Finite Group Databasebeta

← #27156517

Subgroups — #27156517

Order 12416.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#20261571C211yesmaximal, characteristic, complemented
2C2×#15690856C211yesmaximal, characteristic, complemented
3#20261547C218yesmaximal, complemented
4#20261540C218yesmaximal, complemented
5#20261535C216yesmaximal, complemented
6C2×#15690838C216yesmaximal, complemented
7#568×C97C211yesmaximal, characteristic, complemented
8#15690858—21no—
9#15690857C2214yescomplemented
10#15690856—24no—
11#15690856C2218yescomplemented
12#15690856C2214yescomplemented
13#15690855—23no—
14C2×#13095512—21no—
15C2×#13095511C2213yescomplemented
16C2×#13095511C2216yescomplemented
17#41×D97—24no—
18C22×D388C2213yescomplemented
19C2×D4×D97—23no—
20#15690843C2214yescomplemented
21#15690843C22112yescomplemented
22#15690838C22112yescomplemented
23#15690838C22124yescomplemented
24#15690838C22112yescomplemented
25#15690832C22124yescomplemented
26C2×#13095500C22112yescomplemented
27C2×D776C22112yescomplemented
28C2×#41×C97C2211yescharacteristic
29#131×C97C2218yescomplemented
30C2×M16×C97C2216yescomplemented
31C23×D97—23no—
32#13095512—24no—
33#13095512—28no—
34#13095511—26no—
35#13095511—26no—
36#13095511—26no—
37#13095511C23112yes—
38#13095511C2314yes—
39#13095511C23112yescomplemented
40#13095510—212no—
41C2×Dic194C2311yescharacteristic
42C2×Dic194C2316yescomplemented
43Q8×D97—24no—
44C2×D388—23no—
45C2×D388—26no—
46C2×D388C23112yes—
47C2×D388C2313yes—
48C2×D388C2316yescomplemented
49C2×C22⋊D97—23no—
50C2×C22⋊D97—26no—
51D4×D97—224no—
52D4×D97—212no—
53C2×C4×D97—23no—
54C2×C4×D97—26no—
55Dic388C2318yescomplemented
56#13095500C23124yescomplemented
57#13095500C2318yescomplemented
58D776C23124yescomplemented
59#41×C97D414yes—
60#41×C97C2314yes—
61C2×Q8×C97D411yescharacteristic
62C2×D4×C97D413yes—
63C22×C4×C97C2313yes—
64M16×C97C2314yes—
65M16×C97C23112yescomplemented
66C2×C8×C97C23112yescomplemented
67C23×C97C2×D413yes—
68C22×D97—23no—
69C22×D97—26no—
70C22×D97—212no—
71C22×D97—86no—
72Q8×C97C2×D414yes—
73D4×C97C2×D4112yes—
74C2×C4×C97C2×D413yes—
75C2×C4×C97C2×D416yes—
76C2×C4×C97C2416yes—
77C2×C4×C97C2411yescharacteristic
78C22⋊D97—212no—
79C22⋊D97—212no—
80C22⋊D97—24no—
81C2×Dic97—21no—
82C2×Dic97—26no—
83D388—212no—
84D388—26no—
85D388—23no—
86D388C2414yes—
87D388C2416yes—
88C4×D97—24no—
89C4×D97—212no—
90C4×D97—212no—
91Dic194—26no—
92Dic194—21no—
93Dic194C2414yes—
94Dic194C2412yes—
95C8×C97C2418yes—
96C22×C97—43no—
97C22×C97C22×D411yescharacteristic
98C22×C97C22×D416yes—
99C2×D97—24no—
100C2×D97—26no—
101C2×D97—46no—
102C2×D97—86no—
103C2×D97—812no—
104C4×C97C22×D411yescharacteristic
105C4×C97C22×D416yes—
106C4×C97C2511yescharacteristic
107Dic97—24no—
108Dic97—22no—
109C2×C97—21no—
110C2×C97—26no—
111C2×C97C23×D411yescharacteristic
112D97—46no—
113D97—44no—
114#3585—971nomaximal, Sylow
115C97#358511yescharacteristic, complemented, Sylow
116#585—971no—
117C2×H(5,𝔽2)—971no—
118#577—978no—
119#576—978no—
120#575—976no—
121C2×AGL(1,ℤ/8)—976no—
122#568D9711yescharacteristic, complemented
1232−1+4—974no—
1242−1+4—1941no—
125H(5,𝔽2)—974no—
126H(5,𝔽2)—978no—
127H(5,𝔽2)—1944no—
128H(5,𝔽2)—1943no—
129C2×#41—976no—
130C2×#41—973no—
131C2×#41—1944no—
132C2×#41—1941no—
133C2×#41C2×D9711yescharacteristic
134C22×D4—973no—
135C22×D4—1943no—
136#137—974no—
137#137—9712no—
138AGL(1,ℤ/8)—9712no—
139AGL(1,ℤ/8)—9724no—
140AGL(1,ℤ/8)—9712no—
141#135—9724no—
142C2×SD16—9712no—
143C2×D8—9712no—
144#131C2×D9718yescomplemented
145C2×M16C2×D9716yescomplemented
146C24—1943no—
147#41—9712no—
148#41—974no—
149#41—9712no—
150#41—1948no—
151#41—1944no—
152#41—19412no—
153#41—1946no—
154#41—1946no—
155#41—1946no—
156#41D38814yes—
157#41C22×D9714yes—
158C2×Q8—971no—
159C2×Q8—976no—
160C2×Q8—1944no—
161C2×Q8D38811yescharacteristic
162C2×D4—9712no—
163C2×D4—973no—
164C2×D4—976no—
165C2×D4—1943no—
166C2×D4—1946no—
167C2×D4—19424no—
168C2×D4—19412no—
169C2×D4—1943no—
170C2×D4—1946no—
171C2×D4D38813yes—
172C22×C4—1943no—
173C22×C4—1946no—
174C22×C4C22×D9713yes—
175Q16—978no—
176SD16—9724no—
177SD16—978no—
178D8—9724no—
179M16C22×D9714yes—
180M16C22×D97112yescomplemented
181C2×C8C22×D97112yescomplemented
182C23—7766no—
183C23—1946no—
184C23—19412no—
185C23—1943no—
186C23C2×D38813yes—
187Q8—974no—
188Q8—972no—
189Q8—1946no—
190Q8—1941no—
191Q8C2×D38814yes—
192D4—974no—
193D4—976no—
194D4—19412no—
195D4—1944no—
196D4—19412no—
197D4—1943no—
198D4—19412no—
199D4—1946no—
200D4C2×D388112yes—
201C2×C4—1941no—
202C2×C4—1946no—
203C2×C4—1944no—
204C2×C4—19412no—
205C2×C4—19412no—
206C2×C4C2×D38813yes—
207C2×C4C2×D38816yes—
208C2×C4C23×D9716yes—
209C2×C4C23×D9711yescharacteristic
210C8C23×D9718yes—
211C22—43no—
212C22—77612no—
213C22—7766no—
214C22—3886no—
215C22—1944no—
216C22—1946no—
217C22C22×D38811yescharacteristic
218C22C22×D38816yes—
219C4—1942no—
220C4—1944no—
221C4C22×D38811yescharacteristic
222C4C22×D38816yes—
223C4C24×D9711yescharacteristic
224C2—21no—
225C2—26no—
226C2—3886no—
227C2—3884no—
228C2C23×D38811yescharacteristic
subgroupG/H
centerC2C23×D388
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#568×C97C2
socleC2×C97C23×D4
hypercenter#568D97
hypocenterC97#3585
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. C2×#15690856C2semidirect
3. #20261547C2semidirect
4. #20261540C2semidirect
5. #20261535C2semidirect
6. C2×#15690838C2semidirect
7. #568×C97C2semidirect
9. #15690857C22semidirect
11. #15690856C22semidirect
12. #15690856C22semidirect
15. C2×#13095511C22semidirect
16. C2×#13095511C22semidirect
18. C22×D388C22semidirect
20. #15690843C22semidirect
21. #15690843C22semidirect
22. #15690838C22semidirect
23. #15690838C22semidirect
24. #15690838C22semidirect
25. #15690832C22semidirect
26. C2×#13095500C22semidirect
27. C2×D776C22semidirect
29. #131×C97C22semidirect
30. C2×M16×C97C22semidirect
39. #13095511C23semidirect
42. C2×Dic194C23semidirect
48. C2×D388C23semidirect
55. Dic388C23semidirect
56. #13095500C23semidirect
57. #13095500C23semidirect
58. D776C23semidirect
65. M16×C97C23semidirect
66. C2×C8×C97C23semidirect
115. C97#3585semidirect
122. #568D97semidirect
144. #131C2×D97semidirect
145. C2×M16C2×D97semidirect
180. M16C22×D97semidirect
181. C2×C8C22×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
C2×#156908562
#202615472
#202615402
#202615352
C2×#156908382
#568×C972
#358597