Finite Group Databasebeta

← #27153357

Subgroups — #27153357

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×#15690837C212yesmaximal, complemented
2#20261407C211yesmaximal, characteristic, complemented
3D4×D388C211yesmaximal, characteristic, complemented
4#20261112C211yesmaximal, characteristic, complemented
5#20260989C212yesmaximal, complemented
6#20260935C211yesmaximal, characteristic, complemented
7#20260930C211yesmaximal, characteristic, complemented
8#20260883C212yesmaximal, complemented
9#20260870C212yesmaximal, complemented
10#20260814C211yesmaximal, characteristic, complemented
11C4×SD16×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yescomplemented
13C22×D388—21no—
14C2×D4×D97C2212yes—
15#15690837—28no—
16C2×#13095498C2212yescomplemented
17C2×D776C2212yescomplemented
18C2×D776C2212yescomplemented
19C2×#13095496C2212yes—
20C2×#13095494C2212yescomplemented
21#15690807—21no—
22#15690807C2211yescharacteristic
23#15690807C2211yescharacteristic, complemented
24#15690785—22no—
25#15690785C2212yes—
26#15690785C2212yescomplemented
27#15690784—21no—
28#15690780—22no—
29C4×D388—21no—
30C4×D388C2211yescharacteristic
31C4⋊#11715895C2211yescharacteristic, complemented
32#15690730C2212yescomplemented
33#15690730C2212yescomplemented
34#15690722C2212yescomplemented
35C2×SD16×C97C2211yescharacteristic
36C4×Q8×C97C2211yescharacteristic, complemented
37C4×D4×C97C2211yescharacteristic
38#106×C97C2211yescharacteristic, complemented
39#103×C97C2211yescharacteristic, complemented
40#102×C97C2211yescharacteristic
41C4×C8×C97C2211yescharacteristic, complemented
42C23×D97—22no—
43#13095512—24no—
44#13095512—42no—
45C2×D388—22no—
46C2×D388—22no—
47C2×D388—21no—
48C2×D388—21no—
49C2×D388—22no—
50C2×D388—21no—
51C2×D388—21no—
52C2×D388—44no—
53C2×D388D411yescharacteristic
54C2×D388C2312yes—
55C2×D388C2312yes—
56C2×C22⋊D97—22no—
57D4×D97—24no—
58D4×D97—42no—
59C2×C4×D97—22no—
60C2×C4×D97C2312yes—
61#13095498—24no—
62D776—24no—
63D776—42no—
64#13095496—24no—
65#13095494—24no—
66C2×#11715895C2312yes—
67C4⋊Dic97D411yescharacteristic
68#13095488—22no—
69#13095488—22no—
70#13095488D412yes—
71C2×Q8×C97C2311yescharacteristic
72C2×D4×C97C2311yescharacteristic
73C22×C4×C97—21no—
74SD16×C97D414yescomplemented
75C2×C8×C97C2311yescharacteristic
76C2×C8×C97C2311yescharacteristic
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2311yescharacteristic
79C4⋊C4×C97C2311yescharacteristic
80C22⋊C4×C97—21no—
81C42×C97—21no—
82C42×C97C2311yescharacteristic
83C23×C97—21no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—44no—
87C22×D97—44no—
88C22×D97—82no—
89C22×D97C2×D412yes—
90Q8×C97—21no—
91Q8×C97C2×D412yescomplemented
92D4×C97—21no—
93D4×C97C2×D412yes—
94C2×C4×C97—21no—
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—42no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—44no—
102C2×Dic97C2×D412yes—
103D388—22no—
104D388—22no—
105D388—22no—
106D388—22no—
107D388—22no—
108D388—44no—
109D388—44no—
110D388—42no—
111D388—42no—
112D388—81no—
113D388—81no—
114D388—82no—
115C4×D97—22no—
116C4×D97—22no—
117C4×D97—44no—
118C8×C97—21no—
119C8×C97C2×D412yescomplemented
120#11715895—22no—
121C22×C97—22no—
122C22×C97—41no—
123C22×C97—41no—
124C22×C97C22×D411yescharacteristic
125C2×D97—22no—
126C2×D97—22no—
127C2×D97—22no—
128C2×D97—42no—
129C2×D97—42no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—88no—
135C4×C97—22no—
136C4×C97—21no—
137C4×C97—41no—
138C4×C97—41no—
139C4×C97AGL(1,ℤ/8)12yescomplemented
140C4×C97C22×D411yescharacteristic
141C4×C97H(5,𝔽2)11yescharacteristic
142Dic97—22no—
143C2×C97—42no—
144C2×C97D4211yescharacteristic
145C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C97#57611yescharacteristic
147D97—42no—
148D97—82no—
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#460—971no—
158#459—971no—
159#449—972no—
160#447—972no—
161C4×SD16D9711yescharacteristic, complemented
162#436—971no—
163C2×#41—972no—
164C22×D4—972no—
165C22×D4—1941no—
166AGL(1,ℤ/8)—1948no—
167C2×SD16—972no—
168C2×SD16C2×D9711yescharacteristic
169C2×D8—972no—
170C2×D8—972no—
171C2×D8—972no—
172C2×M16—972no—
173#127—971no—
174#127—971no—
175#127—1941no—
176#121—972no—
177#121—972no—
178#121—1942no—
179#121—1941no—
180#120—1942no—
181C4×Q8C2×D9711yescharacteristic, complemented
182C4×D4—971no—
183C4×D4—1941no—
184C4×D4C2×D9711yescharacteristic
185#106C2×D9711yescharacteristic, complemented
186C4⋊C8—971no—
187#103C2×D9711yescharacteristic, complemented
188#102—972no—
189#102—972no—
190#102C2×D9711yescharacteristic
191C22⋊C8—972no—
192C4×C8C2×D9711yescharacteristic, complemented
193C24—1942no—
194#41—3882no—
195#41—1944no—
196C2×Q8C22×D9711yescharacteristic
197C2×D4—971no—
198C2×D4—972no—
199C2×D4—972no—
200C2×D4—3884no—
201C2×D4—3882no—
202C2×D4—1942no—
203C2×D4—1942no—
204C2×D4—1942no—
205C2×D4—1942no—
206C2×D4—1944no—
207C2×D4—1941no—
208C2×D4—1941no—
209C2×D4—1941no—
210C2×D4—1941no—
211C2×D4C22×D9711yescharacteristic
212C22×C4—21no—
213C22×C4—972no—
214C22×C4—1942no—
215SD16—1944no—
216SD16D38814yescomplemented
217D8—3882no—
218D8—1944no—
219D8—1944no—
220M16—1944no—
221C2×C8—972no—
222C2×C8C22×D9711yescharacteristic
223C2×C8C22×D9711yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—971no—
226C4⋊C4C22×D9711yescharacteristic
227C4⋊C4C22×D9711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—972no—
230C22⋊C4—1942no—
231C22⋊C4—1942no—
232C42—21no—
233C42C22×D9711yescharacteristic
234C23—21no—
235C23—7762no—
236C23—972no—
237C23—3884no—
238C23—3884no—
239C23—1942no—
240C23—1942no—
241Q8—21no—
242Q8C2×D38812yescomplemented
243D4—21no—
244D4—7761no—
245D4—7761no—
246D4—7762no—
247D4—3884no—
248D4—3884no—
249D4—3884no—
250D4—3882no—
251D4—3882no—
252D4—1942no—
253D4—1942no—
254D4—1942no—
255D4—1942no—
256D4—1942no—
257D4C2×D38812yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—972no—
263C2×C4—3884no—
264C2×C4—1942no—
265C2×C4—1942no—
266C2×C4D4×D9711yescharacteristic
267C2×C4D4×D9711yescharacteristic
268C2×C4C23×D9711yescharacteristic
269C8—21no—
270C8—1942no—
271C8C2×D38812yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—7768no—
276C22—3882no—
277C22—3882no—
278C22—3882no—
279C22—3882no—
280C22—3882no—
281C22—3882no—
282C22—1942no—
283C22—1942no—
284C22—1942no—
285C22C2×D4×D9711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—1942no—
291C4#1569083712yescomplemented
292C4C22×D38811yescharacteristic
293C4#1569085611yescharacteristic
294C2—42no—
295C2—7762no—
296C2—7762no—
297C2—3882no—
298C2D4×D38811yescharacteristic
299C2C2×#1569083711yescharacteristic
300C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
hypercenterC4×SD16D97
hypocenterC97#3274
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. #20261407C2semidirect
3. D4×D388C2semidirect
4. #20261112C2semidirect
5. #20260989C2semidirect
6. #20260935C2semidirect
7. #20260930C2semidirect
8. #20260883C2semidirect
9. #20260870C2semidirect
10. #20260814C2semidirect
11. C4×SD16×C97C2semidirect
12. C2×#13095512C22semidirect
16. C2×#13095498C22semidirect
17. C2×D776C22semidirect
18. C2×D776C22semidirect
20. C2×#13095494C22semidirect
23. #15690807C22semidirect
26. #15690785C22semidirect
31. C4⋊#11715895C22semidirect
32. #15690730C22semidirect
33. #15690730C22semidirect
34. #15690722C22semidirect
36. C4×Q8×C97C22semidirect
38. #106×C97C22semidirect
39. #103×C97C22semidirect
41. C4×C8×C97C22semidirect
74. SD16×C97D4semidirect
91. Q8×C97C2×D4semidirect
119. C8×C97C2×D4semidirect
139. C4×C97AGL(1,ℤ/8)semidirect
151. C97#3274semidirect
161. C4×SD16D97semidirect
181. C4×Q8C2×D97semidirect
185. #106C2×D97semidirect
187. #103C2×D97semidirect
192. C4×C8C2×D97semidirect
216. SD16D388semidirect
242. Q8C2×D388semidirect
271. C8C2×D388semidirect
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
D4×D3882
#202611122
#202609892
#202609352
#202609302
#202608832
#202608702
#202608142
C4×SD16×C972
#327497