Finite Group Databasebeta

← #27153339

Subgroups — #27153339

Order 12416.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15690837C212yesmaximal, complemented
2#20261388C211yesmaximal, characteristic, complemented
3D4×D388C211yesmaximal, characteristic, complemented
4#20261109C211yesmaximal, characteristic, complemented
5#20260984C212yesmaximal, complemented
6#20260931C211yesmaximal, characteristic, complemented
7#20260884C212yesmaximal, complemented
8#20260883C212yesmaximal, complemented
9C4×#13095496C211yesmaximal, characteristic, complemented
10#20260812C211yesmaximal, characteristic, complemented
11#461×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yescomplemented
13C22×D388—21no—
14C2×D4×D97C2212yes—
15#15690837—28no—
16C2×#13095498C2212yescomplemented
17C2×#13095498C2212yescomplemented
18C2×D776C2212yescomplemented
19C2×#13095496C2211yescharacteristic
20C2×#13095494C2212yescomplemented
21#15690807C2211yescharacteristic
22#15690795—22no—
23#15690786C2212yescomplemented
24#15690785—21no—
25#15690785C2212yes—
26#15690784—21no—
27#15690780—22no—
28C4×D388C2211yescharacteristic
29C4×D388C2211yescharacteristic, complemented
30C4⋊C4×D97—21no—
31#15690743C2211yescharacteristic, complemented
32#15690732C2212yescomplemented
33#15690731C2211yescharacteristic
34#15690730C2211yescharacteristic, complemented
35#15690722C2212yescomplemented
36C4×#11715895C2211yescharacteristic, complemented
37C2×SD16×C97C2212yes—
38C4⋊Q8×C97C2211yescharacteristic, complemented
39C4×D4×C97C2211yescharacteristic
40C4⋊C8×C97C2211yescharacteristic, complemented
41#103×C97C2212yescomplemented
42C23×D97—22no—
43#13095512—24no—
44#13095512—42no—
45C2×D388—22no—
46C2×D388—22no—
47C2×D388—21no—
48C2×D388—21no—
49C2×D388—44no—
50C2×D388D411yescharacteristic
51C2×D388C2312yes—
52C2×D388C2311yescharacteristic
53C2×C22⋊D97—22no—
54D4×D97—24no—
55D4×D97—42no—
56C2×C4×D97—21no—
57C2×C4×D97—22no—
58C2×C4×D97C2312yes—
59#13095498—24no—
60#13095498—42no—
61D776—24no—
62#13095496D414yescomplemented
63#13095494—24no—
64C2×#11715895C2311yescharacteristic
65C2×#11715895C2311yescharacteristic
66C4⋊Dic97D411yescharacteristic
67C4⋊Dic97C2311yescharacteristic
68#13095490—22no—
69#13095488—22no—
70#13095488—21no—
71#13095488—22no—
72#13095488D412yes—
73C2×Q8×C97C2312yes—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76SD16×C97—24no—
77C2×C8×C97C2312yes—
78C4⋊C4×C97—21no—
79C4⋊C4×C97—21no—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C42×C97C2311yescharacteristic
83C23×C97—21no—
84C22×D97—21no—
85C22×D97—22no—
86C22×D97—44no—
87C22×D97—44no—
88C22×D97—82no—
89C22×D97C2×D412yes—
90Q8×C97—22no—
91Q8×C97—22no—
92D4×C97—21no—
93D4×C97C2×D412yes—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—42no—
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2411yescharacteristic
100C22⋊D97—44no—
101C2×Dic97—21no—
102C2×Dic97C2×D412yes—
103D388—22no—
104D388—22no—
105D388—21no—
106D388—22no—
107D388—44no—
108D388—44no—
109D388—42no—
110D388—81no—
111D388C2×D412yescomplemented
112C4×D97—22no—
113C4×D97—22no—
114C4×D97—44no—
115C4×D97—42no—
116C8×C97—22no—
117#11715895—21no—
118#11715895C2×D412yescomplemented
119C22×C97—22no—
120C22×C97—41no—
121C22×C97—41no—
122C22×C97C22×D411yescharacteristic
123C2×D97—22no—
124C2×D97—22no—
125C2×D97—22no—
126C2×D97—22no—
127C2×D97—41no—
128C2×D97—41no—
129C2×D97—42no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—88no—
133C4×C97—21no—
134C4×C97—42no—
135C4×C97—41no—
136C4×C97AGL(1,ℤ/8)12yescomplemented
137C4×C97C22×D411yescharacteristic
138C4×C97H(5,𝔽2)11yescharacteristic
139Dic97—22no—
140Dic97—41no—
141C2×C97—42no—
142C2×C97D4211yescharacteristic
143C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C97#57711yescharacteristic
145D97—42no—
146D97—42no—
147D97—82no—
148#3272—971nomaximal, Sylow
149C97#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—972no—
151#547—971no—
152D42—971no—
153#492—971no—
154#468—972no—
155#461D9711yescharacteristic, complemented
156#459—971no—
157#450—972no—
158#449—972no—
159C4×D8—971no—
160#436—971no—
161C2×#41—972no—
162C22×D4—972no—
163C22×D4—1941no—
164AGL(1,ℤ/8)—1948no—
165C2×SD16—972no—
166C2×SD16—972no—
167C2×SD16C2×D9712yes—
168C2×D8—971no—
169C2×D8—972no—
170C2×M16—972no—
171C4⋊Q8C2×D9711yescharacteristic, complemented
172#127—971no—
173#123—1942no—
174C22⋊Q8—972no—
175#121—972no—
176#121—1941no—
177#121—1941no—
178#120—1942no—
179C4×D4—971no—
180C4×D4—971no—
181C4×D4C2×D9711yescharacteristic
182C2×C4⋊C4—1941no—
183#107—971no—
184C4⋊C8C2×D9711yescharacteristic, complemented
185#103C2×D9712yescomplemented
186#102—971no—
187#102—972no—
188#102—971no—
189C22⋊C8—972no—
190C4×C8—971no—
191C24—1942no—
192#41—3882no—
193#41—1944no—
194C2×Q8C22×D9712yes—
195C2×D4—971no—
196C2×D4—971no—
197C2×D4—972no—
198C2×D4—3884no—
199C2×D4—3882no—
200C2×D4—1942no—
201C2×D4—1942no—
202C2×D4—1942no—
203C2×D4—1944no—
204C2×D4—1941no—
205C2×D4—1941no—
206C2×D4C22×D9711yescharacteristic
207C22×C4—21no—
208C22×C4—972no—
209C22×C4—1942no—
210C22×C4—1941no—
211SD16—24no—
212SD16—3882no—
213SD16—1944no—
214D8—974no—
215D8—1944no—
216M16—1944no—
217C2×C8—971no—
218C2×C8—971no—
219C2×C8C22×D9712yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—971no—
223C4⋊C4—971no—
224C4⋊C4—1942no—
225C4⋊C4C22×D9711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—972no—
228C22⋊C4—1942no—
229C22⋊C4—1942no—
230C22⋊C4—1941no—
231C42C22×D9711yescharacteristic
232C23—21no—
233C23—7762no—
234C23—972no—
235C23—3884no—
236C23—3884no—
237C23—1942no—
238C23—1941no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—7761no—
243D4—972no—
244D4—3884no—
245D4—3884no—
246D4—3884no—
247D4—3882no—
248D4—1942no—
249D4—1942no—
250D4—1942no—
251D4—1941no—
252D4D4×D9712yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—972no—
257C2×C4—3884no—
258C2×C4—3882no—
259C2×C4—1941no—
260C2×C4—1942no—
261C2×C4—1942no—
262C2×C4D4×D9711yescharacteristic
263C2×C4D4×D9711yescharacteristic
264C2×C4C23×D9711yescharacteristic
265C8—22no—
266C8—972no—
267C8—1941no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—7768no—
272C22—3882no—
273C22—3882no—
274C22—3882no—
275C22—3881no—
276C22—3881no—
277C22—1942no—
278C22—1942no—
279C22—1942no—
280C22—1942no—
281C22C2×D4×D9711yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—3881no—
286C4—1942no—
287C4#1569083712yescomplemented
288C4C2×D4×D9711yescharacteristic
289C4#1569085611yescharacteristic
290C2—42no—
291C2—7762no—
292C2—3882no—
293C2—3882no—
294C2#2026136911yescharacteristic
295C2C2×#1569083711yescharacteristic
296C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#461×C97C2
socleC22×C97C22×D4
hypercenter#461D97
hypocenterC97#3272
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. #20261388C2semidirect
3. D4×D388C2semidirect
4. #20261109C2semidirect
5. #20260984C2semidirect
6. #20260931C2semidirect
7. #20260884C2semidirect
8. #20260883C2semidirect
9. C4×#13095496C2semidirect
10. #20260812C2semidirect
11. #461×C97C2semidirect
12. C2×#13095512C22semidirect
16. C2×#13095498C22semidirect
17. C2×#13095498C22semidirect
18. C2×D776C22semidirect
20. C2×#13095494C22semidirect
23. #15690786C22semidirect
29. C4×D388C22semidirect
31. #15690743C22semidirect
32. #15690732C22semidirect
34. #15690730C22semidirect
35. #15690722C22semidirect
36. C4×#11715895C22semidirect
38. C4⋊Q8×C97C22semidirect
40. C4⋊C8×C97C22semidirect
41. #103×C97C22semidirect
62. #13095496D4semidirect
111. D388C2×D4semidirect
118. #11715895C2×D4semidirect
136. C4×C97AGL(1,ℤ/8)semidirect
149. C97#3272semidirect
155. #461D97semidirect
171. C4⋊Q8C2×D97semidirect
184. C4⋊C8C2×D97semidirect
185. #103C2×D97semidirect
287. C4#15690837semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
#202613882
D4×D3882
#202611092
#202609842
#202609312
#202608842
#202608832
C4×#130954962
#202608122
#461×C972
#327297