Finite Group Databasebeta

← C3×#577 #5907

Subgroups — C3×#577 #5907

Order 192.

258 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C3×2−1+4C211yesmaximal, characteristic, complemented
2C3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3C3×#137C213yesmaximal, complemented
4C3×AGL(1,ℤ/8)C213yesmaximal, complemented
5C3×#135C213yesmaximal, complemented
6C2×C3×SD16C213yesmaximal, complemented
7C3×#131C211yesmaximal, characteristic, complemented
8#577C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#41—21no—
10C3×#41—23no—
11C3×#41C2213yes—
12C3×#41C2211yescharacteristic
13C3×#41C2213yescomplemented
14C2×C3×Q8—21no—
15C2×C3×Q8C2213yescomplemented
16C2×C3×D4—23no—
17C2×C3×D4C2213yes—
18C3×Q16C2213yescomplemented
19C3×SD16C2213yescomplemented
20C3×SD16C2216yescomplemented
21C3×SD16C2211yescharacteristic, complemented
22C3×D8C2213yescomplemented
23C3×M16C2213yescomplemented
24C2×C3×C8C2213yescomplemented
252−1+4C2×C311yescharacteristic, complemented
26H(5,𝔽2)C2×C311yescharacteristic, complemented
27#137C2×C313yescomplemented
28AGL(1,ℤ/8)C2×C313yescomplemented
29#135C2×C313yescomplemented
30C2×SD16C2×C313yescomplemented
31#131C2×C311yescharacteristic, complemented
32C23×C3—23no—
33C3×Q8—23no—
34C3×Q8D411yescharacteristic
35C3×Q8C2313yes—
36C3×Q8C2311yescharacteristic
37C3×D4—23no—
38C3×D4—23no—
39C3×D4—23no—
40C3×D4D413yes—
41C3×D4C2311yescharacteristic
42C3×D4C2313yes—
43C2×C3×C4—23no—
44C2×C3×C4—23no—
45C2×C3×C4—23no—
46C2×C3×C4C2313yes—
47C3×C8C2311yescharacteristic
48C3×C8C2313yes—
49#41—21no—
50#41—23no—
51#41C22×C313yes—
52#41C22×C311yescharacteristic
53#41C22×C313yescomplemented
54C2×Q8—21no—
55C2×Q8C22×C313yescomplemented
56C2×D4—23no—
57C2×D4C22×C313yes—
58Q16C22×C313yescomplemented
59SD16C22×C313yescomplemented
60SD16C22×C316yescomplemented
61SD16C22×C311yescharacteristic, complemented
62D8C22×C313yescomplemented
63M16C22×C313yescomplemented
64C2×C8C22×C313yescomplemented
65C22×C3—21no—
66C22×C3—23no—
67C22×C3—83no—
68C22×C3C2×D413yes—
69C3×C4—21no—
70C3×C4—23no—
71C3×C4C2×D413yes—
72C3×C4C2411yescharacteristic
73C23—23no—
74Q8—23no—
75Q8C3×D411yescharacteristic
76Q8C23×C313yes—
77Q8C23×C311yescharacteristic
78D4—23no—
79D4—23no—
80D4—23no—
81D4C3×D413yes—
82D4C23×C311yescharacteristic
83D4C23×C313yes—
84C2×C4—23no—
85C2×C4—23no—
86C2×C4—23no—
87C2×C4C23×C313yes—
88C8C23×C311yescharacteristic
89C8C23×C313yes—
90C2×C3—23no—
91C2×C3—41no—
92C2×C3—43no—
93C2×C3C22×D411yescharacteristic
94C22—21no—
95C22—23no—
96C22—83no—
97C22C2×C3×D413yes—
98C4—21no—
99C4—23no—
100C4C2×C3×D413yes—
101C4C24×C311yescharacteristic
102C3#57711yescharacteristic, direct product factor, Sylow
103C2—23no—
104C2—43no—
105C2—41no—
106C2C22×C3×D411yescharacteristic
subgroupG/H
center, socleC2×C3C22×D4
commutator, Frattini, cosocleC4C24×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×2−1+4C2semidirect
2. C3×H(5,𝔽2)C2semidirect
3. C3×#137C2semidirect
4. C3×AGL(1,ℤ/8)C2semidirect
5. C3×#135C2semidirect
6. C2×C3×SD16C2semidirect
7. C3×#131C2semidirect
8. #577C3direct (N × H)
13. C3×#41C22semidirect
15. C2×C3×Q8C22semidirect
18. C3×Q16C22semidirect
19. C3×SD16C22semidirect
20. C3×SD16C22semidirect
21. C3×SD16C22semidirect
22. C3×D8C22semidirect
23. C3×M16C22semidirect
24. C2×C3×C8C22semidirect
25. 2−1+4C2×C3semidirect
26. H(5,𝔽2)C2×C3semidirect
27. #137C2×C3semidirect
28. AGL(1,ℤ/8)C2×C3semidirect
29. #135C2×C3semidirect
30. C2×SD16C2×C3semidirect
31. #131C2×C3semidirect
53. #41C22×C3semidirect
55. C2×Q8C22×C3semidirect
58. Q16C22×C3semidirect
59. SD16C22×C3semidirect
60. SD16C22×C3semidirect
61. SD16C22×C3semidirect
62. D8C22×C3semidirect
63. M16C22×C3semidirect
64. C2×C8C22×C3semidirect
102. C3#577direct (N × H)

Maximal subgroups

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

subgroupindex
C3×2−1+42
C3×H(5,𝔽2)2
C3×#1372
C3×AGL(1,ℤ/8)2
C3×#1352
C2×C3×SD162
C3×#1312
#5773