Finite Group Databasebeta

← #27156523

Subgroups — #27156523

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
2#20261550C218yesmaximal, complemented
3#20261543C218yesmaximal, complemented
4#20261533C216yesmaximal, complemented
5C2×#15690841C216yesmaximal, complemented
6#20261502C211yesmaximal, characteristic, complemented
7C2×2−1+4×C97C211yesmaximal, characteristic, complemented
8#15690858—21no—
9#15690857C2214yescomplemented
10#15690856C2214yescomplemented
11#15690855—23no—
12C2×#13095511C2216yes—
13#41×D97—24no—
14#15690841C22112yescomplemented
15#15690841C22124yescomplemented
16#15690840C22112yescomplemented
17#15690836C2214yescomplemented
18#15690835C22112yescomplemented
19#15690830C22124yescomplemented
20C2×#13095501C22112yescomplemented
21C2×#13095498C22112yescomplemented
22#15690817C2218yescomplemented
23C2×#13095495C2216yescomplemented
242−1+4×C97—24no—
252−1+4×C97C2218yescomplemented
26C2×#41×C97—23no—
27C2×#41×C97C2211yescharacteristic
28C2×#41×C97C2213yescomplemented
29C22×Q8×C97—21no—
30C22×Q8×C97C2213yescomplemented
31#13095512—24no—
32#13095511—26no—
33#13095511—26no—
34#13095511C23112yes—
35#13095511C2314yes—
36#13095510—212no—
37C2×Dic194C2316yes—
38Q8×D97—24no—
39C2×D388C2316yes—
40C2×C22⋊D97—26no—
41D4×D97—212no—
42C2×C4×D97—26no—
43#13095501C23124yes—
44#13095499C2318yes—
45#13095498C23124yes—
46#13095496C2318yes—
47#13095495C2314yes—
48#13095495C23112yes—
49C2×#11715895C23112yes—
50#41×C97—224no—
51#41×C97—26no—
52#41×C97D414yes—
53#41×C97C23112yes—
54#41×C97C2314yes—
55C2×Q8×C97—28no—
56C2×Q8×C97—23no—
57C2×Q8×C97—26no—
58C2×Q8×C97D411yescharacteristic
59C2×Q8×C97C23112yes—
60C2×Q8×C97C2313yes—
61C2×D4×C97—23no—
62C2×D4×C97D413yes—
63C2×D4×C97C2311yescharacteristic
64C22×C4×C97—23no—
65C22×C4×C97—23no—
66C22×C4×C97C2313yes—
67C23×C97—21no—
68C23×C97C2×D413yes—
69C22×D97—26no—
70Q8×C97—212no—
71Q8×C97—23no—
72Q8×C97C2×D414yes—
73Q8×C97C2416yes—
74D4×C97—212no—
75D4×C97—21no—
76D4×C97C2×D4112yes—
77D4×C97C2412yes—
78C2×C4×C97—23no—
79C2×C4×C97—212no—
80C2×C4×C97—212no—
81C2×C4×C97C2×D413yes—
82C2×C4×C97C2×D416yes—
83C2×C4×C97C2411yescharacteristic
84C2×C4×C97C2416yes—
85C22⋊D97—212no—
86C22⋊D97—24no—
87C2×Dic97—26no—
88D388—26no—
89D388C2414yes—
90C4×D97—24no—
91C4×D97—212no—
92Dic194—26no—
93Dic194C2414yes—
94#11715895C2418yes—
95C22×C97—22no—
96C22×C97—43no—
97C22×C97—42no—
98C22×C97C22×D416yes—
99C22×C97C22×D411yescharacteristic
100C2×D97—24no—
101C2×D97—86no—
102C4×C97—26no—
103C4×C97C22×D416yes—
104C4×C97C22×D411yescharacteristic
105C4×C97C2511yescharacteristic
106Dic97—24no—
107C2×C97—26no—
108C2×C97—21no—
109C2×C97—42no—
110C2×C97C23×D411yescharacteristic
111D97—44no—
112#3586—971nomaximal, Sylow
113C97#358611yescharacteristic, complemented, Sylow
114#585—971no—
115C2×2−1+4D9711yescharacteristic, complemented
116#578—978no—
117#577—978no—
118#575—976no—
119C2×#137—976no—
120#568—971no—
1212−1+4—24no—
1222−1+4—974no—
1232−1+4—1941no—
1242−1+4C2×D9718yescomplemented
125H(5,𝔽2)—974no—
126H(5,𝔽2)—1943no—
127C2×#41—23no—
128C2×#41—976no—
129C2×#41—1944no—
130C2×#41C2×D9711yescharacteristic
131C2×#41C2×D9713yescomplemented
132C22×Q8—21no—
133C22×Q8C2×D9713yescomplemented
134#137—9712no—
135#137—9712no—
136#137—9724no—
137AGL(1,ℤ/8)—9712no—
138AGL(1,ℤ/8)—974no—
139#135—9724no—
140C2×Q16—9712no—
141C2×SD16—9712no—
142#131—978no—
143C2×M16—976no—
144#41—224no—
145#41—26no—
146#41—9712no—
147#41—974no—
148#41—1944no—
149#41—19412no—
150#41—1946no—
151#41—1946no—
152#41C22⋊D9714yes—
153#41C22×D97112yes—
154#41C22×D9714yes—
155C2×Q8—28no—
156C2×Q8—23no—
157C2×Q8—26no—
158C2×Q8—976no—
159C2×Q8—1944no—
160C2×Q8C22⋊D9711yescharacteristic
161C2×Q8C22×D97112yes—
162C2×Q8C22×D9713yes—
163C2×D4—23no—
164C2×D4—976no—
165C2×D4—1946no—
166C2×D4—19412no—
167C2×D4C22⋊D9713yes—
168C2×D4C22×D9711yescharacteristic
169C22×C4—23no—
170C22×C4—23no—
171C22×C4—1946no—
172C22×C4C22×D9713yes—
173Q16—9724no—
174SD16—9724no—
175SD16—978no—
176D8—978no—
177M16—974no—
178M16—9712no—
179C2×C8—9712no—
180C23—21no—
181C23—1946no—
182C23C2×C22⋊D9713yes—
183Q8—212no—
184Q8—23no—
185Q8—974no—
186Q8—1946no—
187Q8C2×C22⋊D9714yes—
188Q8C23×D9716yes—
189D4—212no—
190D4—21no—
191D4—974no—
192D4—1944no—
193D4—19412no—
194D4—1946no—
195D4C2×C22⋊D97112yes—
196D4C23×D9712yes—
197C2×C4—23no—
198C2×C4—212no—
199C2×C4—212no—
200C2×C4—1946no—
201C2×C4—1944no—
202C2×C4—19412no—
203C2×C4C2×C22⋊D9713yes—
204C2×C4C2×C22⋊D9716yes—
205C2×C4C23×D9716yes—
206C2×C4C23×D9711yescharacteristic
207C8—978no—
208C22—22no—
209C22—43no—
210C22—42no—
211C22—7766no—
212C22—1944no—
213C22C22×C22⋊D9711yescharacteristic
214C22C22×C22⋊D9716yes—
215C4—26no—
216C4—1944no—
217C4C22×C22⋊D9716yes—
218C4C22×C22⋊D9711yescharacteristic
219C4C24×D9711yescharacteristic
220C2—21no—
221C2—26no—
222C2—42no—
223C2—3884no—
224C2C23×C22⋊D9711yescharacteristic
subgroupG/H
centerC2C23×C22⋊D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×2−1+4×C97C2
socleC2×C97C23×D4
hypercenterC2×2−1+4D97
hypocenterC97#3586
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. #20261550C2semidirect
3. #20261543C2semidirect
4. #20261533C2semidirect
5. C2×#15690841C2semidirect
6. #20261502C2semidirect
7. C2×2−1+4×C97C2semidirect
9. #15690857C22semidirect
10. #15690856C22semidirect
14. #15690841C22semidirect
15. #15690841C22semidirect
16. #15690840C22semidirect
17. #15690836C22semidirect
18. #15690835C22semidirect
19. #15690830C22semidirect
20. C2×#13095501C22semidirect
21. C2×#13095498C22semidirect
22. #15690817C22semidirect
23. C2×#13095495C22semidirect
25. 2−1+4×C97C22semidirect
28. C2×#41×C97C22semidirect
30. C22×Q8×C97C22semidirect
113. C97#3586semidirect
115. C2×2−1+4D97semidirect
124. 2−1+4C2×D97semidirect
131. C2×#41C2×D97semidirect
133. C22×Q8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
#202615502
#202615432
#202615332
C2×#156908412
#202615022
C2×2−1+4×C972
#358697