Finite Group Databasebeta

← #27153355

Subgroups — #27153355

Order 12416.

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

normal subgroups only

This group has 556 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#20261405C211yesmaximal, characteristic, complemented
3#20261370C211yesmaximal, characteristic, complemented
4#20260985C212yesmaximal, complemented
5#20260932C211yesmaximal, characteristic, complemented
6#20260927C211yesmaximal, characteristic, complemented
7#20260877C212yesmaximal, complemented
8#20260867C212yesmaximal, complemented
9C4×#13095500C211yesmaximal, characteristic, complemented
10#20260814C211yesmaximal, characteristic, complemented
11#493×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2212yescomplemented
13C2×D4×D97—21no—
14C2×D4×D97C2212yes—
15#15690834—28no—
16C2×#13095500C2211yescharacteristic
17C2×#13095499C2212yescomplemented
18C2×#13095496C2212yes—
19C2×#13095494C2212yescomplemented
20#15690806—21no—
21#15690782—22no—
22#15690781—21no—
23#15690781C2212yes—
24#15690781C2212yescomplemented
25#15690778—22no—
26C4×Dic194C2211yescharacteristic, complemented
27C4×D388C2211yescharacteristic
28C4×D388C2211yescharacteristic
29D4×Dic97—21no—
30#15690742C2211yescharacteristic, complemented
31C4⋊#11715895C2211yescharacteristic, complemented
32#15690735C2211yescharacteristic, complemented
33#15690732C2211yescharacteristic
34#15690731C2212yescomplemented
35#15690731C2212yescomplemented
36#15690722C2212yescomplemented
37C2×D8×C97C2212yescomplemented
38C2×D8×C97C2212yescomplemented
39#127×C97C2211yescharacteristic
40#127×C97C2211yescharacteristic, complemented
41C4×C8×C97C2211yescharacteristic, complemented
42C23×D97—22no—
43#13095510—24no—
44#13095510—42no—
45C2×Dic194C2311yescharacteristic
46C2×D388D411yescharacteristic
47C2×D388C2311yescharacteristic
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51C2×C22⋊D97—22no—
52D4×D97—24no—
53D4×D97—42no—
54D4×D97—44no—
55C22×Dic97—22no—
56C2×C4×D97—21no—
57C2×C4×D97C2312yes—
58#13095500D414yescomplemented
59#13095499—24no—
60#13095496—24no—
61#13095494—24no—
62C2×#11715895C2312yes—
63C4⋊Dic97D411yescharacteristic
64C4⋊Dic97C2311yescharacteristic
65C4⋊Dic97C2311yescharacteristic
66#13095490—21no—
67C22⋊Dic97—22no—
68C22⋊Dic97—22no—
69#13095488—21no—
70#13095488D412yes—
71C4×Dic97—21no—
72C2×D4×C97—21no—
73C2×D4×C97—21no—
74C2×D4×C97—21no—
75C2×D4×C97—21no—
76C2×D4×C97C2312yes—
77C2×D4×C97C2312yes—
78D8×C97—24no—
79D8×C97—42no—
80C2×C8×C97C2311yescharacteristic
81C2×C8×C97C2311yescharacteristic
82C42×C97C2311yescharacteristic
83C23×C97—22no—
84C23×C97—22no—
85C22×D97—21no—
86C22×D97—44no—
87C22×D97—44no—
88C22×D97—82no—
89C22×D97C2×D412yes—
90D4×C97—22no—
91D4×C97—22no—
92D4×C97—22no—
93D4×C97—22no—
94D4×C97—42no—
95D4×C97—42no—
96D4×C97—81no—
97D4×C97—81no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—44no—
102C22⋊D97—44no—
103C22⋊D97—44no—
104C22⋊D97—82no—
105C2×Dic97—21no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108C2×Dic97—44no—
109C2×Dic97C2×D412yes—
110D388—22no—
111D388—21no—
112D388C2×D412yes—
113C4×D97—22no—
114C4×D97—22no—
115C4×D97—42no—
116Dic194—21no—
117Dic194C2×D412yescomplemented
118C8×C97—21no—
119C8×C97C2×D412yescomplemented
120#11715895—22no—
121C22×C97—42no—
122C22×C97—42no—
123C22×C97—42no—
124C22×C97—42no—
125C22×C97—42no—
126C22×C97—42no—
127C22×C97C22×D411yescharacteristic
128C2×D97—22no—
129C2×D97—22no—
130C2×D97—22no—
131C2×D97—22no—
132C2×D97—41no—
133C2×D97—41no—
134C2×D97—88no—
135C4×C97—21no—
136C4×C97AGL(1,ℤ/8)12yescomplemented
137C4×C97C22×D411yescharacteristic
138C4×C97H(5,𝔽2)11yescharacteristic
139Dic97—22no—
140Dic97—22no—
141Dic97—41no—
142Dic97—41no—
143C2×C97—82no—
144C2×C97—82no—
145C2×C97D4211yescharacteristic
146C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C97#57611yescharacteristic
148D97—42no—
149D97—42no—
150#3274—971nomaximal, Sylow
151C97#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—972no—
153#550—971no—
154D42—971no—
155#493D9711yescharacteristic, complemented
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—972no—
165C22×D4—1941no—
166AGL(1,ℤ/8)—1948no—
167C2×SD16—972no—
168C2×SD16—971no—
169C2×D8—972no—
170C2×D8C2×D9712yescomplemented
171C2×D8C2×D9712yescomplemented
172C2×M16—972no—
173#127—1941no—
174#127C2×D9711yescharacteristic
175#127C2×D9711yescharacteristic, complemented
176#121—972no—
177#121—972no—
178#121—1942no—
179#121—1941no—
180#120—1942no—
181C4×Q8—971no—
182C4×D4—971no—
183C4×D4—971no—
184C4×D4—1941no—
185#106—971no—
186C4⋊C8—971no—
187#103—971no—
188#102—972no—
189#102—972no—
190#102—971no—
191C22⋊C8—972no—
192C4×C8C2×D9711yescharacteristic, complemented
193C24—1942no—
194#41—3882no—
195#41—1944no—
196C2×Q8—971no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—971no—
202C2×D4—971no—
203C2×D4—3884no—
204C2×D4—3882no—
205C2×D4—1942no—
206C2×D4—1944no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4—1942no—
210C2×D4C22×D9712yes—
211C2×D4C22×D9712yes—
212C22×C4—972no—
213C22×C4—1942no—
214C22×C4—1941no—
215SD16—974no—
216SD16—1944no—
217D8—24no—
218D8—42no—
219D8—1944no—
220M16—1944no—
221C2×C8—972no—
222C2×C8C22×D9711yescharacteristic
223C2×C8C22×D9711yescharacteristic
224C4⋊C4—971no—
225C4⋊C4—971no—
226C4⋊C4—971no—
227C4⋊C4—1941no—
228C22⋊C4—972no—
229C22⋊C4—1942no—
230C22⋊C4—1942no—
231C22⋊C4—1941no—
232C42—1941no—
233C42C22×D9711yescharacteristic
234C23—22no—
235C23—22no—
236C23—7762no—
237C23—972no—
238C23—3884no—
239C23—3884no—
240C23—1941no—
241Q8—972no—
242Q8—1941no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—7762no—
252D4—972no—
253D4—3884no—
254D4—3884no—
255D4—3884no—
256D4—1942no—
257D4—1941no—
258C2×C4—972no—
259C2×C4—3882no—
260C2×C4—3884no—
261C2×C4—1941no—
262C2×C4—1941no—
263C2×C4—1941no—
264C2×C4—1942no—
265C2×C4—1942no—
266C2×C4D4×D9711yescharacteristic
267C2×C4D4×D9711yescharacteristic
268C2×C4C23×D9711yescharacteristic
269C8—21no—
270C8—1942no—
271C8D4×D9712yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—7768no—
279C22—3881no—
280C22—3881no—
281C22—1942no—
282C22—1942no—
283C22—1942no—
284C22—1942no—
285C22C2×D4×D9711yescharacteristic
286C4—21no—
287C4—3881no—
288C4—3881no—
289C4—1942no—
290C4—1942no—
291C4#1569083412yescomplemented
292C4C2×D4×D9711yescharacteristic
293C4#1569085511yescharacteristic
294C2—82no—
295C2—82no—
296C2—3882no—
297C2—3882no—
298C2#2026137011yescharacteristic
299C2C2×#1569083411yescharacteristic
300C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#493×C97C2
socleC22×C97C22×D4
hypercenter#493D97
hypocenterC97#3274
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #20261405C2semidirect
3. #20261370C2semidirect
4. #20260985C2semidirect
5. #20260932C2semidirect
6. #20260927C2semidirect
7. #20260877C2semidirect
8. #20260867C2semidirect
9. C4×#13095500C2semidirect
10. #20260814C2semidirect
11. #493×C97C2semidirect
12. C2×#13095510C22semidirect
17. C2×#13095499C22semidirect
19. C2×#13095494C22semidirect
24. #15690781C22semidirect
26. C4×Dic194C22semidirect
30. #15690742C22semidirect
31. C4⋊#11715895C22semidirect
32. #15690735C22semidirect
34. #15690731C22semidirect
35. #15690731C22semidirect
36. #15690722C22semidirect
37. C2×D8×C97C22semidirect
38. C2×D8×C97C22semidirect
40. #127×C97C22semidirect
41. C4×C8×C97C22semidirect
58. #13095500D4semidirect
117. Dic194C2×D4semidirect
119. C8×C97C2×D4semidirect
136. C4×C97AGL(1,ℤ/8)semidirect
151. C97#3274semidirect
155. #493D97semidirect
170. C2×D8C2×D97semidirect
171. C2×D8C2×D97semidirect
175. #127C2×D97semidirect
192. C4×C8C2×D97semidirect
271. C8D4×D97semidirect
291. C4#15690834semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202614052
#202613702
#202609852
#202609322
#202609272
#202608772
#202608672
C4×#130955002
#202608142
#493×C972
#327497