Finite Group Databasebeta

← #10596154

Subgroups — #10596154

Order 576.

780 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C32×#124C211yesmaximal, characteristic, complemented
2#64156C212yesmaximal, complemented
3#64146C212yesmaximal, complemented
4#64094C214yesmaximal, complemented
5#64091C214yesmaximal, complemented
6#64084C212yesmaximal, complemented
7#5683—34nomaximal
8C23×C3⋊D3C2212yescomplemented
9C2×C32×Q8C2211yescharacteristic, complemented
10C2×C32×D4C2211yescharacteristic
11C2×#638C2214yes—
12C2×C4⋊(C3⋊D3)C2212yes—
13C2×C4⋊(C3⋊D3)C2212yes—
14C2×#634C2212yescomplemented
15C32×C22⋊C4C2214yescomplemented
16C32×C42C2211yescharacteristic, complemented
17#3757C2212yescomplemented
18#3752C2214yescomplemented
19#3752C2214yescomplemented
20#3752C2214yescomplemented
21C4×C3⋊Dic3C2212yescomplemented
22C3×#124D314yescomplemented
23#947—38no—
24#937—38no—
25#885—316no—
26#882—316no—
27#875—38no—
28C23×C32C2312yes—
29C22×C3⋊D3—42no—
30C22×C3⋊D3—44no—
31C22×C3⋊D3C2314yes—
32C32×Q8—41no—
33C32×D4—41no—
34C2×C32×C4C2312yes—
35C2×C32×C4C2311yescharacteristic
36C2×C32×C4C2312yes—
37#638—44no—
38C2×C3⋊Dic3C2314yes—
39C4⋊(C3⋊D3)—42no—
40C4⋊(C3⋊D3)—42no—
41#634—42no—
42H(3,𝔽4)—91noSylow
43C23×D3—38no—
44C2×C3×Q8C2×D314yescomplemented
45C2×C3×D4C2×D314yes—
46C2×#67—316no—
47C2×D12—38no—
48C2×D12—38no—
49C2×Dic6—38no—
50C3×C22⋊C4C2×D3116yescomplemented
51C3×C42C2×D314yescomplemented
52#217—38no—
53(C2×D3)⋊C4—316no—
54(C2×D3)⋊C4—316no—
55(C2×D3)⋊C4—316no—
56C4×Dic3—38no—
57C22×C32—24no—
58C22×C32—22no—
59C22×C32C2411yescharacteristic
60C2×C3⋊D3—24no—
61C2×C3⋊D3—24no—
62C2×C3⋊D3—24no—
63C2×C3⋊D3—48no—
64C32×C4—22no—
65C32×C4—22no—
66C32×C4—21no—
67C3⋊Dic3—24no—
68#124—92no—
69#124—94no—
70#124—92no—
71#124C3⋊D311yescharacteristic, complemented
72#120—92no—
73#120—94no—
74C23×C3C22×D318yes—
75C22×D3—316no—
76C22×D3—128no—
77C22×D3—1216no—
78C3×Q8—44no—
79C3×D4—44no—
80C2×C3×C4C22×D318yes—
81C2×C3×C4C22×D314yes—
82C2×C3×C4C22×D318yes—
83#67—1216no—
84C2×Dic3—316no—
85D12—128no—
86D12—128no—
87Dic6—128no—
88C2×C32—42no—
89C2×C32H(5,𝔽2)11yescharacteristic
90C2×C32H(5,𝔽2)12yes—
91C3⋊D3—44no—
92C24—92no—
93C2×Q8—92no—
94C2×Q8C2×C3⋊D311yescharacteristic, complemented
95C2×D4—94no—
96C2×D4—92no—
97C2×D4—92no—
98C2×D4C2×C3⋊D311yescharacteristic
99C22⋊C4—92no—
100C22⋊C4—94no—
101C22⋊C4—94no—
102C22⋊C4—94no—
103C22⋊C4C2×C3⋊D314yescomplemented
104C42—92no—
105C42C2×C3⋊D311yescharacteristic, complemented
106C22×C3—28no—
107C22×C3—216no—
108C22×C3C23×D314yes—
109C2×D3—616no—
110C2×D3—616no—
111C2×D3—616no—
112C2×D3—1232no—
113C3×C4—28no—
114C3×C4—24no—
115C3×C4—28no—
116Dic3—616no—
117C32H(3,𝔽4)11yescharacteristic, complemented, Sylow
118C23—94no—
119C23—362no—
120C23—364no—
121C23C22×C3⋊D312yes—
122Q8—41no—
123Q8—362no—
124D4—41no—
125D4—362no—
126D4—362no—
127D4—364no—
128C2×C4—94no—
129C2×C4C22×C3⋊D312yes—
130C2×C4C22×C3⋊D311yescharacteristic
131C2×C4C22×C3⋊D312yes—
132C2×C3—48no—
133C2×C3#100414yes—
134C2×C3#100918yes—
135D3—1216no—
136C22—22no—
137C22—24no—
138C22—184no—
139C22—184no—
140C22—184no—
141C22—368no—
142C22C23×C3⋊D311yescharacteristic
143C4—21no—
144C4—22no—
145C4—22no—
146C4—184no—
147C3#568314yescomplemented
148C2—42no—
149C2—364no—
150C2#6436111yescharacteristic
151C2#6436612yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
hypercenter#124C3⋊D3
hypocenterC32H(3,𝔽4)
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C32×#124C2semidirect
2. #64156C2semidirect
3. #64146C2semidirect
4. #64094C2semidirect
5. #64091C2semidirect
6. #64084C2semidirect
8. C23×C3⋊D3C22semidirect
9. C2×C32×Q8C22semidirect
14. C2×#634C22semidirect
15. C32×C22⋊C4C22semidirect
16. C32×C42C22semidirect
17. #3757C22semidirect
18. #3752C22semidirect
19. #3752C22semidirect
20. #3752C22semidirect
21. C4×C3⋊Dic3C22semidirect
22. C3×#124D3semidirect
44. C2×C3×Q8C2×D3semidirect
50. C3×C22⋊C4C2×D3semidirect
51. C3×C42C2×D3semidirect
71. #124C3⋊D3semidirect
94. C2×Q8C2×C3⋊D3semidirect
103. C22⋊C4C2×C3⋊D3semidirect
105. C42C2×C3⋊D3semidirect
117. C32H(3,𝔽4)semidirect
147. C3#5683semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
#641562
#641462
#640942
#640912
#640842
#56833