Finite Group Databasebeta

← #27153331

Subgroups — #27153331

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×#15690834C212yesmaximal, complemented
2#20261389C211yesmaximal, characteristic, complemented
3#20261370C211yesmaximal, characteristic, complemented
4#20261107C211yesmaximal, characteristic, complemented
5#20260980C212yesmaximal, complemented
6#20260941C211yesmaximal, characteristic, complemented
7#20260887C212yesmaximal, complemented
8#20260877C212yesmaximal, complemented
9C4×#13095496C211yesmaximal, characteristic, complemented
10#20260812C211yesmaximal, characteristic, complemented
11#459×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2212yescomplemented
13C2×D4×D97—21no—
14C2×D4×D97C2212yes—
15#15690834—28no—
16C2×#13095500C2212yes—
17C2×#13095499C2212yescomplemented
18C2×#13095499C2212yescomplemented
19C2×#13095496C2211yescharacteristic
20C2×#13095494C2212yescomplemented
21#15690810C2211yescharacteristic, complemented
22#15690794—22no—
23#15690790C2212yescomplemented
24#15690784—21no—
25#15690781—21no—
26#15690781C2212yes—
27#15690778—22no—
28C4×D388C2211yescharacteristic
29C4×D388C2211yescharacteristic
30C2×C4⋊Dic97—21no—
31#15690743C2211yescharacteristic, complemented
32#15690735C2212yescomplemented
33#15690731C2211yescharacteristic, complemented
34#15690730C2211yescharacteristic
35#15690722C2212yescomplemented
36C4×#11715895C2211yescharacteristic, complemented
37C2×D8×C97C2212yescomplemented
38#127×C97C2211yescharacteristic
39C4×D4×C97C2211yescharacteristic, complemented
40C4⋊C8×C97C2211yescharacteristic, complemented
41#102×C97C2212yescomplemented
42C23×D97—22no—
43#13095510—24no—
44#13095510—42no—
45C2×Dic194C2312yes—
46C2×D388D411yescharacteristic
47C2×D388C2311yescharacteristic
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51D4×D97—24no—
52D4×D97—42no—
53D4×D97—44no—
54C22×Dic97—22no—
55C2×C4×D97—21no—
56C2×C4×D97C2312yes—
57#13095500—24no—
58#13095499—24no—
59#13095499—42no—
60#13095496D414yescomplemented
61#13095494—24no—
62C2×#11715895C2311yescharacteristic
63C2×#11715895C2311yescharacteristic
64C4⋊Dic97—21no—
65C4⋊Dic97—21no—
66C4⋊Dic97—22no—
67C4⋊Dic97D411yescharacteristic
68C4⋊Dic97C2311yescharacteristic
69C22⋊Dic97—22no—
70#13095488—22no—
71#13095488—21no—
72#13095488D412yes—
73C2×D4×C97—21no—
74C2×D4×C97—21no—
75C2×D4×C97C2312yes—
76C2×D4×C97C2311yescharacteristic
77C22×C4×C97—21no—
78D8×C97—24no—
79C2×C8×C97C2312yes—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C42×C97C2311yescharacteristic
83C23×C97—21no—
84C23×C97—22no—
85C22×D97—21no—
86C22×D97—44no—
87C22×D97—44no—
88C22×D97—82no—
89C22×D97C2×D412yes—
90D4×C97—22no—
91D4×C97—21no—
92D4×C97—22no—
93D4×C97—42no—
94D4×C97—81no—
95D4×C97C2×D412yescomplemented
96C2×C4×C97—21no—
97C2×C4×C97—42no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—44no—
102C22⋊D97—44no—
103C22⋊D97—44no—
104C2×Dic97—21no—
105C2×Dic97—22no—
106C2×Dic97—44no—
107C2×Dic97C2×D412yes—
108D388—22no—
109D388—21no—
110D388C2×D412yes—
111C4×D97—22no—
112C4×D97—22no—
113C4×D97—42no—
114Dic194—22no—
115Dic194—22no—
116C8×C97—22no—
117#11715895—21no—
118#11715895C2×D412yescomplemented
119C22×C97—22no—
120C22×C97—41no—
121C22×C97—41no—
122C22×C97—42no—
123C22×C97—42no—
124C22×C97—42no—
125C22×C97C22×D411yescharacteristic
126C2×D97—22no—
127C2×D97—22no—
128C2×D97—22no—
129C2×D97—22no—
130C2×D97—41no—
131C2×D97—41no—
132C2×D97—88no—
133C4×C97—21no—
134C4×C97—41no—
135C4×C97AGL(1,ℤ/8)12yescomplemented
136C4×C97C22×D411yescharacteristic
137C4×C97H(5,𝔽2)11yescharacteristic
138Dic97—22no—
139Dic97—42no—
140Dic97—41no—
141C2×C97—42no—
142C2×C97—82no—
143C2×C97D4211yescharacteristic
144C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C97#57711yescharacteristic
146D97—42no—
147D97—42no—
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#461—971no—
156#459D9711yescharacteristic, complemented
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×SD16—972no—
168C2×D8—971no—
169C2×D8C2×D9712yescomplemented
170C2×M16—972no—
171C4⋊Q8—971no—
172#127C2×D9711yescharacteristic
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, complemented
182C2×C4⋊C4—1941no—
183#107—971no—
184C4⋊C8C2×D9711yescharacteristic, complemented
185#103—972no—
186#102—971no—
187#102—971no—
188#102C2×D9712yescomplemented
189C22⋊C8—972no—
190C4×C8—971no—
191C24—1942no—
192#41—3882no—
193#41—1944no—
194C2×Q8—972no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—971no—
198C2×D4—971no—
199C2×D4—3884no—
200C2×D4—3882no—
201C2×D4—1942no—
202C2×D4—1944no—
203C2×D4—1942no—
204C2×D4—1942no—
205C2×D4C22×D9712yes—
206C2×D4C22×D9711yescharacteristic
207C22×C4—21no—
208C22×C4—972no—
209C22×C4—1942no—
210C22×C4—1941no—
211SD16—3882no—
212SD16—1944no—
213SD16—1944no—
214D8—24no—
215D8—974no—
216M16—1944no—
217C2×C8—971no—
218C2×C8—971no—
219C2×C8C22×D9712yes—
220C4⋊C4—971no—
221C4⋊C4—971no—
222C4⋊C4—1942no—
223C4⋊C4—1941no—
224C4⋊C4—1941no—
225C4⋊C4C22×D9711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—972no—
228C22⋊C4—1942no—
229C22⋊C4—1942no—
230C22⋊C4—1941no—
231C42C22×D9711yescharacteristic
232C23—21no—
233C23—22no—
234C23—7762no—
235C23—972no—
236C23—3884no—
237C23—3884no—
238C23—1941no—
239Q8—1942no—
240Q8—1942no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—972no—
247D4—3884no—
248D4—3884no—
249D4—3884no—
250D4—1942no—
251D4—1941no—
252D4D4×D9712yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—972no—
256C2×C4—3884no—
257C2×C4—3882no—
258C2×C4—1941no—
259C2×C4—1942no—
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—42no—
272C22—42no—
273C22—42no—
274C22—7768no—
275C22—3881no—
276C22—3881no—
277C22—1942no—
278C22—1942no—
279C22—1942no—
280C22—1942no—
281C22C2×D4×D9711yescharacteristic
282C4—21no—
283C4—41no—
284C4—3881no—
285C4—3882no—
286C4—1942no—
287C4#1569083412yescomplemented
288C4C2×D4×D9711yescharacteristic
289C4#1569085611yescharacteristic
290C2—42no—
291C2—82no—
292C2—3882no—
293C2—3882no—
294C2#2026136911yescharacteristic
295C2C2×#1569083411yescharacteristic
296C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#459×C97C2
socleC22×C97C22×D4
hypercenter#459D97
hypocenterC97#3272
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #20261389C2semidirect
3. #20261370C2semidirect
4. #20261107C2semidirect
5. #20260980C2semidirect
6. #20260941C2semidirect
7. #20260887C2semidirect
8. #20260877C2semidirect
9. C4×#13095496C2semidirect
10. #20260812C2semidirect
11. #459×C97C2semidirect
12. C2×#13095510C22semidirect
17. C2×#13095499C22semidirect
18. C2×#13095499C22semidirect
20. C2×#13095494C22semidirect
21. #15690810C22semidirect
23. #15690790C22semidirect
31. #15690743C22semidirect
32. #15690735C22semidirect
33. #15690731C22semidirect
35. #15690722C22semidirect
36. C4×#11715895C22semidirect
37. C2×D8×C97C22semidirect
39. C4×D4×C97C22semidirect
40. C4⋊C8×C97C22semidirect
41. #102×C97C22semidirect
60. #13095496D4semidirect
95. D4×C97C2×D4semidirect
118. #11715895C2×D4semidirect
135. C4×C97AGL(1,ℤ/8)semidirect
149. C97#3272semidirect
156. #459D97semidirect
169. C2×D8C2×D97semidirect
181. C4×D4C2×D97semidirect
184. C4⋊C8C2×D97semidirect
188. #102C2×D97semidirect
252. D4D4×D97semidirect
287. C4#15690834semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202613892
#202613702
#202611072
#202609802
#202609412
#202608872
#202608772
C4×#130954962
#202608122
#459×C972
#327297