Finite Group Databasebeta

← #44562158

Subgroups — #44562158

Order 32128.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 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#30632493C214yesmaximal, complemented
2#30631947C214yesmaximal, complemented
3#30631869C212yesmaximal, complemented
4#30631854C212yesmaximal, complemented
5#30631805C212yesmaximal, complemented
6#492×C251C211yesmaximal, characteristic, complemented
7#22548763C2212yes—
8#22548760C2212yescomplemented
9#22548746C2214yescomplemented
10#22548744—24no—
11#22548742—24no—
12#22548741C2214yes—
13#22548737C2212yes—
14#22548736—24no—
15SD16×D251—24no—
16#22548721C2214yescomplemented
17#22548651C2212yescomplemented
18#22548643C2214yescomplemented
19#22548634C2212yescomplemented
20#22548632C2212yescomplemented
21C2×SD16×C251C2214yescomplemented
22C4⋊Q8×C251C2211yescharacteristic, complemented
23#127×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic, complemented
25#16860811—24no—
26#16860810—42no—
27#16860810C2314yes—
28#16860809—24no—
29Q8×D251—24no—
30C2×C22⋊D251—42no—
31D4×D251—24no—
32Dic1004D412yescomplemented
33#16860800—24no—
34#16860799D414yescomplemented
35#16860798—24no—
36#16860797—24no—
37D2008D412yescomplemented
38#16860795—24no—
39#16860794C2314yes—
40#16860793—24no—
41C8×D251—24no—
42C2×Q8×C251C2312yes—
43C2×D4×C251—41no—
44C2×D4×C251C2312yes—
45SD16×C251—24no—
46SD16×C251—24no—
47C2×C8×C251C2312yes—
48C4⋊C4×C251—41no—
49C42×C251C2311yescharacteristic
50C22×D251—42no—
51Dic502—42no—
52Dic502C2×D414yes—
53D1004—42no—
54D1004C2×D414yes—
55C22⋊D251—24no—
56C22⋊D251—44no—
57C2×Dic251—42no—
58C4×D251—24no—
59C4×D251—44no—
60C251⋊C8—24no—
61C23×C251—22no—
62Q8×C251—24no—
63D4×C251—24no—
64D4×C251—82no—
65C2×C4×C251—21no—
66C2×C4×C251—22no—
67C2×C4×C251C2411yescharacteristic
68C8×C251C2×D412yes—
69C8×C251C2×D412yes—
70C22×C251—42no—
71C22×C251—82no—
72C22×C251H(5,𝔽2)11yescharacteristic
73C2×D251—24no—
74C2×D251—162no—
75C4×C251—22no—
76C4×C251—42no—
77C4×C251C22×D412yes—
78Dic251—24no—
79C2×C251—21no—
80C2×C251—82no—
81C2×C251D4211yescharacteristic
82D251—44no—
83C251#329111yescharacteristic, complemented, Sylow
84#3291—2511nomaximal, Sylow
85#577—2514no—
86#492D25111yescharacteristic, complemented
87#471—2514no—
88#456—2512no—
89#453—2512no—
90#443—2512no—
912−1+4—2512no—
92H(5,𝔽2)—2512no—
93#137—5024no—
94#137—2514no—
95AGL(1,ℤ/8)—5024no—
96AGL(1,ℤ/8)—2514no—
97#135—5024no—
98#135—2512no—
99C2×SD16—5024no—
100C2×SD16C2×D25114yescomplemented
101#131—2514no—
102C4⋊Q8C2×D25111yescharacteristic, complemented
103#127C2×D25111yescharacteristic
104#108—2512no—
105#104—2514no—
106#101—2512no—
107#100—2512no—
108C4×C8C2×D25111yescharacteristic, complemented
109#41—10042no—
110#41—5024no—
111#41—5024no—
112#41—2514no—
113C2×Q8—5024no—
114C2×Q8C22×D25112yes—
115C2×D4—41no—
116C2×D4—10042no—
117C2×D4—5024no—
118C2×D4C22×D25112yes—
119Q16—5024no—
120Q16—2512no—
121SD16—24no—
122SD16—24no—
123SD16—5024no—
124SD16—5024no—
125SD16—2514no—
126D8—5024no—
127D8—2512no—
128M16—5024no—
129M16—2514no—
130C2×C8—5024no—
131C2×C8C22×D25112yes—
132C4⋊C4—41no—
133C42C22×D25111yescharacteristic
134C23—22no—
135C23—10042no—
136Q8—24no—
137Q8—10042no—
138Q8—2514no—
139D4—24no—
140D4—82no—
141D4—10042no—
142D4—10044no—
143D4—5024no—
144D4—2514no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—10044no—
148C2×C4—10042no—
149C2×C4—5024no—
150C2×C4C23×D25111yescharacteristic
151C8—5024no—
152C8D4×D25112yes—
153C8D4×D25112yes—
154C22—42no—
155C22—82no—
156C22—40162no—
157C22—5024no—
158C22#2254876011yescharacteristic
159C4—22no—
160C4—42no—
161C4—5024no—
162C4C2×D4×D25112yes—
163C2—21no—
164C2—82no—
165C2—10044no—
166C2#3063232211yescharacteristic
subgroupG/H
centerC2#30632322
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#492×C251C2
socleC2×C251D42
hypercenter#492D251
hypocenterC251#3291
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #30632493C2semidirect
2. #30631947C2semidirect
3. #30631869C2semidirect
4. #30631854C2semidirect
5. #30631805C2semidirect
6. #492×C251C2semidirect
8. #22548760C22semidirect
9. #22548746C22semidirect
16. #22548721C22semidirect
17. #22548651C22semidirect
18. #22548643C22semidirect
19. #22548634C22semidirect
20. #22548632C22semidirect
21. C2×SD16×C251C22semidirect
22. C4⋊Q8×C251C22semidirect
24. C4×C8×C251C22semidirect
32. Dic1004D4semidirect
34. #16860799D4semidirect
37. D2008D4semidirect
83. C251#3291semidirect
86. #492D251semidirect
100. C2×SD16C2×D251semidirect
102. C4⋊Q8C2×D251semidirect
108. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324932
#306319472
#306318692
#306318542
#306318052
#492×C2512
#3291251