Finite Group Databasebeta

← #27156524

Subgroups — #27156524

Order 12416.

1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1412 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×#15690857C211yesmaximal, characteristic, complemented
3#20261552C218yesmaximal, complemented
4#20261547C218yesmaximal, complemented
5#20261535C216yesmaximal, complemented
6C2×#15690843C216yesmaximal, complemented
7#568×C97C211yesmaximal, characteristic, complemented
8#15690858—21no—
9#15690857—24no—
10#15690857C2218yescomplemented
11#15690857C2214yescomplemented
12#15690856C2214yescomplemented
13#15690855—23no—
14C2×#13095511C2216yes—
15C2×#13095511C2213yescomplemented
16C2×#13095510—23no—
17#41×D97—24no—
18C22×Dic194C2213yescomplemented
19C2×Q8×D97—21no—
20#15690843C22112yescomplemented
21#15690843C22124yescomplemented
22#15690843C22112yescomplemented
23#15690838C2214yescomplemented
24#15690838C22112yescomplemented
25#15690832C22124yescomplemented
26C2×Dic388C22112yescomplemented
27C2×#13095500C22112yescomplemented
28C2×#41×C97C2211yescharacteristic
29#131×C97C2218yescomplemented
30C2×M16×C97C2216yescomplemented
31#13095512—24no—
32#13095511—26no—
33#13095511—26no—
34#13095511—26no—
35#13095511C23112yes—
36#13095511C23112yes—
37#13095511C2314yes—
38#13095510—212no—
39#13095510—224no—
40C2×Dic194—23no—
41C2×Dic194—26no—
42C2×Dic194C23112yes—
43C2×Dic194C2316yes—
44C2×Dic194C2313yes—
45Q8×D97—28no—
46Q8×D97—24no—
47C2×D388C2316yes—
48C2×D388C2311yescharacteristic
49C2×C22⋊D97—23no—
50C2×C22⋊D97—26no—
51D4×D97—212no—
52C22×Dic97—23no—
53C2×C4×D97—23no—
54C2×C4×D97—26no—
55Dic388C23124yes—
56#13095500C23124yes—
57#13095500C2318yes—
58D776C2318yes—
59#41×C97D414yes—
60#41×C97C2314yes—
61C2×Q8×C97D411yescharacteristic
62C2×D4×C97D413yes—
63C22×C4×C97C2313yes—
64M16×C97C2314yes—
65M16×C97C23112yes—
66C2×C8×C97C23112yes—
67C23×C97C2×D413yes—
68C22×D97—21no—
69C22×D97—26no—
70Q8×C97C2×D414yes—
71D4×C97C2×D4112yes—
72C2×C4×C97C2×D413yes—
73C2×C4×C97C2×D416yes—
74C2×C4×C97C2416yes—
75C2×C4×C97C2411yescharacteristic
76C22⋊D97—212no—
77C22⋊D97—24no—
78C22⋊D97—212no—
79C2×Dic97—23no—
80C2×Dic97—26no—
81C2×Dic97—212no—
82D388—26no—
83D388—21no—
84D388C2412yes—
85D388C2414yes—
86C4×D97—24no—
87C4×D97—212no—
88C4×D97—212no—
89Dic194—212no—
90Dic194—26no—
91Dic194—23no—
92Dic194C2416yes—
93Dic194C2414yes—
94C8×C97C2418yes—
95C22×C97—43no—
96C22×C97C22×D411yescharacteristic
97C22×C97C22×D416yes—
98C2×D97—22no—
99C2×D97—24no—
100C2×D97—42no—
101C2×D97—86no—
102C4×C97C22×D411yescharacteristic
103C4×C97C22×D416yes—
104C4×C97C2511yescharacteristic
105Dic97—26no—
106Dic97—24no—
107C2×C97—21no—
108C2×C97—26no—
109C2×C97C23×D411yescharacteristic
110D97—42no—
111D97—44no—
112#3586—971nomaximal, Sylow
113C97#358611yescharacteristic, complemented, Sylow
114#585—971no—
115C2×2−1+4—971no—
116#578—978no—
117#577—978no—
118#575—976no—
119C2×#137—976no—
120#568D9711yescharacteristic, complemented
1212−1+4—978no—
1222−1+4—974no—
1232−1+4—1944no—
1242−1+4—1941no—
125H(5,𝔽2)—974no—
126H(5,𝔽2)—1943no—
127C2×#41—976no—
128C2×#41—973no—
129C2×#41—1944no—
130C2×#41—1943no—
131C2×#41C2×D9711yescharacteristic
132C22×Q8—973no—
133C22×Q8—1941no—
134#137—9712no—
135#137—9724no—
136#137—9712no—
137AGL(1,ℤ/8)—974no—
138AGL(1,ℤ/8)—9712no—
139#135—9724no—
140C2×Q16—9712no—
141C2×SD16—9712no—
142#131C2×D9718yescomplemented
143C2×M16C2×D9716yescomplemented
144#41—9712no—
145#41—9712no—
146#41—974no—
147#41—1944no—
148#41—19412no—
149#41—19424no—
150#41—1946no—
151#41—1946no—
152#41—1946no—
153#41D38814yes—
154#41C22×D9714yes—
155C2×Q8—9712no—
156C2×Q8—976no—
157C2×Q8—973no—
158C2×Q8—1948no—
159C2×Q8—1944no—
160C2×Q8—1943no—
161C2×Q8—1946no—
162C2×Q8D38811yescharacteristic
163C2×D4—976no—
164C2×D4—971no—
165C2×D4—1943no—
166C2×D4—1946no—
167C2×D4—19412no—
168C2×D4D38813yes—
169C22×C4—1943no—
170C22×C4—1943no—
171C22×C4—1946no—
172C22×C4C22×D9713yes—
173Q16—9724no—
174SD16—9724no—
175SD16—978no—
176D8—978no—
177M16C22×D9714yes—
178M16C22×D97112yes—
179C2×C8C22×D97112yes—
180C23—1946no—
181C23—1941no—
182C23C2×D38813yes—
183Q8—976no—
184Q8—974no—
185Q8—1943no—
186Q8—19412no—
187Q8—1946no—
188Q8C2×D38814yes—
189D4—972no—
190D4—974no—
191D4—1944no—
192D4—19412no—
193D4—19412no—
194D4—1946no—
195D4—1941no—
196D4C2×D388112yes—
197C2×C4—1943no—
198C2×C4—1946no—
199C2×C4—19412no—
200C2×C4—1944no—
201C2×C4—19412no—
202C2×C4—19412no—
203C2×C4C2×D38813yes—
204C2×C4C2×D38816yes—
205C2×C4C23×D9716yes—
206C2×C4C23×D9711yescharacteristic
207C8C23×D9718yes—
208C22—43no—
209C22—7766no—
210C22—3882no—
211C22—1942no—
212C22—1944no—
213C22C22×D38811yescharacteristic
214C22C22×D38816yes—
215C4—1944no—
216C4—1946no—
217C4C22×D38811yescharacteristic
218C4C22×D38816yes—
219C4C24×D9711yescharacteristic
220C2—21no—
221C2—26no—
222C2—3884no—
223C2—3882no—
224C2C23×D38811yescharacteristic
subgroupG/H
centerC2C23×D388
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#568×C97C2
socleC2×C97C23×D4
hypercenter#568D97
hypocenterC97#3586
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×#15690857C2semidirect
3. #20261552C2semidirect
4. #20261547C2semidirect
5. #20261535C2semidirect
6. C2×#15690843C2semidirect
7. #568×C97C2semidirect
10. #15690857C22semidirect
11. #15690857C22semidirect
12. #15690856C22semidirect
15. C2×#13095511C22semidirect
18. C22×Dic194C22semidirect
20. #15690843C22semidirect
21. #15690843C22semidirect
22. #15690843C22semidirect
23. #15690838C22semidirect
24. #15690838C22semidirect
25. #15690832C22semidirect
26. C2×Dic388C22semidirect
27. C2×#13095500C22semidirect
29. #131×C97C22semidirect
30. C2×M16×C97C22semidirect
113. C97#3586semidirect
120. #568D97semidirect
142. #131C2×D97semidirect
143. C2×M16C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
C2×#156908572
#202615522
#202615472
#202615352
C2×#156908432
#568×C972
#358697