Finite Group Databasebeta

← C23×#3549 #15671168

Subgroups — C23×#3549 #15671168

Order 2944.

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
1C23×H(3,𝔽4)C214yesmaximal, complemented
2C23×#555C216yesmaximal, complemented
3D42×C23C216yesmaximal, complemented
4C23×#535C2112yesmaximal, complemented
5C23×#532C213yesmaximal, complemented
6C2×#41×C23C2216yescomplemented
7C22×D4×C23C22112yescomplemented
8C23×C4⋊Q8C2213yescomplemented
9C23×#127C2213yes—
10C23×#127C2218yescomplemented
11C23×#124C22124yescomplemented
12C23×#124C2216yescomplemented
13C23×#121C22124yescomplemented
14C23×#121C22112yescomplemented
15C23×#120C22124yescomplemented
16C23×#120C2218yescomplemented
17C4×D4×C23C22112yescomplemented
18C23×#117C22112yescomplemented
19C2×C42×C23C2211yescharacteristic, complemented
20C24×C23C2318yes—
21#41×C23—412no—
22C2×Q8×C23C2316yes—
23C2×D4×C23—424no—
24C2×D4×C23C23112yes—
25C2×D4×C23C2316yes—
26C2×D4×C23C23124yes—
27C2×D4×C23C23112yes—
28C22×C4×C23C2313yes—
29C22×C4×C23C23112yes—
30C4⋊C4×C23C23112yes—
31C22⋊C4×C23C23124yes—
32C22⋊C4×C23C23124yes—
33C42×C23C2313yes—
34C42×C23C2311yescharacteristic
35C42×C23C2318yes—
36C23×C23—424no—
37C23×C23C24112yes—
38C23×C23C2411yescharacteristic
39Q8×C23—46no—
40D4×C23—224no—
41D4×C23—424no—
42D4×C23—412no—
43D4×C23—46no—
44C2×C4×C23—224no—
45C2×C4×C23—26no—
46C2×C4×C23C2413yes—
47C2×C4×C23C24112yes—
48C2×C4×C23C2413yes—
49#3549C2311yesmaximal, characteristic, direct product factor, Sylow
50C22×C23—23no—
51C22×C23—212no—
52C22×C23—224no—
53C22×C23—816no—
54C22×C23C2511yescharacteristic
55C4×C23—212no—
56C4×C23—23no—
57C4×C23H(5,𝔽2)16yes—
58H(3,𝔽4)C2×C2314yescomplemented
59#555C2×C2316yescomplemented
60D42C2×C2316yescomplemented
61#535C2×C23112yescomplemented
62#532C2×C2313yescomplemented
63C2×C23—41no—
64C2×C23—412no—
65C2×C23C2×H(5,𝔽2)13yes—
66C2×#41C22×C2316yescomplemented
67C22×D4C22×C23112yescomplemented
68C4⋊Q8C22×C2313yescomplemented
69#127C22×C2313yes—
70#127C22×C2318yescomplemented
71#124C22×C2316yescomplemented
72#124C22×C23124yescomplemented
73#121C22×C23112yescomplemented
74#121C22×C23124yescomplemented
75#120C22×C2318yescomplemented
76#120C22×C23124yescomplemented
77C4×D4C22×C23112yescomplemented
78#117C22×C23112yescomplemented
79C2×C42C22×C2311yescharacteristic, complemented
80C23#354911yescharacteristic, direct product factor, Sylow
81C24C23×C2318yes—
82#41—412no—
83C2×Q8C23×C2316yes—
84C2×D4—424no—
85C2×D4C23×C23112yes—
86C2×D4C23×C2316yes—
87C2×D4C23×C23124yes—
88C2×D4C23×C23112yes—
89C22×C4C23×C2313yes—
90C22×C4C23×C23112yes—
91C4⋊C4C23×C23112yes—
92C22⋊C4C23×C23124yes—
93C22⋊C4C23×C23124yes—
94C42C23×C2313yes—
95C42C23×C2311yescharacteristic
96C42C23×C2318yes—
97C23—424no—
98C23C24×C23112yes—
99C23C24×C2311yescharacteristic
100Q8—46no—
101D4—224no—
102D4—424no—
103D4—412no—
104D4—46no—
105C2×C4—224no—
106C2×C4—26no—
107C2×C4C24×C2313yes—
108C2×C4C24×C23112yes—
109C2×C4C24×C2313yes—
110C22—23no—
111C22—212no—
112C22—224no—
113C22—816no—
114C22C25×C2311yescharacteristic
115C4—212no—
116C4—23no—
117C4C23×H(5,𝔽2)16yes—
118C2—412no—
119C2—41no—
120C2C2×C23×H(5,𝔽2)13yes—
subgroupG/H
center, socleC22×C23C25
commutator, Frattini, cosocleC22C25×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×H(3,𝔽4)C2semidirect
2. C23×#555C2semidirect
3. D42×C23C2semidirect
4. C23×#535C2semidirect
5. C23×#532C2semidirect
6. C2×#41×C23C22semidirect
7. C22×D4×C23C22semidirect
8. C23×C4⋊Q8C22semidirect
10. C23×#127C22semidirect
11. C23×#124C22semidirect
12. C23×#124C22semidirect
13. C23×#121C22semidirect
14. C23×#121C22semidirect
15. C23×#120C22semidirect
16. C23×#120C22semidirect
17. C4×D4×C23C22semidirect
18. C23×#117C22semidirect
19. C2×C42×C23C22semidirect
49. #3549C23direct (N × H)
58. H(3,𝔽4)C2×C23semidirect
59. #555C2×C23semidirect
60. D42C2×C23semidirect
61. #535C2×C23semidirect
62. #532C2×C23semidirect
66. C2×#41C22×C23semidirect
67. C22×D4C22×C23semidirect
68. C4⋊Q8C22×C23semidirect
70. #127C22×C23semidirect
71. #124C22×C23semidirect
72. #124C22×C23semidirect
73. #121C22×C23semidirect
74. #121C22×C23semidirect
75. #120C22×C23semidirect
76. #120C22×C23semidirect
77. C4×D4C22×C23semidirect
78. #117C22×C23semidirect
79. C2×C42C22×C23semidirect
80. C23#3549direct (N × H)

Maximal subgroups

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

subgroupindex
C23×H(3,𝔽4)2
C23×#5552
D42×C232
C23×#5352
C23×#5322
#354923