yescharacteristic
144C2×C83#57712yes—
145D83—81no—
146D83—81no—
147D83—82no—
148#3154—831nomaximal, Sylow
149C83#315411yescharacteristic, complemented, Sylow
150C2×#137—831no—
151C2×AGL(1,ℤ/8)—831no—
152C2×#135—831no—
153C22×SD16—831no—
154#536—831no—
155#535—831no—
156#497—832no—
157#496—832no—
158#495—832no—
159#492—832no—
160#405D8311yescharacteristic, complemented
161C2×#41—831no—
162C2×#41—831no—
163C22×Q8—831no—
164C22×D4—831no—
165#137—1664no—
166AGL(1,ℤ/8)—1664no—
167#135—1664no—
168C2×Q16—831no—
169C2×Q16—832no—
170C2×SD16—832no—
171C2×SD16—832no—
172C2×SD16—832no—
173C2×SD16—831no—
174C2×SD16—831no—
175C2×SD16—832no—
176C2×SD16—1664no—
177C2×D8—832no—
178C2×D8—831no—
179C2×M16C2×D8311yescharacteristic
180C22×C8C2×D8311yescharacteristic
181C4⋊Q8—832no—
182#127—832no—
183#124—832no—
184#124—832no—
185#123—1662no—
186C22⋊Q8—1662no—
187#121—1662no—
188#120—1662no—
189#117C2×D8311yescharacteristic
190#97C2×D8312yescomplemented
191C4×C8C2×D8312yescomplemented
192C24—1661no—
193#41—1662no—
194#41—1662no—
195#41—1662no—
196#41—1662no—
197C2×Q8—831no—
198C2×Q8—832no—
199C2×Q8—831no—
200C2×Q8—1662no—
201C2×Q8—1662no—
202C2×Q8—1661no—
203C2×D4—831no—
204C2×D4—832no—
205C2×D4—831no—
206C2×D4—1661no—
207C2×D4—1661no—
208C2×D4—1661no—
209C2×D4—1662no—
210C2×D4—1662no—
211C2×D4—1664no—
212C22×C4—1661no—
213C22×C4—1661no—
214C22×C4—1661no—
215C22×C4C22×D8311yescharacteristic
216Q16—1662no—
217Q16—1664no—
218SD16—1664no—
219SD16—1664no—
220SD16—1662no—
221SD16—1662no—
222SD16—1664no—
223SD16—1664no—
224D8—1662no—
225D8—1664no—
226M16D33214yescomplemented
227C2×C8D33214yescomplemented
228C2×C8C22×D8311yescharacteristic
229C2×C8C22×D8312yes—
230C2×C8C22×D8311yescharacteristic
231C4⋊C4—1664no—
232C4⋊C4D33212yes—
233C22⋊C4—1664no—
234C22⋊C4—1664no—
235C22⋊C4D33212yes—
236C42C22×D8312yes—
237C23—3322no—
238C23—6642no—
239C23—1661no—
240C23—1662no—
241C23—1661no—
242C23C2×D33211yescharacteristic
243Q8—3322no—
244Q8—1662no—
245Q8—1661no—
246Q8—1662no—
247Q8—1661no—
248Q8—1661no—
249Q8—1661no—
250D4—3322no—
251D4—3322no—
252D4—3322no—
253D4—6644no—
254D4—1661no—
255D4—1662no—
256D4—1661no—
257D4—1661no—
258D4—1662no—
259D4—1661no—
260C2×C4—3322no—
261C2×C4—3322no—
262C2×C4—3322no—
263C2×C4—1661no—
264C2×C4—1662no—
265C2×C4—1661no—
266C2×C4C2×D33214yes—
267C2×C4C2×D33211yescharacteristic
268C2×C4C2×D33212yes—
269C2×C4C2×D33212yes—
270C2×C4C23×D8311yescharacteristic
271C8C2×D33212yes—
272C8C2×D33212yes—
273C8C2×D33214yes—
274C22—22no—
275C22—3321no—
276C22—3321no—
277C22—3322no—
278C22—3322no—
279C22—3324no—
280C22—3322no—
281C22—6644no—
282C22#1458178912yes—
283C22C22×D33211yescharacteristic
284C4—24no—
285C4—3321no—
286C4—3321no—
287C4—3322no—
288C4#1458178912yes—
289C4C22×D33211yescharacteristic
290C4C22×D33211yescharacteristic
291C2—22no—
292C2—6642no—
293C2—6641no—
294C2—6641no—
295C2C2×#1458178911yescharacteristic
296C2#1912851312yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#405×C83C2
socleC22×C83C22×D4
hypercenter#405D83
hypocenterC83#3154
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#14581825C2semidirect
2. C2×#14581820C2semidirect
3. C2×#14581814C2semidirect
4. C22×#13050579C2semidirect
5. #19128277C2semidirect
6. #19128270C2semidirect
7. #19128095C2semidirect
8. #19128090C2semidirect
9. #19128085C2semidirect
10. #19128074C2semidirect
11. #405×C83C2semidirect
19. C2×Dic332C22semidirect
20. C2×Dic332C22semidirect
25. C2×#13050579C22semidirect
26. C2×#13050579C22semidirect
27. C2×#13050579C22semidirect
29. C2×D664C22semidirect
30. #14581792C22semidirect
33. #14581780C22semidirect
41. #97×C83C22semidirect
42. C4×C8×C83C22semidirect
80. M16×C83D4semidirect
81. C2×C8×C83D4semidirect
149. C83#3154semidirect
160. #405D83semidirect
190. #97C2×D83semidirect
191. C4×C8C2×D83semidirect
226. M16D332semidirect
227. C2×C8D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145818252
C2×#145818202
C2×#145818142
C22×#130505792
#191282772
#191282702
#191280952
#191280902
#191280852
#191280742
#405×C832
#315483