Finite Group Databasebeta

← #27153518

Subgroups — #27153518

Order 12416.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261549C211yesmaximal, characteristic, complemented
2#20261546C212yesmaximal, complemented
3#20261118C211yesmaximal, characteristic, complemented
4#20261010C211yesmaximal, characteristic, complemented
5#20261002C211yesmaximal, characteristic, complemented
6#20260996C212yesmaximal, complemented
7#20260913C212yesmaximal, complemented
8#20260906C212yesmaximal, complemented
9#20260852C211yesmaximal, characteristic, complemented
10#20260850C211yesmaximal, characteristic, complemented
11#576×C97C211yesmaximal, characteristic, complemented
12#15690857C2212yescomplemented
13#15690840—22no—
14#15690840C2212yes—
15#15690840C2212yescomplemented
16#15690839—22no—
17#15690836—22no—
18#15690836C2212yescomplemented
19#15690830—22no—
20#15690830C2211yescharacteristic
21#15690829—21no—
22C2×Dic388C2211yescharacteristic, complemented
23Q16×D97—21no—
24C2×#13095499—22no—
25C2×#13095499C2212yescomplemented
26#15690817C2212yescomplemented
27#15690816C2211yescharacteristic, complemented
28#15690797C2212yescomplemented
29#15690746C2211yescharacteristic, complemented
30#15690745C2211yescharacteristic, complemented
31#15690737C2211yescharacteristic
32#15690737C2212yescomplemented
33#15690736C2211yescharacteristic, complemented
34#15690728C2212yescomplemented
35#15690727C2212yescomplemented
36C8×Dic97C2211yescharacteristic, complemented
37H(5,𝔽2)×C97C2212yes—
38AGL(1,ℤ/8)×C97—22no—
39AGL(1,ℤ/8)×C97C2212yes—
40#135×C97—21no—
41#135×C97C2211yescharacteristic
42C2×D8×C97—21no—
43C2×D8×C97C2211yescharacteristic
44#131×C97C2211yescharacteristic
45#13095511—22no—
46#13095511C2311yescharacteristic
47#13095510—22no—
48#13095510—42no—
49C2×Dic194—22no—
50C2×Dic194C2312yes—
51Q8×D97—22no—
52Dic388—21no—
53Dic388—21no—
54#13095501—22no—
55#13095501—22no—
56#13095501D411yescharacteristic
57#13095499—22no—
58#13095499—22no—
59#13095499—22no—
60#13095499—22no—
61#13095499—22no—
62#13095499D411yescharacteristic
63#13095498D411yescharacteristic
64#13095496—22no—
65#13095496D411yescharacteristic
66#13095495—22no—
67#13095495C2311yescharacteristic
68#13095495C2312yes—
69#13095494—21no—
70C2×#11715895—22no—
71C2×#11715895C2311yescharacteristic
72C8×D97—21no—
73C22⋊Dic97—42no—
74C4×Dic97C2311yescharacteristic
75#41×C97—22no—
76#41×C97—22no—
77#41×C97C2312yes—
78#41×C97C2311yescharacteristic
79C2×D4×C97—22no—
80C2×D4×C97—22no—
81C2×D4×C97—42no—
82C2×D4×C97C2312yes—
83Q16×C97D411yescharacteristic
84SD16×C97—22no—
85SD16×C97D412yes—
86D8×C97—22no—
87D8×C97—21no—
88D8×C97—21no—
89D8×C97D411yescharacteristic
90M16×C97—21no—
91M16×C97C2311yescharacteristic
92C2×C8×C97—21no—
93C2×C8×C97C2311yescharacteristic
94C23×C97—22no—
95C23×C97—42no—
96Q8×C97C2×D411yescharacteristic
97Q8×C97C2×D412yes—
98D4×C97—21no—
99D4×C97—22no—
100D4×C97—22no—
101D4×C97—22no—
102D4×C97—42no—
103D4×C97—42no—
104D4×C97C2×D411yescharacteristic
105D4×C97C2×D412yes—
106C2×C4×C97—21no—
107C2×C4×C97—22no—
108C2×C4×C97—42no—
109C2×C4×C97C2411yescharacteristic
110C22⋊D97—21no—
111C22⋊D97—42no—
112C2×Dic97—21no—
113C2×Dic97—22no—
114C2×Dic97—42no—
115D388C2×D411yescharacteristic
116C4×D97—21no—
117C4×D97—42no—
118Dic194—22no—
119Dic194—22no—
120Dic194—42no—
121Dic194C2×D411yescharacteristic
122C8×C97—21no—
123C8×C97C2×D411yescharacteristic
124C8×C97C2×D411yescharacteristic
125#11715895—21no—
126#11715895—22no—
127#11715895C2×D411yescharacteristic
128#11715895C2×D411yescharacteristic
129C22×C97—22no—
130C22×C97—21no—
131C22×C97—42no—
132C22×C97—82no—
133C22×C97—162no—
134C22×C97H(5,𝔽2)11yescharacteristic
135C2×D97—21no—
136C4×C97—22no—
137C4×C97—21no—
138C4×C97C22×D411yescharacteristic
139C4×C97C22×D411yescharacteristic
140Dic97—21no—
141Dic97—41no—
142Dic97—42no—
143C2×C97—21no—
144C2×C97—42no—
145C2×C97—41no—
146C2×C97—82no—
147C2×C97D4211yescharacteristic
148D97—41no—
149#3290—971nomaximal, Sylow
150C97#329011yescharacteristic, complemented, Sylow
151#578—971no—
152#577—972no—
153#576D9711yescharacteristic, complemented
154#495—971no—
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)C2×D9712yes—
164#137—972no—
165#137—972no—
166#137—1942no—
167#137—1942no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—972no—
170AGL(1,ℤ/8)—1942no—
171AGL(1,ℤ/8)C2×D9712yes—
172#135—21no—
173#135—971no—
174#135—1942no—
175#135—1941no—
176#135C2×D9711yescharacteristic
177C2×Q16—971no—
178C2×Q16—1941no—
179C2×SD16—972no—
180C2×SD16—1942no—
181C2×D8—21no—
182C2×D8C2×D9711yescharacteristic
183#131—972no—
184#131—971no—
185#131C2×D9711yescharacteristic
186#124—972no—
187#108—971no—
188#108—971no—
189#104—972no—
190#104—971no—
191#104—971no—
192#101—972no—
193#100—972no—
194C4×C8—971no—
195#41—22no—
196#41—22no—
197#41—971no—
198#41—3882no—
199#41—1942no—
200#41—1942no—
201#41C22×D9711yescharacteristic
202#41C22×D9712yes—
203C2×Q8—972no—
204C2×Q8—1942no—
205C2×Q8—1942no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D9712yes—
210Q16—971no—
211Q16—1942no—
212Q16—1942no—
213Q16—1941no—
214Q16—1941no—
215Q16C22⋊D9711yescharacteristic
216SD16—22no—
217SD16—971no—
218SD16—971no—
219SD16—1942no—
220SD16—1942no—
221SD16—1942no—
222SD16—1942no—
223SD16—1942no—
224SD16C22⋊D9712yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—971no—
229D8—1942no—
230D8C22⋊D9711yescharacteristic
231M16—21no—
232M16—971no—
233M16—972no—
234M16—1942no—
235M16—1941no—
236M16C22×D9711yescharacteristic
237C2×C8—21no—
238C2×C8—971no—
239C2×C8—1942no—
240C2×C8—1941no—
241C2×C8C22×D9711yescharacteristic
242C22⋊C4—3882no—
243C42—971no—
244C23—22no—
245C23—42no—
246Q8—971no—
247Q8—3882no—
248Q8—1942no—
249Q8—1942no—
250Q8D4×D9711yescharacteristic
251Q8C2×C22⋊D9712yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—971no—
259D4—3882no—
260D4—1941no—
261D4D4×D9711yescharacteristic
262D4C2×C22⋊D9712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—3882no—
267C2×C4—3882no—
268C2×C4—1941no—
269C2×C4—1942no—
270C2×C4—1941no—
271C2×C4C23×D9711yescharacteristic
272C8—21no—
273C8—971no—
274C8—971no—
275C8—1942no—
276C8—1941no—
277C8C2×C22⋊D9711yescharacteristic
278C8C2×C22⋊D9711yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—1941no—
285C22#1569085511yescharacteristic
286C4—21no—
287C4—22no—
288C4—3881no—
289C4—3882no—
290C4—1941no—
291C4C2×D4×D9711yescharacteristic
292C4C22×C22⋊D9711yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—3881no—
298C2D4×C22⋊D9711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#576×C97C2
socleC2×C97D42
hypercenter#576D97
hypocenterC97#3290
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20261549C2semidirect
2. #20261546C2semidirect
3. #20261118C2semidirect
4. #20261010C2semidirect
5. #20261002C2semidirect
6. #20260996C2semidirect
7. #20260913C2semidirect
8. #20260906C2semidirect
9. #20260852C2semidirect
10. #20260850C2semidirect
11. #576×C97C2semidirect
12. #15690857C22semidirect
15. #15690840C22semidirect
18. #15690836C22semidirect
22. C2×Dic388C22semidirect
25. C2×#13095499C22semidirect
26. #15690817C22semidirect
27. #15690816C22semidirect
28. #15690797C22semidirect
29. #15690746C22semidirect
30. #15690745C22semidirect
32. #15690737C22semidirect
33. #15690736C22semidirect
34. #15690728C22semidirect
35. #15690727C22semidirect
36. C8×Dic97C22semidirect
150. C97#3290semidirect
153. #576D97semidirect

Maximal subgroups

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

subgroupindex
#202615492
#202615462
#202611182
#202610102
#202610022
#202609962
#202609132
#202609062
#202608522
#202608502
#576×C972
#329097