Finite Group Databasebeta

← #27153338

Subgroups — #27153338

Order 12416.

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

normal subgroups only

This group has 546 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#20261390C211yesmaximal, characteristic, complemented
3D4×D388C211yesmaximal, characteristic, complemented
4#20261108C211yesmaximal, characteristic, complemented
5#20260981C212yesmaximal, complemented
6#20260941C211yesmaximal, characteristic, complemented
7#20260931C211yesmaximal, characteristic, complemented
8#20260888C212yesmaximal, complemented
9#20260882C212yesmaximal, complemented
10C4×D776C211yesmaximal, characteristic, complemented
11#436×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2212yescomplemented
13C22×D388C2212yes—
14C2×D4×D97—21no—
15#15690838—28no—
16C2×#13095500C2212yes—
17C2×#13095500C2212yescomplemented
18C2×#13095500C2212yescomplemented
19C2×D776C2211yescharacteristic
20C2×D776C2212yescomplemented
21#15690810C2211yescharacteristic, complemented
22#15690807C2211yescharacteristic
23#15690792—22no—
24#15690789C2212yescomplemented
25#15690785—21no—
26#15690784C2212yes—
27#15690781—21no—
28#15690780—22no—
29C4×D388C2211yescharacteristic
30C4×D388C2211yescharacteristic, complemented
31C4⋊C4×D97—21no—
32#15690744C2211yescharacteristic, complemented
33#15690735C2212yescomplemented
34#15690732C2211yescharacteristic
35#15690732C2211yescharacteristic, complemented
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×Dic194C2312yes—
46C2×D388—24no—
47C2×D388—22no—
48C2×D388—21no—
49C2×D388—21no—
50C2×D388—42no—
51C2×D388C2312yes—
52C2×D388C2311yescharacteristic
53C2×D388C2311yescharacteristic
54C2×C22⋊D97—22no—
55C2×C22⋊D97—22no—
56D4×D97—44no—
57C2×C4×D97—21no—
58C2×C4×D97—22no—
59C2×C4×D97—21no—
60#13095500—24no—
61#13095500—24no—
62#13095500—42no—
63D776—24no—
64D776D414yescomplemented
65C4⋊Dic97—21no—
66C4⋊Dic97—21no—
67C4⋊Dic97C2311yescharacteristic
68C4⋊Dic97C2311yescharacteristic
69#13095490—22no—
70C22⋊Dic97—22no—
71#13095488—21no—
72#13095488—21no—
73#13095488—22no—
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—21no—
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—
99C2×Dic97—21no—
100C2×Dic97—21no—
101C2×Dic97—22no—
102D388—22no—
103D388—21no—
104D388—21no—
105D388—22no—
106D388—44no—
107D388—42no—
108D388—81no—
109D388C2×D412yes—
110D388C2×D412yescomplemented
111C4×D97—44no—
112C4×D97—42no—
113C4×D97—42no—
114Dic194—22no—
115Dic194—22no—
116C8×C97—22no—
117C8×C97—21no—
118C8×C97C2×D412yescomplemented
119C22×C97—22no—
120C22×C97—22no—
121C22×C97—22no—
122C22×C97C22×D411yescharacteristic
123C2×D97—22no—
124C2×D97—22no—
125C2×D97—41no—
126C2×D97—41no—
127C2×D97—41no—
128C2×D97—41no—
129C2×D97—42no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—88no—
133C4×C97—22no—
134C4×C97—21no—
135C4×C97AGL(1,ℤ/8)12yescomplemented
136C4×C97C22×D411yescharacteristic
137C4×C97H(5,𝔽2)11yescharacteristic
138Dic97—42no—
139Dic97—41no—
140Dic97—41no—
141C2×C97—42no—
142C2×C97D4211yescharacteristic
143C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C97#57711yescharacteristic
145D97—42no—
146D97—42no—
147D97—82no—
148#3272—971nomaximal, Sylow
149C97#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—972no—
151#547—971no—
152D42—971no—
153#492—971no—
154#468—972no—
155#461—971no—
156#459—971no—
157#450—972no—
158#449—972no—
159C4×D8—971no—
160#436D9711yescharacteristic, complemented
161C2×#41—972no—
162C22×D4—972no—
163C22×D4—1941no—
164AGL(1,ℤ/8)—1948no—
165C2×SD16—972no—
166C2×SD16—972no—
167C2×SD16—972no—
168C2×D8—971no—
169C2×D8—972no—
170C2×M16C2×D9712yescomplemented
171C4⋊Q8—971no—
172#127—971no—
173#123—1942no—
174C22⋊Q8—972no—
175#121—972no—
176#121—1941no—
177#121—1941no—
178#120—1942no—
179C4×D4—971no—
180C4×D4—971no—
181C4×D4C2×D9711yescharacteristic
182C2×C4⋊C4—1941no—
183#107—971no—
184C4⋊C8C2×D9711yescharacteristic, complemented
185#103—972no—
186#102—971no—
187#102—971no—
188#102—972no—
189C22⋊C8C2×D9712yescomplemented
190C4×C8C2×D9711yescharacteristic, complemented
191C24—1942no—
192#41—3882no—
193#41—1944no—
194C2×Q8—972no—
195C2×D4—971no—
196C2×D4—971no—
197C2×D4—972no—
198C2×D4—3882no—
199C2×D4—3884no—
200C2×D4—1944no—
201C2×D4—1942no—
202C2×D4—1942no—
203C2×D4—1941no—
204C2×D4—1942no—
205C2×D4—1941no—
206C2×D4D38811yescharacteristic
207C22×C4—1941no—
208C22×C4—1942no—
209C22×C4—1941no—
210C22×C4C22×D9712yes—
211SD16—3882no—
212SD16—1944no—
213SD16—1944no—
214D8—974no—
215D8—1944no—
216M16—24no—
217C2×C8C22×D9711yescharacteristic
218C2×C8C22×D9712yes—
219C2×C8C22×D9711yescharacteristic
220C4⋊C4—971no—
221C4⋊C4—971no—
222C4⋊C4—1942no—
223C4⋊C4—1941no—
224C4⋊C4—1941no—
225C4⋊C4D38811yescharacteristic
226C22⋊C4—1942no—
227C22⋊C4—1941no—
228C22⋊C4—1941no—
229C22⋊C4—1942no—
230C22⋊C4D38812yes—
231C42C22×D9711yescharacteristic
232C23—7762no—
233C23—3884no—
234C23—3884no—
235C23—1942no—
236C23—1941no—
237C23—1941no—
238C23C2×D38812yes—
239Q8—1942no—
240Q8—1942no—
241D4—22no—
242D4—7761no—
243D4—972no—
244D4—972no—
245D4—3884no—
246D4—3884no—
247D4—3884no—
248D4—3882no—
249D4—1942no—
250D4—1942no—
251D4—1941no—
252D4—1941no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3884no—
256C2×C4—3882no—
257C2×C4—3882no—
258C2×C4—1941no—
259C2×C4—1941no—
260C2×C4—1942no—
261C2×C4C2×D38811yescharacteristic
262C2×C4C2×D38812yes—
263C2×C4C2×D38811yescharacteristic
264C2×C4C23×D9711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D9712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—7768no—
272C22—3882no—
273C22—3882no—
274C22—3882no—
275C22—3881no—
276C22—3881no—
277C22—3881no—
278C22—3881no—
279C22—1942no—
280C22—1942no—
281C22C22×D38811yescharacteristic
282C4—21no—
283C4—22no—
284C4—3881no—
285C4—3881no—
286C4—3882no—
287C4#1569083812yescomplemented
288C4C2×D4×D9711yescharacteristic
289C4#1569085611yescharacteristic
290C2—42no—
291C2—7762no—
292C2—3882no—
293C2—3882no—
294C2D4×D38811yescharacteristic
295C2C2×#1569083811yescharacteristic
296C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#436×C97C2
socleC22×C97C22×D4
hypercenter#436D97
hypocenterC97#3272
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. #20261390C2semidirect
3. D4×D388C2semidirect
4. #20261108C2semidirect
5. #20260981C2semidirect
6. #20260941C2semidirect
7. #20260931C2semidirect
8. #20260888C2semidirect
9. #20260882C2semidirect
10. C4×D776C2semidirect
11. #436×C97C2semidirect
12. C2×#13095511C22semidirect
17. C2×#13095500C22semidirect
18. C2×#13095500C22semidirect
20. C2×D776C22semidirect
21. #15690810C22semidirect
24. #15690789C22semidirect
30. C4×D388C22semidirect
32. #15690744C22semidirect
33. #15690735C22semidirect
35. #15690732C22semidirect
36. #15690732C22semidirect
37. C2×M16×C97C22semidirect
39. C4⋊C8×C97C22semidirect
40. C22⋊C8×C97C22semidirect
41. C4×C8×C97C22semidirect
64. D776D4semidirect
110. D388C2×D4semidirect
118. C8×C97C2×D4semidirect
135. C4×C97AGL(1,ℤ/8)semidirect
149. C97#3272semidirect
160. #436D97semidirect
170. C2×M16C2×D97semidirect
184. C4⋊C8C2×D97semidirect
189. C22⋊C8C2×D97semidirect
190. C4×C8C2×D97semidirect
267. C8D4×D97semidirect
287. C4#15690838semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
#202613902
D4×D3882
#202611082
#202609812
#202609412
#202609312
#202608882
#202608822
C4×D7762
#436×C972
#327297