Finite Group Databasebeta

← #27152218

Subgroups — #27152218

Order 12416.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15690833C211yesmaximal, characteristic, complemented
3C2×D8×D97C211yesmaximal, characteristic, complemented
4#20261303C212yesmaximal, complemented
5#20261125C212yesmaximal, complemented
6#20261121C212yesmaximal, complemented
7#20261120C211yesmaximal, characteristic, complemented
8#20261112C211yesmaximal, characteristic, complemented
9#20261110C211yesmaximal, characteristic, complemented
10#20260689C211yesmaximal, characteristic, complemented
11#495×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×D4×D97C2212yes—
14#15690837—28no—
15#15690833—24no—
16C2×#13095498C2212yescomplemented
17C2×#13095498C2212yescomplemented
18C2×D776C2211yescharacteristic
19C2×D776C2212yescomplemented
20C2×D776C2211yescharacteristic, complemented
21C2×#13095496C2212yes—
22C2×#13095496C2212yescomplemented
23D8×D97—24no—
24C2×#13095494C2211yescharacteristic
25C2×C8×D97C2211yescharacteristic
26#15690807C2212yescomplemented
27#15690806C2212yescomplemented
28#15690799C2212yescomplemented
29#15690783—24no—
30#15690780—24no—
31#15690767C2211yescharacteristic
32#15690721C2211yescharacteristic, complemented
33#15690720C2211yescharacteristic, complemented
34C8×Dic97C2211yescharacteristic, complemented
35C2×Q16×C97C2211yescharacteristic, complemented
36C2×SD16×C97C2212yescomplemented
37C2×D8×C97C2211yescharacteristic
38#124×C97C2212yescomplemented
39C4×C8×C97C2211yescharacteristic, complemented
40C23×D97—22no—
41#13095512—24no—
42#13095512—24no—
43C2×D388—22no—
44C2×D388—24no—
45C2×D388C2312yes—
46C2×D388C2312yes—
47C2×C22⋊D97—22no—
48C2×C22⋊D97—24no—
49D4×D97—24no—
50D4×D97—24no—
51C2×C4×D97—22no—
52C2×C4×D97C2311yescharacteristic
53#13095498—24no—
54#13095498—24no—
55D776—22no—
56D776—22no—
57D776—24no—
58#13095496—24no—
59#13095496—24no—
60#13095494D414yescomplemented
61C2×#11715895C2311yescharacteristic
62C2×#11715895C2311yescharacteristic
63C8×D97D414yescomplemented
64#13095490D412yes—
65#13095488—24no—
66#13095488D412yes—
67C4×Dic97C2311yescharacteristic
68C2×Q8×C97C2312yes—
69C2×D4×C97C2312yes—
70Q16×C97—22no—
71SD16×C97—24no—
72D8×C97—22no—
73C2×C8×C97C2311yescharacteristic
74C2×C8×C97C2311yescharacteristic
75C22⋊C4×C97—24no—
76C42×C97C2311yescharacteristic
77C23×C97—22no—
78C22×D97—22no—
79C22×D97—22no—
80C22×D97—44no—
81C22×D97—84no—
82C22×D97C2×D411yescharacteristic
83Q8×C97—22no—
84Q8×C97—22no—
85D4×C97—22no—
86D4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97C2×D412yes—
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—44no—
91C22⋊D97—84no—
92C2×Dic97C2×D412yes—
93C2×Dic97C2×D411yescharacteristic
94D388—22no—
95D388—22no—
96D388—22no—
97D388—22no—
98D388—44no—
99D388—84no—
100C4×D97—44no—
101C4×D97C2×D412yes—
102C4×D97C2×D412yes—
103C8×C97C2×D412yes—
104C8×C97C2×D412yes—
105#11715895C2×D412yes—
106#11715895C2×D412yes—
107C22×C97—42no—
108C22×C97—44no—
109C22×C97C22×D411yescharacteristic
110C2×D97—22no—
111C2×D97—42no—
112C2×D97—42no—
113C2×D97—44no—
114C2×D97—44no—
115C2×D97—88no—
116C2×D97#12712yes—
117C4×C97—22no—
118C4×C97—42no—
119C4×C97C22×D411yescharacteristic
120C4×C97C22×D411yescharacteristic
121Dic97—22no—
122Dic97#12712yes—
123C2×C97—82no—
124C2×C97C2×#12711yescharacteristic
125C2×C97#57612yes—
126D97—22no—
127D97—82no—
128D97—82no—
129#3155—971nomaximal, Sylow
130C97#315511yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—972no—
132C2×#135—971no—
133C22×D8—971no—
134#535—972no—
135#496—972no—
136#496—972no—
137#495—971no—
138#495D9711yescharacteristic, complemented
139#493—971no—
140#493—971no—
141#405—971no—
142C2×#41—972no—
143C22×D4—972no—
144AGL(1,ℤ/8)—1948no—
145#135—1944no—
146C2×Q16C2×D9711yescharacteristic, complemented
147C2×SD16—972no—
148C2×SD16—972no—
149C2×SD16C2×D9712yescomplemented
150C2×D8—972no—
151C2×D8—972no—
152C2×D8—971no—
153C2×D8—971no—
154C2×D8—972no—
155C2×D8—1944no—
156C2×D8C2×D9711yescharacteristic
157C2×M16—971no—
158C22×C8—971no—
159#127—972no—
160#127—972no—
161#124—972no—
162#124C2×D9712yescomplemented
163#121—1944no—
164#120—1944no—
165#117—971no—
166#97—971no—
167#97—971no—
168C4×C8—971no—
169C4×C8C2×D9711yescharacteristic, complemented
170C24—1942no—
171#41—1944no—
172#41—1944no—
173C2×Q8C22×D9712yes—
174C2×D4—972no—
175C2×D4—972no—
176C2×D4—1942no—
177C2×D4—1944no—
178C2×D4—1944no—
179C2×D4—1944no—
180C2×D4—1942no—
181C2×D4—1944no—
182C2×D4C22×D9712yes—
183C22×C4—971no—
184C22×C4—1942no—
185Q16—22no—
186SD16—24no—
187SD16—1944no—
188SD16—1944no—
189D8—22no—
190D8—1944no—
191D8—1944no—
192D8—1942no—
193D8—1942no—
194D8—1944no—
195M16—974no—
196C2×C8—971no—
197C2×C8—971no—
198C2×C8—974no—
199C2×C8C22×D9711yescharacteristic
200C2×C8C22×D9711yescharacteristic
201C4⋊C4—972no—
202C22⋊C4—24no—
203C22⋊C4—972no—
204C22⋊C4—1944no—
205C42—971no—
206C42C22×D9711yescharacteristic
207C23—22no—
208C23—7764no—
209C23—971no—
210C23—3884no—
211C23—1942no—
212C23—1942no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—7764no—
218D4—7764no—
219D4—3884no—
220D4—3884no—
221D4—1942no—
222D4—1942no—
223D4—1942no—
224D4—1942no—
225C2×C4—22no—
226C2×C4—971no—
227C2×C4—972no—
228C2×C4—972no—
229C2×C4—972no—
230C2×C4—3884no—
231C2×C4D4×D9712yes—
232C2×C4C23×D9711yescharacteristic
233C8—972no—
234C8—972no—
235C8D4×D9712yes—
236C8D4×D9712yes—
237C22—42no—
238C22—44no—
239C22—7768no—
240C22—972no—
241C22—3882no—
242C22—3882no—
243C22—3884no—
244C22—3884no—
245C22—1942no—
246C22C2×D4×D9711yescharacteristic
247C4—22no—
248C4—42no—
249C4—972no—
250C4—1942no—
251C4C2×D4×D9711yescharacteristic
252C4C2×D4×D9711yescharacteristic
253C2—82no—
254C2—7762no—
255C2—7762no—
256C2—1942no—
257C2#127×D9711yescharacteristic
258C2#2026153712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC22×C97C22×D4
hypercenter#495D97
hypocenterC97#3155
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690833C2semidirect
3. C2×D8×D97C2semidirect
4. #20261303C2semidirect
5. #20261125C2semidirect
6. #20261121C2semidirect
7. #20261120C2semidirect
8. #20261112C2semidirect
9. #20261110C2semidirect
10. #20260689C2semidirect
11. #495×C97C2semidirect
16. C2×#13095498C22semidirect
17. C2×#13095498C22semidirect
19. C2×D776C22semidirect
20. C2×D776C22semidirect
22. C2×#13095496C22semidirect
26. #15690807C22semidirect
27. #15690806C22semidirect
28. #15690799C22semidirect
32. #15690721C22semidirect
33. #15690720C22semidirect
34. C8×Dic97C22semidirect
35. C2×Q16×C97C22semidirect
36. C2×SD16×C97C22semidirect
38. #124×C97C22semidirect
39. C4×C8×C97C22semidirect
60. #13095494D4semidirect
63. C8×D97D4semidirect
130. C97#3155semidirect
138. #495D97semidirect
146. C2×Q16C2×D97semidirect
149. C2×SD16C2×D97semidirect
162. #124C2×D97semidirect
169. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908332
C2×D8×D972
#202613032
#202611252
#202611212
#202611202
#202611122
#202611102
#202606892
#495×C972
#315597