Finite Group Databasebeta

← #27153358

Subgroups — #27153358

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×#15690838C212yesmaximal, complemented
2#20261406C211yesmaximal, characteristic, complemented
3D4×D388C211yesmaximal, characteristic, complemented
4#20261112C211yesmaximal, characteristic, complemented
5#20260988C212yesmaximal, complemented
6#20260936C211yesmaximal, characteristic, complemented
7#20260931C211yesmaximal, characteristic, complemented
8#20260882C212yesmaximal, complemented
9#20260871C212yesmaximal, complemented
10C4×#13095500C211yesmaximal, characteristic, complemented
11#436×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2212yescomplemented
13C22×D388C2212yes—
14C2×D4×D97—21no—
15#15690838—28no—
16C2×#13095500C2211yescharacteristic
17C2×#13095500C2212yescomplemented
18C2×D776C2212yes—
19C2×D776C2212yescomplemented
20C2×D776C2212yescomplemented
21#15690807C2211yescharacteristic
22#15690807C2211yescharacteristic, complemented
23#15690806—21no—
24#15690785—21no—
25#15690784C2212yes—
26#15690784C2212yescomplemented
27#15690783—22no—
28#15690780—22no—
29C4×Dic194C2211yescharacteristic, complemented
30C4×D388C2211yescharacteristic
31#15690772—21no—
32#15690742C2211yescharacteristic, complemented
33#15690735C2211yescharacteristic, complemented
34#15690732C2211yescharacteristic
35#15690732C2212yescomplemented
36#15690732C2212yescomplemented
37C2×M16×C97C2212yescomplemented
38C4×D4×C97C2211yescharacteristic
39C4⋊C8×C97C2211yescharacteristic, complemented
40C22⋊C8×C97C2212yescomplemented
41C4×C8×C97C2211yescharacteristic, complemented
42C23×D97—22no—
43#13095511—24no—
44#13095511—42no—
45C2×Dic194C2311yescharacteristic
46C2×D388—24no—
47C2×D388—21no—
48C2×D388—22no—
49C2×D388—21no—
50C2×D388—21no—
51C2×D388—21no—
52C2×D388—42no—
53C2×D388C2312yes—
54C2×D388C2312yes—
55C2×D388C2311yescharacteristic
56C2×C22⋊D97—22no—
57C2×C22⋊D97—22no—
58C2×C22⋊D97—22no—
59D4×D97—44no—
60C2×C4×D97—22no—
61C2×C4×D97—21no—
62#13095500—24no—
63#13095500D414yescomplemented
64D776—24no—
65D776—24no—
66D776—42no—
67C4⋊Dic97C2311yescharacteristic
68C4⋊Dic97C2311yescharacteristic
69#13095490—21no—
70#13095488—22no—
71#13095488—21no—
72#13095488—22no—
73C4×Dic97—21no—
74C2×D4×C97D411yescharacteristic
75C22×C4×C97C2312yes—
76M16×C97—24no—
77C2×C8×C97C2312yes—
78C2×C8×C97C2311yescharacteristic
79C2×C8×C97C2311yescharacteristic
80C4⋊C4×C97D411yescharacteristic
81C22⋊C4×C97D412yes—
82C42×C97C2311yescharacteristic
83C23×C97C2×D412yes—
84C22×D97—21no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—44no—
88C22×D97—44no—
89C22×D97—82no—
90D4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D412yes—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—44no—
98C22⋊D97—44no—
99C22⋊D97—82no—
100C2×Dic97—21no—
101C2×Dic97—21no—
102C2×Dic97—21no—
103D388—22no—
104D388—22no—
105D388—21no—
106D388—22no—
107D388—22no—
108D388—44no—
109D388—42no—
110D388—42no—
111D388—81no—
112D388—81no—
113D388C2×D412yes—
114C4×D97—44no—
115C4×D97—42no—
116Dic194—21no—
117Dic194C2×D412yescomplemented
118C8×C97—22no—
119C8×C97—21no—
120C8×C97C2×D412yescomplemented
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97C22×D411yescharacteristic
125C2×D97—22no—
126C2×D97—41no—
127C2×D97—41no—
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×C97AGL(1,ℤ/8)12yescomplemented
138C4×C97C22×D411yescharacteristic
139C4×C97H(5,𝔽2)11yescharacteristic
140Dic97—22no—
141Dic97—41no—
142Dic97—41no—
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×SD16—971no—
162#436D9711yescharacteristic, complemented
163C2×#41—972no—
164C22×D4—972no—
165C22×D4—1941no—
166AGL(1,ℤ/8)—1948no—
167C2×SD16—971no—
168C2×SD16—972no—
169C2×D8—972no—
170C2×D8—972no—
171C2×D8—972no—
172C2×M16C2×D9712yescomplemented
173#127—971no—
174#127—971no—
175#127—1941no—
176#121—972no—
177#121—972no—
178#121—1941no—
179#121—1942no—
180#120—1942no—
181C4×Q8—971no—
182C4×D4—971no—
183C4×D4—1941no—
184C4×D4C2×D9711yescharacteristic
185#106—971no—
186C4⋊C8C2×D9711yescharacteristic, complemented
187#103—971no—
188#102—971no—
189#102—972no—
190#102—972no—
191C22⋊C8C2×D9712yescomplemented
192C4×C8C2×D9711yescharacteristic, complemented
193C24—1942no—
194#41—3882no—
195#41—1944no—
196C2×Q8—971no—
197C2×D4—971no—
198C2×D4—972no—
199C2×D4—972no—
200C2×D4—3882no—
201C2×D4—3884no—
202C2×D4—1944no—
203C2×D4—1942no—
204C2×D4—1942no—
205C2×D4—1942no—
206C2×D4—1941no—
207C2×D4—1941no—
208C2×D4—1941no—
209C2×D4—1942no—
210C2×D4—1941no—
211C2×D4D38811yescharacteristic
212C22×C4—1941no—
213C22×C4—1942no—
214C22×C4C22×D9712yes—
215SD16—974no—
216SD16—1944no—
217D8—3882no—
218D8—1944no—
219D8—1944no—
220M16—24no—
221C2×C8C22×D9711yescharacteristic
222C2×C8C22×D9712yes—
223C2×C8C22×D9711yescharacteristic
224C4⋊C4—971no—
225C4⋊C4—971no—
226C4⋊C4—1941no—
227C4⋊C4D38811yescharacteristic
228C22⋊C4—1941no—
229C22⋊C4—1942no—
230C22⋊C4—1942no—
231C22⋊C4D38812yes—
232C42—1941no—
233C42C22×D9711yescharacteristic
234C23—7762no—
235C23—3884no—
236C23—3884no—
237C23—1942no—
238C23—1942no—
239C23—1941no—
240C23C2×D38812yes—
241Q8—972no—
242Q8—1941no—
243D4—22no—
244D4—7761no—
245D4—7761no—
246D4—7762no—
247D4—972no—
248D4—3884no—
249D4—3884no—
250D4—3884no—
251D4—3882no—
252D4—3882no—
253D4—1942no—
254D4—1942no—
255D4—1942no—
256D4—1942no—
257D4—1941no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3884no—
261C2×C4—3882no—
262C2×C4—1941no—
263C2×C4—1941no—
264C2×C4—1941no—
265C2×C4C2×D38811yescharacteristic
266C2×C4C2×D38812yes—
267C2×C4C2×D38811yescharacteristic
268C2×C4C23×D9711yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D9712yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—7768no—
276C22—3882no—
277C22—3882no—
278C22—3882no—
279C22—3882no—
280C22—3882no—
281C22—3882no—
282C22—3881no—
283C22—3881no—
284C22—1942no—
285C22C22×D38811yescharacteristic
286C4—21no—
287C4—22no—
288C4—3881no—
289C4—3881no—
290C4—1942no—
291C4#1569083812yescomplemented
292C4C2×D4×D9711yescharacteristic
293C4#1569085611yescharacteristic
294C2—42no—
295C2—7762no—
296C2—7762no—
297C2—3882no—
298C2D4×D38811yescharacteristic
299C2C2×#1569083811yescharacteristic
300C2#2026154011yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#436×C97C2
socleC22×C97C22×D4
hypercenter#436D97
hypocenterC97#3274
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. #20261406C2semidirect
3. D4×D388C2semidirect
4. #20261112C2semidirect
5. #20260988C2semidirect
6. #20260936C2semidirect
7. #20260931C2semidirect
8. #20260882C2semidirect
9. #20260871C2semidirect
10. C4×#13095500C2semidirect
11. #436×C97C2semidirect
12. C2×#13095511C22semidirect
17. C2×#13095500C22semidirect
19. C2×D776C22semidirect
20. C2×D776C22semidirect
22. #15690807C22semidirect
26. #15690784C22semidirect
29. C4×Dic194C22semidirect
32. #15690742C22semidirect
33. #15690735C22semidirect
35. #15690732C22semidirect
36. #15690732C22semidirect
37. C2×M16×C97C22semidirect
39. C4⋊C8×C97C22semidirect
40. C22⋊C8×C97C22semidirect
41. C4×C8×C97C22semidirect
63. #13095500D4semidirect
117. Dic194C2×D4semidirect
120. C8×C97C2×D4semidirect
137. C4×C97AGL(1,ℤ/8)semidirect
151. C97#3274semidirect
162. #436D97semidirect
172. C2×M16C2×D97semidirect
186. C4⋊C8C2×D97semidirect
191. C22⋊C8C2×D97semidirect
192. C4×C8C2×D97semidirect
271. C8D4×D97semidirect
291. C4#15690838semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
#202614062
D4×D3882
#202611122
#202609882
#202609362
#202609312
#202608822
#202608712
C4×#130955002
#436×C972
#327497