Finite Group Databasebeta

← C3×#3416 #85280

Subgroups — C3×#3416 #85280

Order 384.

360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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×#578C211yesmaximal, characteristic, complemented
2C3×#576C211yesmaximal, characteristic, complemented
3C3×#510C213yesmaximal, complemented
4C3×#509C213yesmaximal, complemented
5C3×#508C213yesmaximal, complemented
6C2×C3×SD32C213yesmaximal, complemented
7C3×#504C211yesmaximal, characteristic, complemented
8#3416C311yesmaximal, characteristic, direct product factor, Sylow
9C3×2−1+4—21no—
10C3×H(5,𝔽2)—21no—
11C3×#137—23no—
12C3×AGL(1,ℤ/8)—23no—
13C3×#135C2213yes—
14C3×#135C2213yescomplemented
15C2×C3×Q16C2213yescomplemented
16C2×C3×D8C2213yes—
17C3×#131C2211yescharacteristic
18C3×Q32C2213yescomplemented
19C3×SD32C2213yescomplemented
20C3×SD32C2216yescomplemented
21C3×SD32C2211yescharacteristic, complemented
22C3×D16C2213yescomplemented
23C3×M32C2213yescomplemented
24C2×C3×C16C2213yescomplemented
25#578C2×C311yescharacteristic, complemented
26#576C2×C311yescharacteristic, complemented
27#510C2×C313yescomplemented
28#509C2×C313yescomplemented
29#508C2×C313yescomplemented
30C2×SD32C2×C313yescomplemented
31#504C2×C311yescharacteristic, complemented
32C3×#41—23no—
33C3×#41—23no—
34C3×#41—41no—
35C3×#41—43no—
36C3×#41D411yescharacteristic
37C2×C3×Q8—23no—
38C2×C3×Q8—41no—
39C2×C3×D4—23no—
40C2×C3×D4—43no—
41C3×Q16—23no—
42C3×Q16C2313yes—
43C3×Q16C2311yescharacteristic
44C3×SD16—23no—
45C3×SD16—23no—
46C3×D8—23no—
47C3×D8C2311yescharacteristic
48C3×D8C2313yes—
49C3×M16D413yes—
50C2×C3×C8C2313yes—
51C3×C16C2311yescharacteristic
52C3×C16C2313yes—
532−1+4—21no—
54H(5,𝔽2)—21no—
55#137—23no—
56AGL(1,ℤ/8)—23no—
57#135C22×C313yes—
58#135C22×C313yescomplemented
59C2×Q16C22×C313yescomplemented
60C2×D8C22×C313yes—
61#131C22×C311yescharacteristic
62Q32C22×C313yescomplemented
63SD32C22×C313yescomplemented
64SD32C22×C316yescomplemented
65SD32C22×C311yescharacteristic, complemented
66D16C22×C313yescomplemented
67M32C22×C313yescomplemented
68C2×C16C22×C313yescomplemented
69C23×C3—43no—
70C3×Q8—21no—
71C3×Q8—23no—
72C3×Q8—43no—
73C3×Q8D811yescharacteristic
74C3×D4—23no—
75C3×D4—21no—
76C3×D4—43no—
77C3×D4—43no—
78C3×D4—43no—
79C3×D4D813yes—
80C2×C3×C4—43no—
81C2×C3×C4—43no—
82C2×C3×C4—43no—
83C2×C3×C4C2×D413yes—
84C3×C8C2×D413yes—
85C3×C8C2411yescharacteristic
86#41—23no—
87#41—23no—
88#41—41no—
89#41—43no—
90#41C3×D411yescharacteristic
91C2×Q8—23no—
92C2×Q8—41no—
93C2×D4—23no—
94C2×D4—43no—
95Q16—23no—
96Q16C23×C313yes—
97Q16C23×C311yescharacteristic
98SD16—23no—
99SD16—23no—
100D8—23no—
101D8C23×C311yescharacteristic
102D8C23×C313yes—
103M16C3×D413yes—
104C2×C8C23×C313yes—
105C16C23×C311yescharacteristic
106C16C23×C313yes—
107C22×C3—41no—
108C22×C3—43no—
109C22×C3—163no—
110C22×C3C2×D813yes—
111C3×C4—41no—
112C3×C4—43no—
113C3×C4C2×D813yes—
114C3×C4C22×D411yescharacteristic
115C23—43no—
116Q8—21no—
117Q8—23no—
118Q8—43no—
119Q8C3×D811yescharacteristic
120D4—23no—
121D4—21no—
122D4—43no—
123D4—43no—
124D4—43no—
125D4C3×D813yes—
126C2×C4—43no—
127C2×C4—43no—
128C2×C4—43no—
129C2×C4C2×C3×D413yes—
130C8C2×C3×D413yes—
131C8C24×C311yescharacteristic
132C2×C3—23no—
133C2×C3—81no—
134C2×C3—83no—
135C2×C3C22×D811yescharacteristic
136C22—41no—
137C22—43no—
138C22—163no—
139C22C2×C3×D813yes—
140C4—41no—
141C4—43no—
142C4C2×C3×D813yes—
143C4C22×C3×D411yescharacteristic
144C3#341611yescharacteristic, direct product factor, Sylow
145C2—23no—
146C2—83no—
147C2—81no—
148C2C22×C3×D811yescharacteristic
subgroupG/H
center, socleC2×C3C22×D8
commutator, Frattini, cosocleC8C24×C3
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#578C2semidirect
2. C3×#576C2semidirect
3. C3×#510C2semidirect
4. C3×#509C2semidirect
5. C3×#508C2semidirect
6. C2×C3×SD32C2semidirect
7. C3×#504C2semidirect
8. #3416C3direct (N × H)
14. C3×#135C22semidirect
15. C2×C3×Q16C22semidirect
18. C3×Q32C22semidirect
19. C3×SD32C22semidirect
20. C3×SD32C22semidirect
21. C3×SD32C22semidirect
22. C3×D16C22semidirect
23. C3×M32C22semidirect
24. C2×C3×C16C22semidirect
25. #578C2×C3semidirect
26. #576C2×C3semidirect
27. #510C2×C3semidirect
28. #509C2×C3semidirect
29. #508C2×C3semidirect
30. C2×SD32C2×C3semidirect
31. #504C2×C3semidirect
58. #135C22×C3semidirect
59. C2×Q16C22×C3semidirect
62. Q32C22×C3semidirect
63. SD32C22×C3semidirect
64. SD32C22×C3semidirect
65. SD32C22×C3semidirect
66. D16C22×C3semidirect
67. M32C22×C3semidirect
68. C2×C16C22×C3semidirect
144. C3#3416direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5782
C3×#5762
C3×#5102
C3×#5092
C3×#5082
C2×C3×SD322
C3×#5042
#34163