Finite Group Databasebeta

← #27153351

Subgroups — #27153351

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×#15690837C212yesmaximal, complemented
2#20261407C211yesmaximal, characteristic, complemented
3#20261370C211yesmaximal, characteristic, complemented
4#20261111C211yesmaximal, characteristic, complemented
5#20260987C212yesmaximal, complemented
6#20260931C211yesmaximal, characteristic, complemented
7#20260879C212yesmaximal, complemented
8#20260870C212yesmaximal, complemented
9C4×#13095498C211yesmaximal, characteristic, complemented
10#20260812C211yesmaximal, characteristic, complemented
11#460×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yescomplemented
13C2×D4×D97—21no—
14C2×D4×D97C2212yes—
15#15690837—28no—
16C2×#13095498C2211yescharacteristic
17C2×D776C2212yes—
18C2×#13095496C2212yescomplemented
19C2×#13095496C2212yescomplemented
20C2×#13095494C2212yescomplemented
21#15690807—21no—
22#15690807C2211yescharacteristic, complemented
23#15690785—22no—
24#15690785C2212yescomplemented
25#15690781—21no—
26#15690781C2212yes—
27#15690778—22no—
28C4×D388—21no—
29C4×D388C2211yescharacteristic
30C4×D388C2211yescharacteristic
31#15690741C2211yescharacteristic, complemented
32#15690734C2211yescharacteristic, complemented
33#15690732C2212yescomplemented
34#15690730C2211yescharacteristic
35#15690722C2212yescomplemented
36C4×#11715895C2211yescharacteristic, complemented
37C2×SD16×C97C2212yescomplemented
38#127×C97C2211yescharacteristic
39C4×Q8×C97C2211yescharacteristic, complemented
40C4⋊C8×C97C2211yescharacteristic, complemented
41#102×C97C2212yescomplemented
42C23×D97—22no—
43#13095512—24no—
44#13095512—42no—
45C2×D388—22no—
46C2×D388—21no—
47C2×D388—22no—
48C2×D388—21no—
49C2×D388D411yescharacteristic
50C2×D388C2312yes—
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—22no—
53C2×C22⋊D97—22no—
54D4×D97—24no—
55D4×D97—42no—
56D4×D97—44no—
57C2×C4×D97—21no—
58C2×C4×D97—22no—
59C2×C4×D97C2312yes—
60#13095498D414yescomplemented
61D776—24no—
62#13095496—24no—
63#13095496—42no—
64#13095494—24no—
65C2×#11715895C2311yescharacteristic
66C2×#11715895C2311yescharacteristic
67C4⋊Dic97D411yescharacteristic
68C4⋊Dic97C2311yescharacteristic
69C22⋊Dic97—22no—
70#13095488—21no—
71#13095488—22no—
72#13095488D412yes—
73C2×Q8×C97C2311yescharacteristic
74C2×D4×C97—21no—
75C2×D4×C97—21no—
76C2×D4×C97C2312yes—
77SD16×C97—24no—
78C2×C8×C97C2312yes—
79C4⋊C4×C97—21no—
80C4⋊C4×C97C2311yescharacteristic
81C42×C97—21no—
82C42×C97C2311yescharacteristic
83C23×C97—22no—
84C22×D97—21no—
85C22×D97—22no—
86C22×D97—44no—
87C22×D97—44no—
88C22×D97—82no—
89C22×D97C2×D412yes—
90Q8×C97—21no—
91Q8×C97C2×D412yescomplemented
92D4×C97—22no—
93D4×C97—22no—
94D4×C97—42no—
95D4×C97—81no—
96C2×C4×C97—21no—
97C2×C4×C97—21no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—44no—
102C22⋊D97—44no—
103C2×Dic97—21no—
104C2×Dic97C2×D412yes—
105D388—22no—
106D388—22no—
107D388—21no—
108D388—22no—
109D388—44no—
110D388—42no—
111D388—81no—
112D388—82no—
113D388C2×D412yes—
114C4×D97—22no—
115C4×D97—22no—
116C4×D97—44no—
117C4×D97—42no—
118C8×C97—22no—
119#11715895—21no—
120#11715895C2×D412yescomplemented
121C22×C97—42no—
122C22×C97—42no—
123C22×C97—42no—
124C22×C97C22×D411yescharacteristic
125C2×D97—22no—
126C2×D97—22no—
127C2×D97—22no—
128C2×D97—22no—
129C2×D97—41no—
130C2×D97—41no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—88no—
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—82no—
144C2×C97D4211yescharacteristic
145C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C97#57611yescharacteristic
147D97—42no—
148D97—42no—
149D97—82no—
150#3274—971nomaximal, Sylow
151C97#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—972no—
153#550—971no—
154D42—971no—
155#493—971no—
156#469—972no—
157#460D9711yescharacteristic, complemented
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—971no—
168C2×SD16C2×D9712yescomplemented
169C2×D8—972no—
170C2×D8—972no—
171C2×D8—972no—
172C2×M16—972no—
173#127—971no—
174#127—1941no—
175#127C2×D9711yescharacteristic
176#121—972no—
177#121—972no—
178#121—1942no—
179#121—1941no—
180#120—1942no—
181C4×Q8C2×D9711yescharacteristic, complemented
182C4×D4—971no—
183C4×D4—971no—
184C4×D4—1941no—
185#106—971no—
186C4⋊C8C2×D9711yescharacteristic, complemented
187#103—971no—
188#102—972no—
189#102—971no—
190#102C2×D9712yescomplemented
191C22⋊C8—972no—
192C4×C8—971no—
193C24—1942no—
194#41—3882no—
195#41—1944no—
196C2×Q8C22×D9711yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—971no—
200C2×D4—971no—
201C2×D4—972no—
202C2×D4—3884no—
203C2×D4—3882no—
204C2×D4—1942no—
205C2×D4—1942no—
206C2×D4—1942no—
207C2×D4—1944no—
208C2×D4—1942no—
209C2×D4—1941no—
210C2×D4—1941no—
211C2×D4C22×D9712yes—
212C22×C4—972no—
213C22×C4—1942no—
214C22×C4—1941no—
215SD16—24no—
216SD16—974no—
217D8—3882no—
218D8—1944no—
219D8—1944no—
220M16—1944no—
221C2×C8—971no—
222C2×C8—971no—
223C2×C8C22×D9712yes—
224C4⋊C4—21no—
225C4⋊C4—971no—
226C4⋊C4—971no—
227C4⋊C4C22×D9711yescharacteristic
228C22⋊C4—972no—
229C22⋊C4—1942no—
230C22⋊C4—1942no—
231C22⋊C4—1941no—
232C42—21no—
233C42C22×D9711yescharacteristic
234C23—22no—
235C23—7762no—
236C23—972no—
237C23—3884no—
238C23—3884no—
239C23—1942no—
240C23—1941no—
241Q8—21no—
242Q8D4×D9712yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—7761no—
248D4—7762no—
249D4—972no—
250D4—3884no—
251D4—3884no—
252D4—3884no—
253D4—3882no—
254D4—1942no—
255D4—1942no—
256D4—1942no—
257D4—1941no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—972no—
261C2×C4—3884no—
262C2×C4—3882no—
263C2×C4—1941no—
264C2×C4—1942no—
265C2×C4—1942no—
266C2×C4D4×D9711yescharacteristic
267C2×C4D4×D9711yescharacteristic
268C2×C4C23×D9711yescharacteristic
269C8—22no—
270C8—972no—
271C8—1941no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—7768no—
276C22—3882no—
277C22—3882no—
278C22—3882no—
279C22—3881no—
280C22—3881no—
281C22—1942no—
282C22—1942no—
283C22—1942no—
284C22—1942no—
285C22C2×D4×D9711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—3881no—
290C4—1942no—
291C4#1569083712yescomplemented
292C4C2×D4×D9711yescharacteristic
293C4#1569085611yescharacteristic
294C2—82no—
295C2—7762no—
296C2—3882no—
297C2—3882no—
298C2#2026136911yescharacteristic
299C2C2×#1569083711yescharacteristic
300C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#460×C97C2
socleC22×C97C22×D4
hypercenter#460D97
hypocenterC97#3274
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. #20261407C2semidirect
3. #20261370C2semidirect
4. #20261111C2semidirect
5. #20260987C2semidirect
6. #20260931C2semidirect
7. #20260879C2semidirect
8. #20260870C2semidirect
9. C4×#13095498C2semidirect
10. #20260812C2semidirect
11. #460×C97C2semidirect
12. C2×#13095512C22semidirect
18. C2×#13095496C22semidirect
19. C2×#13095496C22semidirect
20. C2×#13095494C22semidirect
22. #15690807C22semidirect
24. #15690785C22semidirect
31. #15690741C22semidirect
32. #15690734C22semidirect
33. #15690732C22semidirect
35. #15690722C22semidirect
36. C4×#11715895C22semidirect
37. C2×SD16×C97C22semidirect
39. C4×Q8×C97C22semidirect
40. C4⋊C8×C97C22semidirect
41. #102×C97C22semidirect
60. #13095498D4semidirect
91. Q8×C97C2×D4semidirect
120. #11715895C2×D4semidirect
138. C4×C97AGL(1,ℤ/8)semidirect
151. C97#3274semidirect
157. #460D97semidirect
168. C2×SD16C2×D97semidirect
181. C4×Q8C2×D97semidirect
186. C4⋊C8C2×D97semidirect
190. #102C2×D97semidirect
242. Q8D4×D97semidirect
291. C4#15690837semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
#202614072
#202613702
#202611112
#202609872
#202609312
#202608792
#202608702
C4×#130954982
#202608122
#460×C972
#327497