Finite Group Databasebeta

← #27152185

Subgroups — #27152185

Order 12416.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C2×#15690834C212yesmaximal, complemented
3#20261284C211yesmaximal, characteristic, complemented
4#127×D97C211yesmaximal, characteristic, complemented
5#20261125C211yesmaximal, characteristic, complemented
6#20261124C211yesmaximal, characteristic, complemented
7#20261122C211yesmaximal, characteristic, complemented
8#20261110C212yesmaximal, complemented
9#20261106C212yesmaximal, complemented
10#20260687C211yesmaximal, characteristic, complemented
11#496×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2211yescharacteristic
13C2×#13095510C2211yescharacteristic
14C2×D4×D97—22no—
15C2×D4×D97C2212yes—
16#15690837—28no—
17#15690834—28no—
18C2×#13095500C2212yescomplemented
19C2×#13095499C2212yescomplemented
20C2×#13095498C2212yescomplemented
21C2×D776C2212yescomplemented
22C2×#13095496C2212yescomplemented
23C2×#13095496C2212yescomplemented
24C2×#13095494C2212yes—
25#15690808C2211yescharacteristic, complemented
26#15690807C2211yescharacteristic
27#15690806—22no—
28#15690806C2212yes—
29#15690806C2211yescharacteristic, complemented
30#15690798C2211yescharacteristic, complemented
31#15690783—22no—
32#15690770—22no—
33C42×D97C2211yescharacteristic
34#15690721C2211yescharacteristic, complemented
35C8×Dic97C2212yescomplemented
36C2×SD16×C97C2212yescomplemented
37C2×D8×C97C2212yescomplemented
38#127×C97C2211yescharacteristic
39#124×C97C2211yescharacteristic, complemented
40#97×C97C2211yescharacteristic, complemented
41C23×D97—22no—
42#13095512—24no—
43#13095510—24no—
44C2×Dic194C2311yescharacteristic
45C2×D388—21no—
46C2×D388—22no—
47C2×D388C2311yescharacteristic
48C2×D388C2312yes—
49C2×C22⋊D97—21no—
50C2×C22⋊D97—22no—
51C2×C22⋊D97—22no—
52C2×C22⋊D97—24no—
53D4×D97—24no—
54D4×D97—24no—
55D4×D97—48no—
56C22×Dic97—21no—
57C2×C4×D97—21no—
58C2×C4×D97D412yes—
59C2×C4×D97C2311yescharacteristic
60#13095500—24no—
61#13095499—24no—
62#13095498—24no—
63D776—24no—
64#13095496—24no—
65#13095496—24no—
66#13095494D418yescomplemented
67C2×#11715895C2312yes—
68#13095490—22no—
69#13095488—22no—
70C4×Dic97D412yes—
71C4×Dic97C2311yescharacteristic
72C2×Q8×C97C2311yescharacteristic
73C2×D4×C97—22no—
74C2×D4×C97C2312yes—
75C2×D4×C97C2311yescharacteristic
76SD16×C97—24no—
77D8×C97—24no—
78C2×C8×C97C2312yes—
79C22⋊C4×C97—22no—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C23×C97—22no—
83C22×D97—21no—
84C22×D97—22no—
85C22×D97—48no—
86C22×D97—44no—
87C22×D97C2×D411yescharacteristic
88Q8×C97—22no—
89D4×C97—22no—
90D4×C97—22no—
91D4×C97—22no—
92D4×C97—44no—
93C2×C4×C97—21no—
94C2×C4×C97C2×D412yes—
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—42no—
97C22⋊D97—44no—
98C22⋊D97—48no—
99C22⋊D97—44no—
100C2×Dic97—21no—
101C2×Dic97—42no—
102C2×Dic97C2×D411yescharacteristic
103C2×Dic97C2×D412yes—
104D388—22no—
105D388—22no—
106D388—22no—
107D388—44no—
108D388—42no—
109C4×D97—24no—
110C4×D97—42no—
111C4×D97C2×D414yes—
112Dic194—22no—
113C8×C97C2×D414yescomplemented
114#11715895C2×D414yescomplemented
115C22×C97—41no—
116C22×C97—42no—
117C22×C97—44no—
118C22×C97—42no—
119C22×C97C22×D411yescharacteristic
120C2×D97—22no—
121C2×D97—41no—
122C2×D97—42no—
123C2×D97—44no—
124C2×D97—42no—
125C2×D97—88no—
126C2×D97#12712yes—
127C4×C97—22no—
128C4×C97—41no—
129C4×C97C22×D412yes—
130Dic97—41no—
131Dic97#12712yes—
132Dic97AGL(1,ℤ/8)14yescomplemented
133C2×C97—82no—
134C2×C97—81no—
135C2×C97C2×#12711yescharacteristic
136C2×C97C2×AGL(1,ℤ/8)12yes—
137D97—22no—
138D97—81no—
139D97—82no—
140#3151—971nomaximal, Sylow
141C97#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—972no—
143C2×AGL(1,ℤ/8)—972no—
144#532—971no—
145C2×#127—971no—
146#496—971no—
147#496—971no—
148#496—971no—
149#496D9711yescharacteristic, complemented
150#493—972no—
151#492—972no—
152C4×M16—971no—
153C2×#41—971no—
154C2×#41—971no—
155C22×D4—972no—
156C22×D4—1942no—
157AGL(1,ℤ/8)—1948no—
158AGL(1,ℤ/8)—1948no—
159C2×SD16—972no—
160C2×SD16—972no—
161C2×SD16—972no—
162C2×SD16C2×D9712yescomplemented
163C2×D8—972no—
164C2×D8—972no—
165C2×D8—972no—
166C2×D8C2×D9712yescomplemented
167C2×M16—972no—
168C4⋊Q8—971no—
169#127—972no—
170#127—971no—
171#127—971no—
172#127—1942no—
173#127C2×D9711yescharacteristic
174#124—971no—
175#124C2×D9711yescharacteristic, complemented
176#121—1942no—
177C4×D4—1942no—
178C2×C42—971no—
179#97—971no—
180#97C2×D9711yescharacteristic, complemented
181C4×C8—972no—
182C24—1942no—
183#41—1944no—
184#41—1944no—
185C2×Q8—971no—
186C2×Q8C22×D9711yescharacteristic
187C2×D4—22no—
188C2×D4—971no—
189C2×D4—972no—
190C2×D4—3888no—
191C2×D4—1941no—
192C2×D4—1944no—
193C2×D4—1944no—
194C2×D4—1942no—
195C2×D4—1944no—
196C2×D4—1942no—
197C2×D4—1941no—
198C2×D4—1942no—
199C2×D4C22×D9711yescharacteristic
200C2×D4C22×D9712yes—
201C22×C4—971no—
202C22×C4—972no—
203C22×C4—1941no—
204C22×C4—1941no—
205SD16—24no—
206SD16—1944no—
207SD16—1944no—
208SD16—1944no—
209D8—24no—
210D8—1944no—
211D8—1944no—
212D8—1944no—
213M16—978no—
214C2×C8—972no—
215C2×C8C22×D9712yes—
216C4⋊C4—1942no—
217C22⋊C4—22no—
218C22⋊C4—1942no—
219C42—972no—
220C42—971no—
221C42C22×D9711yescharacteristic
222C23—21no—
223C23—22no—
224C23—971no—
225C23—3884no—
226C23—3888no—
227C23—1941no—
228C23—1942no—
229Q8—22no—
230Q8—1942no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3884no—
236D4—3882no—
237D4—3888no—
238D4—3884no—
239D4—3884no—
240D4—3882no—
241D4—1942no—
242D4—1942no—
243D4—1942no—
244C2×C4—21no—
245C2×C4—972no—
246C2×C4—971no—
247C2×C4—974no—
248C2×C4—3882no—
249C2×C4—3882no—
250C2×C4—1944no—
251C2×C4—1941no—
252C2×C4D4×D9712yes—
253C2×C4C23×D9711yescharacteristic
254C8—974no—
255C8D4×D9714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—7768no—
261C22—972no—
262C22—3884no—
263C22—3882no—
264C22—3881no—
265C22—3882no—
266C22—1942no—
267C22C2×D4×D9711yescharacteristic
268C4—22no—
269C4—41no—
270C4—974no—
271C4—972no—
272C4—3881no—
273C4C2×D4×D9712yes—
274C2—82no—
275C2—81no—
276C2—7762no—
277C2—7761no—
278C2—1942no—
279C2#127×D9711yescharacteristic
280C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#496×C97C2
socleC22×C97C22×D4
hypercenter#496D97
hypocenterC97#3151
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690834C2semidirect
3. #20261284C2semidirect
4. #127×D97C2semidirect
5. #20261125C2semidirect
6. #20261124C2semidirect
7. #20261122C2semidirect
8. #20261110C2semidirect
9. #20261106C2semidirect
10. #20260687C2semidirect
11. #496×C97C2semidirect
18. C2×#13095500C22semidirect
19. C2×#13095499C22semidirect
20. C2×#13095498C22semidirect
21. C2×D776C22semidirect
22. C2×#13095496C22semidirect
23. C2×#13095496C22semidirect
25. #15690808C22semidirect
29. #15690806C22semidirect
30. #15690798C22semidirect
34. #15690721C22semidirect
35. C8×Dic97C22semidirect
36. C2×SD16×C97C22semidirect
37. C2×D8×C97C22semidirect
39. #124×C97C22semidirect
40. #97×C97C22semidirect
66. #13095494D4semidirect
113. C8×C97C2×D4semidirect
114. #11715895C2×D4semidirect
132. Dic97AGL(1,ℤ/8)semidirect
141. C97#3151semidirect
149. #496D97semidirect
162. C2×SD16C2×D97semidirect
166. C2×D8C2×D97semidirect
175. #124C2×D97semidirect
180. #97C2×D97semidirect
255. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908342
#202612842
#127×D972
#202611252
#202611242
#202611222
#202611102
#202611062
#202606872
#496×C972
#315197