Finite Group Databasebeta

← #10596228

Subgroups — #10596228

Order 576.

804 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#64361C212yesmaximal, complemented
2C2×C32×D8C211yesmaximal, characteristic, complemented
3#64141C212yesmaximal, complemented
4#64121C212yesmaximal, complemented
5#64120C214yesmaximal, complemented
6D8×C3⋊D3C212yesmaximal, complemented
7#64113C211yesmaximal, characteristic, complemented
8#64110C211yesmaximal, characteristic, complemented
9#5757—34nomaximal
10C2×C32×D4C2212yescomplemented
11C2×#638—24no—
12#3830—22no—
13#3830C2214yescomplemented
14D4×C3⋊D3—22no—
15D4×C3⋊D3C2214yescomplemented
16#3828C2211yescharacteristic
17#3828C2212yescomplemented
18C32×D8C2212yescomplemented
19C32×D8C2212yescomplemented
20C2×C32×C8C2211yescharacteristic, complemented
21#3754C2214yescomplemented
22#3753C2214yescomplemented
23#3748C2211yescharacteristic, complemented
24#3746C2211yescharacteristic, complemented
25#3745C2211yescharacteristic, complemented
26#3744C2212yescomplemented
27#3743C2212yescomplemented
28C8×C3⋊D3C2212yescomplemented
29#1004—38no—
30C2×C3×D8D314yescomplemented
31#932—38no—
32#912—38no—
33#911—316no—
34D3×D8—38no—
35#904—34no—
36#901—34no—
37C23×C32—22no—
38C22×C3⋊D3—24no—
39C32×D4—22no—
40C32×D4C2314yes—
41C2×C32×C4C2311yescharacteristic
42#638—22no—
43#638—24no—
44#638—24no—
45#638D412yes—
46C2×C3⋊Dic3—24no—
47C4⋊(C3⋊D3)D411yescharacteristic
48C4⋊(C3⋊D3)C2312yes—
49C4×C3⋊D3—22no—
50C4×C3⋊D3C2312yes—
51#634D411yescharacteristic
52#634C2312yes—
53C32×C8C2311yescharacteristic
54C32×C8C2311yescharacteristic
55C3⋊(C3⋊C8)C2312yes—
56#576—91noSylow
57C2×C3×D4C2×D318yescomplemented
58C2×#67—616no—
59#237—316no—
60#237—68no—
61D3×D4—316no—
62D3×D4—68no—
63#235—34no—
64#235—38no—
65C3×D8C2×D318yescomplemented
66C3×D8C2×D318yescomplemented
67C2×C3×C8C2×D314yescomplemented
68#214—316no—
69#213—316no—
70#208—34no—
71Dic12—34no—
72D24—34no—
73#204—38no—
74ΓL(1,𝔽25)—38no—
75D3×C8—38no—
76C22×C32—24no—
77C22×C32—82no—
78C22×C32C2×D411yescharacteristic
79C2×C3⋊D3—22no—
80C2×C3⋊D3—84no—
81C2×C3⋊D3C2×D412yes—
82C32×C4C2×D411yescharacteristic
83C32×C4C2411yescharacteristic
84C3⋊Dic3—22no—
85C3⋊Dic3C2×D412yes—
86H(5,𝔽2)—92no—
87AGL(1,ℤ/8)—92no—
88AGL(1,ℤ/8)—94no—
89#135—92no—
90#135—91no—
91C2×D8—92no—
92C2×D8C3⋊D311yescharacteristic, complemented
93#131—91no—
94C23×C3—28no—
95C22×D3—616no—
96C3×D4—28no—
97C3×D4C22×D3116yes—
98C2×C3×C4C22×D314yes—
99#67—38no—
100#67—616no—
101#67—616no—
102#67—68no—
103C2×Dic3—616no—
104D12—34no—
105D12—38no—
106C4×D3—38no—
107C4×D3—68no—
108Dic6—34no—
109Dic6—38no—
110C3×C8C22×D314yes—
111C3×C8C22×D314yes—
112C3⋊C8—38no—
113C2×C32—21no—
114C2×C32—44no—
115C2×C32C22×D411yescharacteristic
116C3⋊D3—22no—
117C3⋊D3—42no—
118#41—91no—
119#41—94no—
120#41—92no—
121#41—182no—
122C2×D4—94no—
123C2×D4—182no—
124C2×D4—184no—
125C2×D4C2×C3⋊D312yescomplemented
126Q16—91no—
127SD16—94no—
128SD16—92no—
129D8—94no—
130D8—91no—
131D8C2×C3⋊D312yescomplemented
132D8C2×C3⋊D312yescomplemented
133M16—91no—
134M16—92no—
135C2×C8—92no—
136C2×C8C2×C3⋊D311yescharacteristic, complemented
137C22×C3—216no—
138C22×C3—88no—
139C22×C3D3×D414yes—
140C2×D3—38no—
141C2×D3—68no—
142C2×D3—2416no—
143C3×C4D3×D414yes—
144C3×C4C23×D314yes—
145Dic3—38no—
146Dic3—68no—
147C32#57611yescharacteristic, complemented, Sylow
148C23—22no—
149C23—184no—
150Q8—91no—
151Q8—92no—
152D4—22no—
153D4—92no—
154D4—92no—
155D4—91no—
156D4—184no—
157D4—184no—
158D4—182no—
159D4C22×C3⋊D314yes—
160C2×C4—92no—
161C2×C4—184no—
162C2×C4—182no—
163C2×C4C22×C3⋊D311yescharacteristic
164C8—92no—
165C8C22×C3⋊D311yescharacteristic
166C8C22×C3⋊D311yescharacteristic
167C2×C3—24no—
168C2×C3—416no—
169C2×C3C2×D3×D414yes—
170D3—68no—
171D3—128no—
172C22—24no—
173C22—82no—
174C22—92no—
175C22—182no—
176C22—724no—
177C22D4×C3⋊D311yescharacteristic
178C4—92no—
179C4—182no—
180C4D4×C3⋊D311yescharacteristic
181C4C23×C3⋊D311yescharacteristic
182C3#575714yescomplemented
183C2—21no—
184C2—44no—
185C2—182no—
186C2—362no—
187C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC2×C32×D8C2
socleC2×C32C22×D4
hypercenterC2×D8C3⋊D3
hypocenterC32#576
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. #64361C2semidirect
2. C2×C32×D8C2semidirect
3. #64141C2semidirect
4. #64121C2semidirect
5. #64120C2semidirect
6. D8×C3⋊D3C2semidirect
7. #64113C2semidirect
8. #64110C2semidirect
10. C2×C32×D4C22semidirect
13. #3830C22semidirect
15. D4×C3⋊D3C22semidirect
17. #3828C22semidirect
18. C32×D8C22semidirect
19. C32×D8C22semidirect
20. C2×C32×C8C22semidirect
21. #3754C22semidirect
22. #3753C22semidirect
23. #3748C22semidirect
24. #3746C22semidirect
25. #3745C22semidirect
26. #3744C22semidirect
27. #3743C22semidirect
28. C8×C3⋊D3C22semidirect
30. C2×C3×D8D3semidirect
57. C2×C3×D4C2×D3semidirect
65. C3×D8C2×D3semidirect
66. C3×D8C2×D3semidirect
67. C2×C3×C8C2×D3semidirect
92. C2×D8C3⋊D3semidirect
125. C2×D4C2×C3⋊D3semidirect
131. D8C2×C3⋊D3semidirect
132. D8C2×C3⋊D3semidirect
136. C2×C8C2×C3⋊D3semidirect
147. C32#576semidirect
182. C3#5757semidirect

Maximal subgroups

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

subgroupindex
#643612
C2×C32×D82
#641412
#641212
#641202
D8×C3⋊D32
#641132
#641102
#57573