Finite Group Databasebeta

← #27182925

Subgroups — #27182925

Order 12928.

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#20265983C214yesmaximal, complemented
2#20265437C214yesmaximal, complemented
3#20265359C212yesmaximal, complemented
4#20265344C212yesmaximal, complemented
5#20265295C212yesmaximal, complemented
6#492×C101C211yesmaximal, characteristic, complemented
7#15692303C2212yes—
8#15692300C2212yescomplemented
9#15692286C2214yescomplemented
10#15692284—24no—
11#15692282—24no—
12#15692281C2214yes—
13#15692277C2212yes—
14#15692276—24no—
15SD16×D101—24no—
16#15692261C2214yescomplemented
17#15692191C2212yescomplemented
18#15692183C2214yescomplemented
19#15692174C2212yescomplemented
20#15692172C2212yescomplemented
21C2×SD16×C101C2214yescomplemented
22C4⋊Q8×C101C2211yescharacteristic, complemented
23#127×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic, complemented
25#13108075—24no—
26#13108074—42no—
27#13108074C2314yes—
28#13108073—24no—
29Q8×D101—24no—
30C2×C22⋊D101—42no—
31D4×D101—24no—
32Dic404D412yescomplemented
33#13108064—24no—
34#13108063D414yescomplemented
35#13108062—24no—
36#13108061—24no—
37D808D412yescomplemented
38#13108059—24no—
39#13108058C2314yes—
40#13108057—24no—
41C8×D101—24no—
42C2×Q8×C101C2312yes—
43C2×D4×C101—41no—
44C2×D4×C101C2312yes—
45SD16×C101—24no—
46SD16×C101—24no—
47C2×C8×C101C2312yes—
48C4⋊C4×C101—41no—
49C42×C101C2311yescharacteristic
50C23×C101—22no—
51C22×D101—42no—
52Q8×C101—24no—
53D4×C101—24no—
54D4×C101—82no—
55C2×C4×C101—21no—
56C2×C4×C101—22no—
57C2×C4×C101C2411yescharacteristic
58C22⋊D101—24no—
59C22⋊D101—44no—
60C2×Dic101—42no—
61D404—42no—
62D404C2×D414yes—
63C4×D101—24no—
64C4×D101—44no—
65Dic202—42no—
66Dic202C2×D414yes—
67C8×C101C2×D412yes—
68C8×C101C2×D412yes—
69#11717579—24no—
70C22×C101—42no—
71C22×C101—82no—
72C22×C101H(5,𝔽2)11yescharacteristic
73C2×D101—24no—
74C2×D101—162no—
75C4×C101—22no—
76C4×C101—42no—
77C4×C101C22×D412yes—
78Dic101—24no—
79C2×C101—21no—
80C2×C101—82no—
81C2×C101D4211yescharacteristic
82D101—44no—
83#3291—1011nomaximal, Sylow
84C101#329111yescharacteristic, complemented, Sylow
85#577—1014no—
86#492D10111yescharacteristic, complemented
87#471—1014no—
88#456—1012no—
89#453—1012no—
90#443—1012no—
912−1+4—1012no—
92H(5,𝔽2)—1012no—
93#137—1014no—
94#137—2024no—
95AGL(1,ℤ/8)—1014no—
96AGL(1,ℤ/8)—2024no—
97#135—1012no—
98#135—2024no—
99C2×SD16—2024no—
100C2×SD16C2×D10114yescomplemented
101#131—1014no—
102C4⋊Q8C2×D10111yescharacteristic, complemented
103#127C2×D10111yescharacteristic
104#108—1012no—
105#104—1014no—
106#101—1012no—
107#100—1012no—
108C4×C8C2×D10111yescharacteristic, complemented
109#41—1014no—
110#41—4042no—
111#41—2024no—
112#41—2024no—
113C2×Q8—2024no—
114C2×Q8C22×D10112yes—
115C2×D4—41no—
116C2×D4—4042no—
117C2×D4—2024no—
118C2×D4C22×D10112yes—
119Q16—1012no—
120Q16—2024no—
121SD16—24no—
122SD16—24no—
123SD16—1014no—
124SD16—2024no—
125SD16—2024no—
126D8—1012no—
127D8—2024no—
128M16—1014no—
129M16—2024no—
130C2×C8—2024no—
131C2×C8C22×D10112yes—
132C4⋊C4—41no—
133C42C22×D10111yescharacteristic
134C23—22no—
135C23—4042no—
136Q8—24no—
137Q8—1014no—
138Q8—4042no—
139D4—24no—
140D4—82no—
141D4—1014no—
142D4—4042no—
143D4—4044no—
144D4—2024no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—4044no—
148C2×C4—4042no—
149C2×C4—2024no—
150C2×C4C23×D10111yescharacteristic
151C8—2024no—
152C8D4×D10112yes—
153C8D4×D10112yes—
154C22—42no—
155C22—82no—
156C22—16162no—
157C22—2024no—
158C22#1569230011yescharacteristic
159C4—22no—
160C4—42no—
161C4—2024no—
162C4C2×D4×D10112yes—
163C2—21no—
164C2—82no—
165C2—4044no—
166C2#2026581211yescharacteristic
subgroupG/H
centerC2#20265812
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#492×C101C2
socleC2×C101D42
hypercenter#492D101
hypocenterC101#3291
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20265983C2semidirect
2. #20265437C2semidirect
3. #20265359C2semidirect
4. #20265344C2semidirect
5. #20265295C2semidirect
6. #492×C101C2semidirect
8. #15692300C22semidirect
9. #15692286C22semidirect
16. #15692261C22semidirect
17. #15692191C22semidirect
18. #15692183C22semidirect
19. #15692174C22semidirect
20. #15692172C22semidirect
21. C2×SD16×C101C22semidirect
22. C4⋊Q8×C101C22semidirect
24. C4×C8×C101C22semidirect
32. Dic404D4semidirect
34. #13108063D4semidirect
37. D808D4semidirect
84. C101#3291semidirect
86. #492D101semidirect
100. C2×SD16C2×D101semidirect
102. C4⋊Q8C2×D101semidirect
108. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659832
#202654372
#202653592
#202653442
#202652952
#492×C1012
#3291101