Finite Group Databasebeta

← #3292×C241 #44504959

Subgroups — #3292×C241 #44504959

Order 30848.

476 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 184 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#576×C241C214yesmaximal, complemented
2#493×C241C211yesmaximal, characteristic, complemented
3#472×C241C214yesmaximal, complemented
4#453×C241C214yesmaximal, complemented
5#443×C241C212yesmaximal, complemented
6H(5,𝔽2)×C241C2214yescomplemented
7AGL(1,ℤ/8)×C241—28no—
8AGL(1,ℤ/8)×C241C2218yescomplemented
9#135×C241—24no—
10#135×C241C2212yes—
11C2×D8×C241—24no—
12C2×D8×C241C2214yescomplemented
13#131×C241C2214yescomplemented
14#127×C241C2212yescomplemented
15#108×C241C2212yescomplemented
16#104×C241C2214yescomplemented
17#100×C241C2214yescomplemented
18C4×C8×C241C2211yescharacteristic, complemented
19#41×C241—28no—
20#41×C241C2314yes—
21C2×D4×C241—28no—
22C2×D4×C241—42no—
23C2×D4×C241—44no—
24C2×D4×C241C2314yes—
25Q16×C241D412yescomplemented
26SD16×C241—28no—
27SD16×C241D414yescomplemented
28D8×C241—28no—
29D8×C241—24no—
30D8×C241—24no—
31D8×C241D412yescomplemented
32M16×C241—24no—
33M16×C241C2314yes—
34C2×C8×C241—24no—
35C2×C8×C241C2312yes—
36C42×C241C2311yescharacteristic
37C23×C241—24no—
38C23×C241—44no—
39Q8×C241C2×D414yes—
40D4×C241—24no—
41D4×C241—28no—
42D4×C241—48no—
43D4×C241—84no—
44D4×C241C2×D414yes—
45C2×C4×C241—21no—
46C2×C4×C241—24no—
47C2×C4×C241—44no—
48C2×C4×C241C2411yescharacteristic
49C8×C241—24no—
50C8×C241C2×D412yes—
51C8×C241C2×D412yes—
52C22×C241—24no—
53C22×C241—44no—
54C22×C241—84no—
55C22×C241—164no—
56C22×C241H(5,𝔽2)11yescharacteristic
57C4×C241—22no—
58C4×C241—24no—
59C4×C241C22×D412yes—
60C2×C241—21no—
61C2×C241—44no—
62C2×C241—84no—
63C2×C241D4211yescharacteristic
64C241#329211yescharacteristic, direct product factor, Sylow
65#3292C24111yesmaximal, characteristic, direct product factor, Sylow
66#576C2×C24114yescomplemented
67#493C2×C24111yescharacteristic, complemented
68#472C2×C24114yescomplemented
69#453C2×C24114yescomplemented
70#443C2×C24112yescomplemented
71H(5,𝔽2)C22×C24114yescomplemented
72AGL(1,ℤ/8)—28no—
73AGL(1,ℤ/8)C22×C24118yescomplemented
74#135—24no—
75#135C22×C24112yes—
76C2×D8—24no—
77C2×D8C22×C24114yescomplemented
78#131C22×C24114yescomplemented
79#127C22×C24112yescomplemented
80#108C22×C24112yescomplemented
81#104C22×C24114yescomplemented
82#100C22×C24114yescomplemented
83C4×C8C22×C24111yescharacteristic, complemented
84#41—28no—
85#41C23×C24114yes—
86C2×D4—28no—
87C2×D4—42no—
88C2×D4—44no—
89C2×D4C23×C24114yes—
90Q16D4×C24112yescomplemented
91SD16—28no—
92SD16D4×C24114yescomplemented
93D8—28no—
94D8—24no—
95D8—24no—
96D8D4×C24112yescomplemented
97M16—24no—
98M16C23×C24114yes—
99C2×C8—24no—
100C2×C8C23×C24112yes—
101C42C23×C24111yescharacteristic
102C23—24no—
103C23—44no—
104Q8C2×D4×C24114yes—
105D4—24no—
106D4—28no—
107D4—48no—
108D4—84no—
109D4C2×D4×C24114yes—
110C2×C4—21no—
111C2×C4—24no—
112C2×C4—44no—
113C2×C4C24×C24111yescharacteristic
114C8—24no—
115C8C2×D4×C24112yes—
116C8C2×D4×C24112yes—
117C22—24no—
118C22—44no—
119C22—84no—
120C22—164no—
121C22H(5,𝔽2)×C24111yescharacteristic
122C4—22no—
123C4—24no—
124C4C22×D4×C24112yes—
125C2—21no—
126C2—44no—
127C2—84no—
128C2D42×C24111yescharacteristic
subgroupG/H
center, socleC2×C241D42
commutator, Frattini, cosocleC2×C4C24×C241
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C241 ◁ C2×C4×C241 ◁ G

Complements

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

NH ≅ G/Ntype
1. #576×C241C2semidirect
2. #493×C241C2semidirect
3. #472×C241C2semidirect
4. #453×C241C2semidirect
5. #443×C241C2semidirect
6. H(5,𝔽2)×C241C22semidirect
8. AGL(1,ℤ/8)×C241C22semidirect
12. C2×D8×C241C22semidirect
13. #131×C241C22semidirect
14. #127×C241C22semidirect
15. #108×C241C22semidirect
16. #104×C241C22semidirect
17. #100×C241C22semidirect
18. C4×C8×C241C22semidirect
25. Q16×C241D4semidirect
27. SD16×C241D4semidirect
31. D8×C241D4semidirect
64. C241#3292direct (N × H)
65. #3292C241direct (N × H)
66. #576C2×C241semidirect
67. #493C2×C241semidirect
68. #472C2×C241semidirect
69. #453C2×C241semidirect
70. #443C2×C241semidirect
71. H(5,𝔽2)C22×C241semidirect
73. AGL(1,ℤ/8)C22×C241semidirect
77. C2×D8C22×C241semidirect
78. #131C22×C241semidirect
79. #127C22×C241semidirect
80. #108C22×C241semidirect
81. #104C22×C241semidirect
82. #100C22×C241semidirect
83. C4×C8C22×C241semidirect
90. Q16D4×C241semidirect
92. SD16D4×C241semidirect
96. D8D4×C241semidirect

Maximal subgroups

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

subgroupindex
#576×C2412
#493×C2412
#472×C2412
#453×C2412
#443×C2412
#3292241