Finite Group Databasebeta

← C3×#3549 #87433

Subgroups — C3×#3549 #87433

Order 384.

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
1C3×H(3,𝔽4)C214yesmaximal, complemented
2C3×#555C216yesmaximal, complemented
3C3×D42C216yesmaximal, complemented
4C3×#535C2112yesmaximal, complemented
5C3×#532C213yesmaximal, complemented
6#3549C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C2216yescomplemented
8C22×C3×D4C22112yescomplemented
9C3×C4⋊Q8C2213yescomplemented
10C3×#127C2213yes—
11C3×#127C2218yescomplemented
12C3×#124C2216yescomplemented
13C3×#124C22124yescomplemented
14C3×#121C22112yescomplemented
15C3×#121C22124yescomplemented
16C3×#120C22124yescomplemented
17C3×#120C2218yescomplemented
18C3×C4×D4C22112yescomplemented
19C3×#117C22112yescomplemented
20C2×C3×C42C2211yescharacteristic, complemented
21H(3,𝔽4)C2×C314yescomplemented
22#555C2×C316yescomplemented
23D42C2×C316yescomplemented
24#535C2×C3112yescomplemented
25#532C2×C313yescomplemented
26C24×C3C2318yes—
27C3×#41—412no—
28C2×C3×Q8C2316yes—
29C2×C3×D4—424no—
30C2×C3×D4C23112yes—
31C2×C3×D4C2316yes—
32C2×C3×D4C23124yes—
33C2×C3×D4C23112yes—
34C22×C3×C4C2313yes—
35C22×C3×C4C23112yes—
36C3×C4⋊C4C23112yes—
37C3×C22⋊C4C23124yes—
38C3×C22⋊C4C23124yes—
39C3×C42C2313yes—
40C3×C42C2311yescharacteristic
41C3×C42C2318yes—
42C2×#41C22×C316yescomplemented
43C22×D4C22×C3112yescomplemented
44C4⋊Q8C22×C313yescomplemented
45#127C22×C313yes—
46#127C22×C318yescomplemented
47#124C22×C316yescomplemented
48#124C22×C3124yescomplemented
49#121C22×C3112yescomplemented
50#121C22×C3124yescomplemented
51#120C22×C318yescomplemented
52#120C22×C3124yescomplemented
53C4×D4C22×C3112yescomplemented
54#117C22×C3112yescomplemented
55C2×C42C22×C311yescharacteristic, complemented
56C23×C3—424no—
57C23×C3C24112yes—
58C23×C3C2411yescharacteristic
59C3×Q8—46no—
60C3×D4—224no—
61C3×D4—424no—
62C3×D4—412no—
63C3×D4—46no—
64C2×C3×C4—224no—
65C2×C3×C4—26no—
66C2×C3×C4C2413yes—
67C2×C3×C4C24112yes—
68C2×C3×C4C2413yes—
69C24C23×C318yes—
70#41—412no—
71C2×Q8C23×C316yes—
72C2×D4—424no—
73C2×D4C23×C3112yes—
74C2×D4C23×C316yes—
75C2×D4C23×C3124yes—
76C2×D4C23×C3112yes—
77C22×C4C23×C313yes—
78C22×C4C23×C3112yes—
79C4⋊C4C23×C3112yes—
80C22⋊C4C23×C3124yes—
81C22⋊C4C23×C3124yes—
82C42C23×C313yes—
83C42C23×C311yescharacteristic
84C42C23×C318yes—
85C22×C3—23no—
86C22×C3—212no—
87C22×C3—224no—
88C22×C3—816no—
89C22×C3C2511yescharacteristic
90C3×C4—212no—
91C3×C4—23no—
92C3×C4H(5,𝔽2)16yes—
93C23—424no—
94C23C24×C3112yes—
95C23C24×C311yescharacteristic
96Q8—46no—
97D4—224no—
98D4—424no—
99D4—412no—
100D4—46no—
101C2×C4—224no—
102C2×C4—26no—
103C2×C4C24×C313yes—
104C2×C4C24×C3112yes—
105C2×C4C24×C313yes—
106C2×C3—41no—
107C2×C3—412no—
108C2×C3C2×H(5,𝔽2)13yes—
109C22—23no—
110C22—212no—
111C22—224no—
112C22—816no—
113C22C25×C311yescharacteristic
114C4—212no—
115C4—23no—
116C4C3×H(5,𝔽2)16yes—
117C3#354911yescharacteristic, direct product factor, Sylow
118C2—412no—
119C2—41no—
120C2C2×C3×H(5,𝔽2)13yes—
subgroupG/H
center, socleC22×C3C25
commutator, Frattini, cosocleC22C25×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×H(3,𝔽4)C2semidirect
2. C3×#555C2semidirect
3. C3×D42C2semidirect
4. C3×#535C2semidirect
5. C3×#532C2semidirect
6. #3549C3direct (N × H)
7. C2×C3×#41C22semidirect
8. C22×C3×D4C22semidirect
9. C3×C4⋊Q8C22semidirect
11. C3×#127C22semidirect
12. C3×#124C22semidirect
13. C3×#124C22semidirect
14. C3×#121C22semidirect
15. C3×#121C22semidirect
16. C3×#120C22semidirect
17. C3×#120C22semidirect
18. C3×C4×D4C22semidirect
19. C3×#117C22semidirect
20. C2×C3×C42C22semidirect
21. H(3,𝔽4)C2×C3semidirect
22. #555C2×C3semidirect
23. D42C2×C3semidirect
24. #535C2×C3semidirect
25. #532C2×C3semidirect
42. C2×#41C22×C3semidirect
43. C22×D4C22×C3semidirect
44. C4⋊Q8C22×C3semidirect
46. #127C22×C3semidirect
47. #124C22×C3semidirect
48. #124C22×C3semidirect
49. #121C22×C3semidirect
50. #121C22×C3semidirect
51. #120C22×C3semidirect
52. #120C22×C3semidirect
53. C4×D4C22×C3semidirect
54. #117C22×C3semidirect
55. C2×C42C22×C3semidirect
117. C3#3549direct (N × H)

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽4)2
C3×#5552
C3×D422
C3×#5352
C3×#5322
#35493