Finite Group Databasebeta

← #27153521

Subgroups — #27153521

Order 12416.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#20261548C211yesmaximal, characteristic, complemented
2#20261541C212yesmaximal, complemented
3#20261536C211yesmaximal, characteristic, complemented
4#20261007C211yesmaximal, characteristic, complemented
5#20261002C211yesmaximal, characteristic, complemented
6#20260995C212yesmaximal, complemented
7#20260912C212yesmaximal, complemented
8#20260907C212yesmaximal, complemented
9#20260853C211yesmaximal, characteristic, complemented
10#20260853C211yesmaximal, characteristic, complemented
11#495×C97C211yesmaximal, characteristic, complemented
12#15690858C2212yescomplemented
13#15690855C2212yes—
14#15690842—22no—
15#15690841C2212yes—
16#15690841C2212yescomplemented
17#15690839—22no—
18#15690837—22no—
19#15690836C2212yes—
20#15690836C2212yescomplemented
21#15690834—22no—
22#15690833—21no—
23#15690832C2211yescharacteristic
24#15690832C2211yescharacteristic
25#15690831—22no—
26#15690829—21no—
27Q16×D97—21no—
28SD16×D97—22no—
29D8×D97—21no—
30#15690816C2211yescharacteristic
31#15690816C2212yescomplemented
32#15690816C2211yescharacteristic, complemented
33#15690746C2211yescharacteristic, complemented
34#15690746C2211yescharacteristic, complemented
35#15690738C2211yescharacteristic
36#15690738C2211yescharacteristic, complemented
37#15690738C2212yescomplemented
38#15690729C2212yescomplemented
39#15690727C2212yescomplemented
40C2×Q16×C97C2211yescharacteristic, complemented
41C2×SD16×C97C2212yescomplemented
42C2×D8×C97C2211yescharacteristic
43#124×C97C2212yescomplemented
44C4×C8×C97C2211yescharacteristic, complemented
45#13095512—22no—
46#13095512—22no—
47#13095511—42no—
48#13095511C2311yescharacteristic
49#13095511C2312yes—
50#13095511C2311yescharacteristic
51#13095510—22no—
52#13095510—22no—
53Q8×D97—22no—
54Q8×D97—22no—
55C2×C22⋊D97—42no—
56D4×D97—22no—
57D4×D97—22no—
58Dic388D411yescharacteristic
59Dic388D411yescharacteristic
60#13095501—22no—
61#13095501—22no—
62#13095500D411yescharacteristic
63#13095500D411yescharacteristic
64#13095500D412yes—
65#13095499—22no—
66#13095499—22no—
67#13095498—22no—
68#13095498—22no—
69D776D411yescharacteristic
70D776D411yescharacteristic
71#13095496—22no—
72#13095496—22no—
73#13095495C2311yescharacteristic
74#13095495C2312yes—
75#13095495C2311yescharacteristic
76#13095494—21no—
77#13095494—22no—
78#13095494—21no—
79C8×D97—21no—
80C8×D97—22no—
81C8×D97—21no—
82C2×Q8×C97C2312yes—
83C2×D4×C97C2312yes—
84Q16×C97—21no—
85Q16×C97—21no—
86SD16×C97—22no—
87SD16×C97—22no—
88D8×C97—21no—
89D8×C97—21no—
90C2×C8×C97C2311yescharacteristic
91C2×C8×C97C2311yescharacteristic
92C22⋊C4×C97—42no—
93C42×C97C2311yescharacteristic
94C23×C97—22no—
95C22×D97—42no—
96Q8×C97—22no—
97Q8×C97—22no—
98D4×C97—22no—
99D4×C97—22no—
100C2×C4×C97—21no—
101C2×C4×C97—22no—
102C2×C4×C97C2411yescharacteristic
103C22⋊D97—21no—
104C22⋊D97—22no—
105C22⋊D97—21no—
106C22⋊D97—42no—
107C22⋊D97—42no—
108C2×Dic97—42no—
109D388—42no—
110D388C2×D411yescharacteristic
111D388C2×D412yes—
112D388C2×D411yescharacteristic
113C4×D97—21no—
114C4×D97—22no—
115C4×D97—21no—
116C4×D97—42no—
117C4×D97—42no—
118Dic194—42no—
119Dic194C2×D411yescharacteristic
120Dic194C2×D412yes—
121Dic194C2×D411yescharacteristic
122C8×C97C2×D411yescharacteristic
123C8×C97C2×D411yescharacteristic
124C8×C97C2×D411yescharacteristic
125C8×C97C2×D411yescharacteristic
126#11715895—21no—
127#11715895—22no—
128#11715895—21no—
129C22×C97—42no—
130C22×C97—82no—
131C22×C97H(5,𝔽2)11yescharacteristic
132C2×D97—21no—
133C2×D97—22no—
134C2×D97—21no—
135C2×D97—162no—
136C4×C97—41no—
137C4×C97—42no—
138C4×C97C22×D411yescharacteristic
139C4×C97C22×D411yescharacteristic
140Dic97—21no—
141Dic97—22no—
142Dic97—21no—
143C2×C97—21no—
144C2×C97—82no—
145C2×C97D4211yescharacteristic
146D97—41no—
147D97—42no—
148D97—41no—
149#3290—971nomaximal, Sylow
150C97#329011yescharacteristic, complemented, Sylow
151#578—971no—
152#577—972no—
153#576—971no—
154#495D9711yescharacteristic, complemented
155#473—971no—
156#472—971no—
157#471—972no—
158#455—972no—
159#454—972no—
160#443—971no—
161#443—971no—
1622−1+4—972no—
163H(5,𝔽2)—972no—
164#137—972no—
165#137—972no—
166#137—1942no—
167#137—1942no—
168AGL(1,ℤ/8)—972no—
169AGL(1,ℤ/8)—972no—
170AGL(1,ℤ/8)—1942no—
171AGL(1,ℤ/8)—1942no—
172#135—971no—
173#135—971no—
174#135—1941no—
175#135—1941no—
176#135—1942no—
177C2×Q16—1941no—
178C2×Q16C2×D9711yescharacteristic, complemented
179C2×SD16—1942no—
180C2×SD16C2×D9712yescomplemented
181C2×D8—1941no—
182C2×D8C2×D9711yescharacteristic
183#131—971no—
184#131—972no—
185#131—971no—
186#124C2×D9712yescomplemented
187#108—971no—
188#108—971no—
189#104—971no—
190#104—972no—
191#104—971no—
192#101—972no—
193#100—972no—
194C4×C8C2×D9711yescharacteristic, complemented
195#41—971no—
196#41—972no—
197#41—971no—
198#41—3882no—
199#41—1942no—
200#41—1942no—
201#41—1942no—
202#41—1942no—
203C2×Q8—1942no—
204C2×Q8—1942no—
205C2×Q8C22×D9712yes—
206C2×D4—3882no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4C22×D9712yes—
210Q16—21no—
211Q16—21no—
212Q16—971no—
213Q16—971no—
214Q16—1942no—
215Q16—1942no—
216SD16—22no—
217SD16—22no—
218SD16—971no—
219SD16—971no—
220SD16—972no—
221SD16—1942no—
222SD16—1942no—
223SD16—1942no—
224SD16—1942no—
225D8—21no—
226D8—21no—
227D8—971no—
228D8—971no—
229D8—1942no—
230D8—1942no—
231M16—971no—
232M16—972no—
233M16—971no—
234M16—1941no—
235M16—1942no—
236M16—1941no—
237C2×C8—1941no—
238C2×C8—1942no—
239C2×C8—1941no—
240C2×C8C22×D9711yescharacteristic
241C2×C8C22×D9711yescharacteristic
242C22⋊C4—42no—
243C42C22×D9711yescharacteristic
244C23—22no—
245C23—3882no—
246Q8—22no—
247Q8—22no—
248Q8—971no—
249Q8—972no—
250Q8—971no—
251Q8—3882no—
252D4—22no—
253D4—22no—
254D4—971no—
255D4—972no—
256D4—971no—
257D4—3882no—
258D4—3882no—
259D4—3882no—
260D4—1941no—
261D4—1942no—
262D4—1941no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—3882no—
266C2×C4—3882no—
267C2×C4—3882no—
268C2×C4—1941no—
269C2×C4—1942no—
270C2×C4—1941no—
271C2×C4C23×D9711yescharacteristic
272C8—1941no—
273C8—1942no—
274C8—1941no—
275C8D4×D9711yescharacteristic
276C8D4×D9711yescharacteristic
277C8D4×D9711yescharacteristic
278C8D4×D9711yescharacteristic
279C22—42no—
280C22—82no—
281C22—15522no—
282C22—1941no—
283C22—1942no—
284C22—1941no—
285C22#1569085511yescharacteristic
286C4—41no—
287C4—42no—
288C4—1941no—
289C4—1942no—
290C4—1941no—
291C4C2×D4×D9711yescharacteristic
292C4C2×D4×D9711yescharacteristic
293C2—21no—
294C2—82no—
295C2—3881no—
296C2—3882no—
297C2—3881no—
298C2#2026137011yescharacteristic
subgroupG/H
centerC2#20261370
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC2×C97D42
hypercenter#495D97
hypocenterC97#3290
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20261548C2semidirect
2. #20261541C2semidirect
3. #20261536C2semidirect
4. #20261007C2semidirect
5. #20261002C2semidirect
6. #20260995C2semidirect
7. #20260912C2semidirect
8. #20260907C2semidirect
9. #20260853C2semidirect
10. #20260853C2semidirect
11. #495×C97C2semidirect
12. #15690858C22semidirect
16. #15690841C22semidirect
20. #15690836C22semidirect
31. #15690816C22semidirect
32. #15690816C22semidirect
33. #15690746C22semidirect
34. #15690746C22semidirect
36. #15690738C22semidirect
37. #15690738C22semidirect
38. #15690729C22semidirect
39. #15690727C22semidirect
40. C2×Q16×C97C22semidirect
41. C2×SD16×C97C22semidirect
43. #124×C97C22semidirect
44. C4×C8×C97C22semidirect
150. C97#3290semidirect
154. #495D97semidirect
178. C2×Q16C2×D97semidirect
180. C2×SD16C2×D97semidirect
186. #124C2×D97semidirect
194. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615482
#202615412
#202615362
#202610072
#202610022
#202609952
#202609122
#202609072
#202608532
#495×C972
#329097