Finite Group Databasebeta

← #27153517

Subgroups — #27153517

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#20261552C211yesmaximal, characteristic, complemented
2#20261545C212yesmaximal, complemented
3#20261536C211yesmaximal, characteristic, complemented
4#20261118C211yesmaximal, characteristic, complemented
5#20261011C211yesmaximal, characteristic, complemented
6#20260997C212yesmaximal, complemented
7#20260915C212yesmaximal, complemented
8#20260906C212yesmaximal, complemented
9#20260851C211yesmaximal, characteristic, complemented
10#20260850C211yesmaximal, characteristic, complemented
11#472×C97C211yesmaximal, characteristic, complemented
12#15690857C2212yescomplemented
13#15690855C2212yes—
14#15690843—22no—
15#15690843C2212yes—
16#15690843C2212yescomplemented
17#15690839—22no—
18#15690836C2212yes—
19#15690834—22no—
20#15690834—22no—
21#15690832—21no—
22#15690832C2211yescharacteristic
23#15690830C2211yescharacteristic
24#15690829—21no—
25#15690829—22no—
26C2×Dic388—21no—
27C2×Dic388C2211yescharacteristic, complemented
28C2×#13095499C2212yescomplemented
29SD16×D97—22no—
30D8×D97—21no—
31#15690818C2212yescomplemented
32#15690816C2211yescharacteristic
33#15690797C2212yescomplemented
34#15690745C2211yescharacteristic, complemented
35#15690737C2211yescharacteristic, complemented
36#15690736C2211yescharacteristic
37#15690736C2212yescomplemented
38#15690728C2212yescomplemented
39C8×Dic97C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C97C2212yescomplemented
41C2×D8×C97C2211yescharacteristic
42#131×C97C2211yescharacteristic, complemented
43#108×C97C2211yescharacteristic, complemented
44#100×C97C2212yescomplemented
45#13095511—22no—
46#13095511C2311yescharacteristic
47#13095511C2312yes—
48#13095510—22no—
49#13095510—22no—
50#13095510—22no—
51#13095510—42no—
52C2×Dic194—22no—
53C2×Dic194C2312yes—
54Q8×D97—22no—
55C2×C22⋊D97—42no—
56D4×D97—22no—
57D4×D97—22no—
58Dic388—22no—
59Dic388—22no—
60Dic388—21no—
61Dic388—21no—
62Dic388D411yescharacteristic
63#13095501D411yescharacteristic
64#13095500—22no—
65#13095500—22no—
66#13095500D412yes—
67#13095499—22no—
68#13095499—22no—
69#13095499—22no—
70#13095499D411yescharacteristic
71#13095498D411yescharacteristic
72D776D411yescharacteristic
73#13095496—22no—
74#13095496D411yescharacteristic
75#13095495C2311yescharacteristic
76#13095494—21no—
77#13095494—22no—
78C2×#11715895C2311yescharacteristic
79C8×D97—21no—
80C8×D97—22no—
81C22⋊Dic97—42no—
82C4×Dic97C2311yescharacteristic
83#41×C97C2311yescharacteristic
84C2×D4×C97C2312yes—
85SD16×C97—22no—
86D8×C97—22no—
87D8×C97—21no—
88D8×C97—21no—
89M16×C97—21no—
90M16×C97C2312yes—
91M16×C97C2311yescharacteristic
92C2×C8×C97—21no—
93C2×C8×C97C2311yescharacteristic
94C23×C97—22no—
95C22×D97—42no—
96Q8×C97C2×D411yescharacteristic
97D4×C97—21no—
98D4×C97—22no—
99D4×C97—22no—
100D4×C97C2×D411yescharacteristic
101C2×C4×C97—21no—
102C2×C4×C97C2411yescharacteristic
103C22⋊D97—21no—
104C22⋊D97—22no—
105C22⋊D97—42no—
106C22⋊D97—42no—
107C22⋊D97—42no—
108C2×Dic97—21no—
109C2×Dic97—22no—
110C2×Dic97—42no—
111C2×Dic97—42no—
112D388C2×D412yes—
113D388C2×D411yescharacteristic
114C4×D97—21no—
115C4×D97—22no—
116C4×D97—42no—
117Dic194—22no—
118Dic194—22no—
119Dic194—42no—
120Dic194C2×D412yes—
121Dic194C2×D411yescharacteristic
122C8×C97—21no—
123C8×C97—22no—
124C8×C97C2×D411yescharacteristic
125C8×C97C2×D411yescharacteristic
126#11715895—21no—
127#11715895C2×D411yescharacteristic
128#11715895C2×D411yescharacteristic
129C22×C97—21no—
130C22×C97—42no—
131C22×C97—82no—
132C22×C97H(5,𝔽2)11yescharacteristic
133C2×D97—22no—
134C2×D97—21no—
135C2×D97—162no—
136C4×C97—21no—
137C4×C97C22×D411yescharacteristic
138C4×C97C22×D411yescharacteristic
139Dic97—22no—
140Dic97—21no—
141Dic97—41no—
142Dic97—42no—
143C2×C97—21no—
144C2×C97—41no—
145C2×C97—82no—
146C2×C97D4211yescharacteristic
147D97—42no—
148D97—41no—
149#3290—971nomaximal, Sylow
150C97#329011yescharacteristic, complemented, Sylow
151#578—971no—
152#577—972no—
153#576—971no—
154#495—971no—
155#473—971no—
156#472D9711yescharacteristic, complemented
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)—1942no—
170AGL(1,ℤ/8)—1942no—
171AGL(1,ℤ/8)C2×D9712yescomplemented
172#135—971no—
173#135—971no—
174#135—1941no—
175#135—1942no—
176#135—1941no—
177C2×Q16—971no—
178C2×Q16—1941no—
179C2×SD16—972no—
180C2×SD16—1942no—
181C2×D8—1941no—
182C2×D8C2×D9711yescharacteristic
183#131—972no—
184#131—971no—
185#131C2×D9711yescharacteristic, complemented
186#124—972no—
187#108—971no—
188#108C2×D9711yescharacteristic, complemented
189#104—971no—
190#104—972no—
191#104—971no—
192#101—972no—
193#100C2×D9712yescomplemented
194C4×C8—971no—
195#41—972no—
196#41—971no—
197#41—3882no—
198#41—1942no—
199#41—1942no—
200#41—1942no—
201#41—1942no—
202#41C22×D9711yescharacteristic
203C2×Q8—972no—
204C2×Q8—1942no—
205C2×Q8—1942no—
206C2×D4—3882no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4C22×D9712yes—
210Q16—971no—
211Q16—971no—
212Q16—1942no—
213Q16—1942no—
214Q16—1941no—
215Q16—1941no—
216SD16—22no—
217SD16—971no—
218SD16—971no—
219SD16—972no—
220SD16—1942no—
221SD16—1942no—
222SD16—1942no—
223SD16—1942no—
224SD16—1942no—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—971no—
229D8—971no—
230D8—1942no—
231M16—21no—
232M16—971no—
233M16—1942no—
234M16—1941no—
235M16C22×D9711yescharacteristic
236M16C22×D9712yes—
237C2×C8—21no—
238C2×C8—971no—
239C2×C8—1942no—
240C2×C8—1941no—
241C2×C8C22×D9711yescharacteristic
242C22⋊C4—3882no—
243C42—971no—
244C23—22no—
245C23—3882no—
246Q8—972no—
247Q8—971no—
248Q8—3882no—
249Q8—1942no—
250Q8—1942no—
251Q8D4×D9711yescharacteristic
252D4—21no—
253D4—22no—
254D4—22no—
255D4—972no—
256D4—971no—
257D4—3882no—
258D4—3882no—
259D4—3882no—
260D4—1941no—
261D4—1942no—
262D4D4×D9711yescharacteristic
263C2×C4—21no—
264C2×C4—3882no—
265C2×C4—3882no—
266C2×C4—3882no—
267C2×C4—1941no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1942no—
271C2×C4C23×D9711yescharacteristic
272C8—21no—
273C8—22no—
274C8—971no—
275C8—971no—
276C8—1941no—
277C8D4×D9711yescharacteristic
278C8D4×D9711yescharacteristic
279C22—21no—
280C22—42no—
281C22—82no—
282C22—15522no—
283C22—1942no—
284C22—1941no—
285C22#1569085611yescharacteristic
286C4—21no—
287C4—3881no—
288C4—3882no—
289C4—1941no—
290C4—1942no—
291C4C2×D4×D9711yescharacteristic
292C4C2×D4×D9711yescharacteristic
293C2—21no—
294C2—41no—
295C2—82no—
296C2—3881no—
297C2—3882no—
298C2#2026136911yescharacteristic
subgroupG/H
centerC2#20261369
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#472×C97C2
socleC2×C97D42
hypercenter#472D97
hypocenterC97#3290
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20261552C2semidirect
2. #20261545C2semidirect
3. #20261536C2semidirect
4. #20261118C2semidirect
5. #20261011C2semidirect
6. #20260997C2semidirect
7. #20260915C2semidirect
8. #20260906C2semidirect
9. #20260851C2semidirect
10. #20260850C2semidirect
11. #472×C97C2semidirect
12. #15690857C22semidirect
16. #15690843C22semidirect
27. C2×Dic388C22semidirect
28. C2×#13095499C22semidirect
31. #15690818C22semidirect
33. #15690797C22semidirect
34. #15690745C22semidirect
35. #15690737C22semidirect
37. #15690736C22semidirect
38. #15690728C22semidirect
39. C8×Dic97C22semidirect
40. AGL(1,ℤ/8)×C97C22semidirect
42. #131×C97C22semidirect
43. #108×C97C22semidirect
44. #100×C97C22semidirect
150. C97#3290semidirect
156. #472D97semidirect
171. AGL(1,ℤ/8)C2×D97semidirect
185. #131C2×D97semidirect
188. #108C2×D97semidirect
193. #100C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615522
#202615452
#202615362
#202611182
#202610112
#202609972
#202609152
#202609062
#202608512
#202608502
#472×C972
#329097