Finite Group Databasebeta

← #27153352

Subgroups — #27153352

Order 12416.

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

normal subgroups only

This group has 556 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#20261406C211yesmaximal, characteristic, complemented
3#20261111C211yesmaximal, characteristic, complemented
4#20260986C212yesmaximal, complemented
5#20260932C211yesmaximal, characteristic, complemented
6#20260930C211yesmaximal, characteristic, complemented
7#20260878C212yesmaximal, complemented
8#20260869C212yesmaximal, complemented
9C4×#13095499C211yesmaximal, characteristic, complemented
10#20260813C211yesmaximal, characteristic, complemented
11D42×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2212yescomplemented
13#15690836—28no—
14C2×#13095499C2211yescharacteristic
15C2×#13095499C2212yescomplemented
16C2×#13095496C2212yes—
17C2×#13095496C2212yescomplemented
18C2×#13095496C2212yescomplemented
19C2×#13095495C2212yescomplemented
20#15690807C2211yescharacteristic, complemented
21#15690806—21no—
22#15690784C2212yescomplemented
23#15690783—22no—
24C4×Dic194C2211yescharacteristic, complemented
25#15690772—21no—
26#15690741C2211yescharacteristic, complemented
27C4⋊#11715895C2211yescharacteristic, complemented
28#15690733C2211yescharacteristic, complemented
29#15690731C2211yescharacteristic
30#15690731C2212yescomplemented
31#15690730C2212yescomplemented
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×Dic194C2311yescharacteristic
45C2×D388—21no—
46C2×D388—21no—
47C2×D388C2312yes—
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C4×D97—22no—
51#13095499—24no—
52#13095499D414yescomplemented
53#13095496—24no—
54#13095496—24no—
55#13095496—42no—
56#13095495—24no—
57C2×#11715895C2312yes—
58C2×#11715895C2311yescharacteristic
59C2×#11715895C2311yescharacteristic
60C4⋊Dic97C2311yescharacteristic
61#13095490—21no—
62#13095488—22no—
63C4×Dic97—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—22no—
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—
108C22⋊D97—82no—
109C2×Dic97—21no—
110C2×Dic97—21no—
111D388—22no—
112D388—22no—
113D388—42no—
114D388—81no—
115C4×D97—44no—
116Dic194—21no—
117Dic194C2×D412yescomplemented
118#11715895—22no—
119#11715895—21no—
120#11715895C2×D412yescomplemented
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—22no—
125C22×C97—41no—
126C22×C97—41no—
127C22×C97—42no—
128C22×C97—42no—
129C22×C97—42no—
130C22×C97—88no—
131C22×C97C22×D411yescharacteristic
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—42no—
135C4×C97—22no—
136C4×C97—21no—
137C4×C97—41no—
138C4×C97AGL(1,ℤ/8)12yescomplemented
139C4×C97C22×D411yescharacteristic
140C4×C97H(5,𝔽2)11yescharacteristic
141Dic97—22no—
142Dic97—41no—
143C2×C97—42no—
144C2×C97—42no—
145C2×C97—82no—
146C2×C97D4211yescharacteristic
147C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C97#57611yescharacteristic
149D97—82no—
150#3274—971nomaximal, Sylow
151C97#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—972no—
153#550—971no—
154D42D9711yescharacteristic, complemented
155#493—971no—
156#469—972no—
157#460—971no—
158#459—971no—
159#449—972no—
160#447—972no—
161C4×SD16—971no—
162#436—971no—
163C2×#41—972no—
164C22×D4—21no—
165C22×D4C2×D9712yes—
166AGL(1,ℤ/8)—1948no—
167C2×SD16—971no—
168C2×SD16—972no—
169C2×D8—972no—
170C2×D8—972no—
171C2×D8—972no—
172C2×M16—972no—
173#127—971no—
174#127—1941no—
175#127C2×D9711yescharacteristic
176#121—21no—
177#121—972no—
178#121—1942no—
179#121C2×D9712yes—
180#120—22no—
181C4×Q8—971no—
182C4×D4—1941no—
183C4×D4C2×D9711yescharacteristic
184C4×D4C2×D9711yescharacteristic
185#106—971no—
186C4⋊C8—971no—
187#103—971no—
188#102—971no—
189#102—972no—
190#102—972no—
191C22⋊C8—972no—
192C4×C8—971no—
193C24—22no—
194#41—3882no—
195#41—1944no—
196C2×Q8—971no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—972no—
205C2×D4—1942no—
206C2×D4—1942no—
207C2×D4—1941no—
208C2×D4—1941no—
209C2×D4C22⋊D9711yescharacteristic
210C2×D4C22×D9712yes—
211C2×D4C22×D9711yescharacteristic
212C22×C4—21no—
213C22×C4—1942no—
214C22×C4C22×D9712yes—
215SD16—974no—
216SD16—1944no—
217D8—3882no—
218D8—1944no—
219D8—1944no—
220M16—1944no—
221C2×C8—971no—
222C2×C8—971no—
223C2×C8—972no—
224C4⋊C4—971no—
225C4⋊C4—1941no—
226C4⋊C4C22⋊D9711yescharacteristic
227C4⋊C4C22×D9711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—1942no—
231C22⋊C4C22⋊D9712yes—
232C42—1941no—
233C42C22×D9711yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—1942no—
240C23C2×C22⋊D9712yes—
241Q8—972no—
242Q8—1941no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—7761no—
252D4—7762no—
253D4—3884no—
254D4—3882no—
255D4—1942no—
256D4—1942no—
257D4D4×D9712yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3884no—
263C2×C4—1941no—
264C2×C4—1941no—
265C2×C4C2×C22⋊D9711yescharacteristic
266C2×C4C2×C22⋊D9712yes—
267C2×C4C2×C22⋊D9711yescharacteristic
268C2×C4C23×D9711yescharacteristic
269C8—972no—
270C8—1942no—
271C8—1941no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—3882no—
283C22—3882no—
284C22—3882no—
285C22C22×C22⋊D9711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—3881no—
290C4—1942no—
291C4#1569083612yescomplemented
292C4C2×D4×D9711yescharacteristic
293C4#1569085511yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—7762no—
298C2D4×C22⋊D9711yescharacteristic
299C2C2×#1569083611yescharacteristic
300C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD42×C97C2
socleC22×C97C22×D4
hypercenterD42D97
hypocenterC97#3274
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. #20261406C2semidirect
3. #20261111C2semidirect
4. #20260986C2semidirect
5. #20260932C2semidirect
6. #20260930C2semidirect
7. #20260878C2semidirect
8. #20260869C2semidirect
9. C4×#13095499C2semidirect
10. #20260813C2semidirect
11. D42×C97C2semidirect
12. C2×#13095511C22semidirect
15. C2×#13095499C22semidirect
17. C2×#13095496C22semidirect
18. C2×#13095496C22semidirect
19. C2×#13095495C22semidirect
20. #15690807C22semidirect
22. #15690784C22semidirect
24. C4×Dic194C22semidirect
26. #15690741C22semidirect
27. C4⋊#11715895C22semidirect
28. #15690733C22semidirect
30. #15690731C22semidirect
31. #15690730C22semidirect
32. C22⋊#11715895C22semidirect
33. C4×#11715895C22semidirect
52. #13095499D4semidirect
117. Dic194C2×D4semidirect
120. #11715895C2×D4semidirect
138. C4×C97AGL(1,ℤ/8)semidirect
151. C97#3274semidirect
154. D42D97semidirect
291. C4#15690836semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
#202614062
#202611112
#202609862
#202609322
#202609302
#202608782
#202608692
C4×#130954992
#202608132
D42×C972
#327497