Finite Group Databasebeta

← C31×#3549 #16843482

Subgroups — C31×#3549 #16843482

Order 3968.

1258 conjugacy classes of subgroups; only the 768 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 768 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
1C31×H(3,𝔽4)C214yesmaximal, complemented
2C31×#555C216yesmaximal, complemented
3D42×C31C216yesmaximal, complemented
4C31×#535C2112yesmaximal, complemented
5C31×#532C213yesmaximal, complemented
6C2×#41×C31C2216yescomplemented
7C22×D4×C31C22112yescomplemented
8C31×C4⋊Q8C2213yescomplemented
9C31×#127C2213yes—
10C31×#127C2218yescomplemented
11C31×#124C2216yescomplemented
12C31×#124C22124yescomplemented
13C31×#121C22112yescomplemented
14C31×#121C22124yescomplemented
15C31×#120C2218yescomplemented
16C31×#120C22124yescomplemented
17C4×D4×C31C22112yescomplemented
18C31×#117C22112yescomplemented
19C2×C42×C31C2211yescharacteristic, complemented
20C24×C31C2318yes—
21#41×C31—412no—
22C2×Q8×C31C2316yes—
23C2×D4×C31—424no—
24C2×D4×C31C23112yes—
25C2×D4×C31C2316yes—
26C2×D4×C31C23124yes—
27C2×D4×C31C23112yes—
28C22×C4×C31C2313yes—
29C22×C4×C31C23112yes—
30C4⋊C4×C31C23112yes—
31C22⋊C4×C31C23124yes—
32C22⋊C4×C31C23124yes—
33C42×C31C2313yes—
34C42×C31C2311yescharacteristic
35C42×C31C2318yes—
36C23×C31—424no—
37C23×C31C24112yes—
38C23×C31C2411yescharacteristic
39Q8×C31—46no—
40D4×C31—224no—
41D4×C31—424no—
42D4×C31—412no—
43D4×C31—46no—
44C2×C4×C31—224no—
45C2×C4×C31—26no—
46C2×C4×C31C2413yes—
47C2×C4×C31C24112yes—
48C2×C4×C31C2413yes—
49#3549C3111yesmaximal, characteristic, direct product factor, Sylow
50C22×C31—23no—
51C22×C31—212no—
52C22×C31—224no—
53C22×C31—816no—
54C22×C31C2511yescharacteristic
55C4×C31—212no—
56C4×C31—23no—
57C4×C31H(5,𝔽2)16yes—
58H(3,𝔽4)C2×C3114yescomplemented
59#555C2×C3116yescomplemented
60D42C2×C3116yescomplemented
61#535C2×C31112yescomplemented
62#532C2×C3113yescomplemented
63C2×C31—41no—
64C2×C31—412no—
65C2×C31C2×H(5,𝔽2)13yes—
66C2×#41C22×C3116yescomplemented
67C22×D4C22×C31112yescomplemented
68C4⋊Q8C22×C3113yescomplemented
69#127C22×C3113yes—
70#127C22×C3118yescomplemented
71#124C22×C3116yescomplemented
72#124C22×C31124yescomplemented
73#121C22×C31112yescomplemented
74#121C22×C31124yescomplemented
75#120C22×C3118yescomplemented
76#120C22×C31124yescomplemented
77C4×D4C22×C31112yescomplemented
78#117C22×C31112yescomplemented
79C2×C42C22×C3111yescharacteristic, complemented
80C31#354911yescharacteristic, direct product factor, Sylow
81C24C23×C3118yes—
82#41—412no—
83C2×Q8C23×C3116yes—
84C2×D4—424no—
85C2×D4C23×C31112yes—
86C2×D4C23×C3116yes—
87C2×D4C23×C31124yes—
88C2×D4C23×C31112yes—
89C22×C4C23×C3113yes—
90C22×C4C23×C31112yes—
91C4⋊C4C23×C31112yes—
92C22⋊C4C23×C31124yes—
93C22⋊C4C23×C31124yes—
94C42C23×C3113yes—
95C42C23×C3111yescharacteristic
96C42C23×C3118yes—
97C23—424no—
98C23C24×C31112yes—
99C23C24×C3111yescharacteristic
100Q8—46no—
101D4—224no—
102D4—424no—
103D4—412no—
104D4—46no—
105C2×C4—224no—
106C2×C4—26no—
107C2×C4C24×C3113yes—
108C2×C4C24×C31112yes—
109C2×C4C24×C3113yes—
110C22—23no—
111C22—212no—
112C22—224no—
113C22—816no—
114C22C25×C3111yescharacteristic
115C4—212no—
116C4—23no—
117C4C31×H(5,𝔽2)16yes—
118C2—412no—
119C2—41no—
120C2C2×C31×H(5,𝔽2)13yes—
subgroupG/H
center, socleC22×C31C25
commutator, Frattini, cosocleC22C25×C31
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×H(3,𝔽4)C2semidirect
2. C31×#555C2semidirect
3. D42×C31C2semidirect
4. C31×#535C2semidirect
5. C31×#532C2semidirect
6. C2×#41×C31C22semidirect
7. C22×D4×C31C22semidirect
8. C31×C4⋊Q8C22semidirect
10. C31×#127C22semidirect
11. C31×#124C22semidirect
12. C31×#124C22semidirect
13. C31×#121C22semidirect
14. C31×#121C22semidirect
15. C31×#120C22semidirect
16. C31×#120C22semidirect
17. C4×D4×C31C22semidirect
18. C31×#117C22semidirect
19. C2×C42×C31C22semidirect
49. #3549C31direct (N × H)
58. H(3,𝔽4)C2×C31semidirect
59. #555C2×C31semidirect
60. D42C2×C31semidirect
61. #535C2×C31semidirect
62. #532C2×C31semidirect
66. C2×#41C22×C31semidirect
67. C22×D4C22×C31semidirect
68. C4⋊Q8C22×C31semidirect
70. #127C22×C31semidirect
71. #124C22×C31semidirect
72. #124C22×C31semidirect
73. #121C22×C31semidirect
74. #121C22×C31semidirect
75. #120C22×C31semidirect
76. #120C22×C31semidirect
77. C4×D4C22×C31semidirect
78. #117C22×C31semidirect
79. C2×C42C22×C31semidirect
80. C31#3549direct (N × H)

Maximal subgroups

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

subgroupindex
C31×H(3,𝔽4)2
C31×#5552
D42×C312
C31×#5352
C31×#5322
#354931