Finite Group Databasebeta

← #27153332

Subgroups — #27153332

Order 12416.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690836C212yesmaximal, complemented
2#20261390C211yesmaximal, characteristic, complemented
3#20261107C211yesmaximal, characteristic, complemented
4#20260979C212yesmaximal, complemented
5#20260940C211yesmaximal, characteristic, complemented
6#20260927C211yesmaximal, characteristic, complemented
7#20260886C212yesmaximal, complemented
8#20260878C212yesmaximal, complemented
9C4×#13095496C211yesmaximal, characteristic, complemented
10#20260813C211yesmaximal, characteristic, complemented
11D42×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2212yescomplemented
13#15690836—28no—
14C2×#13095499C2212yes—
15C2×#13095499C2212yescomplemented
16C2×#13095499C2212yescomplemented
17C2×#13095496C2211yescharacteristic
18C2×#13095496C2212yescomplemented
19C2×#13095495C2212yescomplemented
20#15690810C2211yescharacteristic, complemented
21#15690792—22no—
22#15690789C2212yescomplemented
23#15690781—21no—
24C4×D388C2211yescharacteristic, complemented
25C4⋊C4×D97—21no—
26#15690743C2211yescharacteristic, complemented
27C4⋊#11715895C2211yescharacteristic, complemented
28#15690733C2212yescomplemented
29#15690731C2211yescharacteristic
30#15690731C2212yescomplemented
31#15690730C2211yescharacteristic, complemented
32C22⋊#11715895C2212yescomplemented
33C4×#11715895C2211yescharacteristic, complemented
34C22×D4×C97—21no—
35C22×D4×C97C2212yes—
36#127×C97C2211yescharacteristic
37#121×C97—21no—
38#121×C97C2212yes—
39#120×C97—22no—
40C4×D4×C97C2211yescharacteristic
41C4×D4×C97C2211yescharacteristic
42#13095511—24no—
43#13095511—42no—
44C2×Dic194C2312yes—
45C2×D388C2311yescharacteristic
46C2×C22⋊D97—22no—
47C2×C4×D97—21no—
48C2×C4×D97—22no—
49#13095499—24no—
50#13095499—24no—
51#13095499—42no—
52#13095496—24no—
53#13095496D414yescomplemented
54#13095495—24no—
55C2×#11715895C2312yes—
56C2×#11715895C2311yescharacteristic
57C2×#11715895C2311yescharacteristic
58C4⋊Dic97—21no—
59C4⋊Dic97—21no—
60C4⋊Dic97C2311yescharacteristic
61#13095490—22no—
62C22⋊Dic97—22no—
63#13095488—21no—
64C24×C97—22no—
65C2×D4×C97—24no—
66C2×D4×C97—22no—
67C2×D4×C97—22no—
68C2×D4×C97—21no—
69C2×D4×C97—21no—
70C2×D4×C97—42no—
71C2×D4×C97—44no—
72C2×D4×C97D411yescharacteristic
73C2×D4×C97C2312yes—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97C2312yes—
77C4⋊C4×C97D411yescharacteristic
78C4⋊C4×C97C2311yescharacteristic
79C22⋊C4×C97—21no—
80C22⋊C4×C97—22no—
81C22⋊C4×C97D412yes—
82C42×C97C2311yescharacteristic
83C23×C97—21no—
84C23×C97—22no—
85C23×C97—44no—
86C23×C97—44no—
87C23×C97—82no—
88C23×C97C2×D412yes—
89C22×D97—21no—
90D4×C97—22no—
91D4×C97—22no—
92D4×C97—21no—
93D4×C97—22no—
94D4×C97—44no—
95D4×C97—44no—
96D4×C97—42no—
97D4×C97—81no—
98D4×C97C2×D412yes—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—21no—
102C2×C4×C97—42no—
103C2×C4×C97C2×D411yescharacteristic
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D411yescharacteristic
106C2×C4×C97C2411yescharacteristic
107C22⋊D97—44no—
108C2×Dic97—21no—
109C2×Dic97—22no—
110D388—21no—
111D388C2×D412yescomplemented
112C4×D97—44no—
113C4×D97—42no—
114Dic194—22no—
115Dic194—22no—
116#11715895—22no—
117#11715895—21no—
118#11715895C2×D412yescomplemented
119C22×C97—22no—
120C22×C97—22no—
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—41no—
124C22×C97—41no—
125C22×C97—42no—
126C22×C97—42no—
127C22×C97—42no—
128C22×C97—88no—
129C22×C97C22×D411yescharacteristic
130C2×D97—22no—
131C2×D97—41no—
132C2×D97—41no—
133C4×C97—22no—
134C4×C97—21no—
135C4×C97—41no—
136C4×C97AGL(1,ℤ/8)12yescomplemented
137C4×C97C22×D411yescharacteristic
138C4×C97H(5,𝔽2)11yescharacteristic
139Dic97—42no—
140Dic97—41no—
141C2×C97—42no—
142C2×C97—42no—
143C2×C97—82no—
144C2×C97D4211yescharacteristic
145C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C97#57711yescharacteristic
147D97—42no—
148#3272—971nomaximal, Sylow
149C97#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—972no—
151#547—971no—
152D42D9711yescharacteristic, complemented
153#492—971no—
154#468—972no—
155#461—971no—
156#459—971no—
157#450—972no—
158#449—972no—
159C4×D8—971no—
160#436—971no—
161C2×#41—972no—
162C22×D4—21no—
163C22×D4C2×D9712yes—
164AGL(1,ℤ/8)—1948no—
165C2×SD16—972no—
166C2×SD16—972no—
167C2×SD16—972no—
168C2×D8—971no—
169C2×D8—972no—
170C2×M16—972no—
171C4⋊Q8—971no—
172#127C2×D9711yescharacteristic
173#123—1942no—
174C22⋊Q8—972no—
175#121—21no—
176#121—1941no—
177#121C2×D9712yes—
178#120—22no—
179C4×D4—971no—
180C4×D4C2×D9711yescharacteristic
181C4×D4C2×D9711yescharacteristic
182C2×C4⋊C4—1941no—
183#107—971no—
184C4⋊C8—971no—
185#103—972no—
186#102—971no—
187#102—972no—
188#102—971no—
189C22⋊C8—972no—
190C4×C8—971no—
191C24—22no—
192#41—3882no—
193#41—1944no—
194C2×Q8—972no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—971no—
203C2×D4—1942no—
204C2×D4C22⋊D9711yescharacteristic
205C2×D4C22×D9712yes—
206C2×D4C22×D9711yescharacteristic
207C22×C4—21no—
208C22×C4—1941no—
209C22×C4—1942no—
210C22×C4C22×D9712yes—
211SD16—3882no—
212SD16—1944no—
213SD16—1944no—
214D8—974no—
215D8—1944no—
216M16—1944no—
217C2×C8—971no—
218C2×C8—971no—
219C2×C8—972no—
220C4⋊C4—971no—
221C4⋊C4—1942no—
222C4⋊C4—1941no—
223C4⋊C4—1941no—
224C4⋊C4C22⋊D9711yescharacteristic
225C4⋊C4C22×D9711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—1942no—
229C22⋊C4—1941no—
230C22⋊C4C22⋊D9712yes—
231C42C22×D9711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—1941no—
238C23C2×C22⋊D9712yes—
239Q8—1942no—
240Q8—1942no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—972no—
250D4—3884no—
251D4—1941no—
252D4D4×D9712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3884no—
258C2×C4—3882no—
259C2×C4—1941no—
260C2×C4—1942no—
261C2×C4C2×C22⋊D9711yescharacteristic
262C2×C4C2×C22⋊D9712yes—
263C2×C4C2×C22⋊D9711yescharacteristic
264C2×C4C23×D9711yescharacteristic
265C8—972no—
266C8—1942no—
267C8—1941no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3881no—
279C22—3881no—
280C22—1942no—
281C22C22×C22⋊D9711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—3881no—
286C4—3882no—
287C4#1569083612yescomplemented
288C4C2×D4×D9711yescharacteristic
289C4#1569085511yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—3882no—
294C2D4×C22⋊D9711yescharacteristic
295C2C2×#1569083611yescharacteristic
296C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD42×C97C2
socleC22×C97C22×D4
hypercenterD42D97
hypocenterC97#3272
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. #20261390C2semidirect
3. #20261107C2semidirect
4. #20260979C2semidirect
5. #20260940C2semidirect
6. #20260927C2semidirect
7. #20260886C2semidirect
8. #20260878C2semidirect
9. C4×#13095496C2semidirect
10. #20260813C2semidirect
11. D42×C97C2semidirect
12. C2×#13095511C22semidirect
15. C2×#13095499C22semidirect
16. C2×#13095499C22semidirect
18. C2×#13095496C22semidirect
19. C2×#13095495C22semidirect
20. #15690810C22semidirect
22. #15690789C22semidirect
24. C4×D388C22semidirect
26. #15690743C22semidirect
27. C4⋊#11715895C22semidirect
28. #15690733C22semidirect
30. #15690731C22semidirect
31. #15690730C22semidirect
32. C22⋊#11715895C22semidirect
33. C4×#11715895C22semidirect
53. #13095496D4semidirect
111. D388C2×D4semidirect
118. #11715895C2×D4semidirect
136. C4×C97AGL(1,ℤ/8)semidirect
149. C97#3272semidirect
152. D42D97semidirect
287. C4#15690836semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
#202613902
#202611072
#202609792
#202609402
#202609272
#202608862
#202608782
C4×#130954962
#202608132
D42×C972
#327297