Finite Group Databasebeta

← #10596240

Subgroups — #10596240

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#64366C212yesmaximal, complemented
2C32×#135C211yesmaximal, characteristic, complemented
3#64158C212yesmaximal, complemented
4#64128C212yesmaximal, complemented
5#64123C214yesmaximal, complemented
6D8×C3⋊D3C212yesmaximal, complemented
7C2×#3745C211yesmaximal, characteristic, complemented
8#64110C211yesmaximal, characteristic, complemented
9#5769—34nomaximal
10C32×#41C2212yescomplemented
11#3832C2214yescomplemented
12D4×C3⋊D3—24no—
13D4×C3⋊D3C2214yescomplemented
14#3828—22no—
15#3828C2211yescharacteristic
16C2×C4⋊(C3⋊D3)—22no—
17C2×C4⋊(C3⋊D3)C2212yescomplemented
18C32×Q16C2211yescharacteristic, complemented
19C32×SD16C2212yescomplemented
20C32×D8C2211yescharacteristic, complemented
21C2×C32×C8C2211yescharacteristic, complemented
22#3755C2214yescomplemented
23#3753C2214yescomplemented
24#3748C2211yescharacteristic, complemented
25#3745C2212yescomplemented
26#3745C2212yescomplemented
27#3743C2212yescomplemented
28C8×C3⋊D3C2212yescomplemented
29#1009—38no—
30C3×#135D314yescomplemented
31#949—38no—
32#919—38no—
33#914—316no—
34D3×D8—38no—
35C2×D24—34no—
36#901—34no—
37C22×C3⋊D3—22no—
38C22×C3⋊D3—24no—
39C32×Q8C2312yes—
40C32×D4—22no—
41C32×D4C2312yes—
42C2×C32×C4—22no—
43C2×C32×C4C2311yescharacteristic
44#638—24no—
45#638D412yes—
46C4⋊(C3⋊D3)—24no—
47C4⋊(C3⋊D3)—22no—
48C4⋊(C3⋊D3)D411yescharacteristic
49C4⋊(C3⋊D3)C2314yes—
50C4×C3⋊D3—24no—
51C4×C3⋊D3C2312yes—
52#634D411yescharacteristic
53C32×C8C2311yescharacteristic
54C32×C8C2311yescharacteristic
55C3⋊(C3⋊C8)C2312yes—
56#576—91noSylow
57C3×#41C2×D318yescomplemented
58#239—316no—
59D3×D4—316no—
60D3×D4—616no—
61#235—34no—
62#235—68no—
63C2×D12—38no—
64C2×D12—68no—
65C3×Q16C2×D314yescomplemented
66C3×SD16C2×D318yescomplemented
67C3×D8C2×D314yescomplemented
68C2×C3×C8C2×D314yescomplemented
69#215—316no—
70#213—316no—
71#208—34no—
72D24—38no—
73D24—38no—
74ΓL(1,𝔽25)—38no—
75D3×C8—38no—
76C22×C32—22no—
77C22×C32C2×D411yescharacteristic
78C2×C3⋊D3—24no—
79C2×C3⋊D3—84no—
80C2×C3⋊D3—82no—
81C2×C3⋊D3C2×D412yes—
82C32×C4—22no—
83C32×C4C2×D411yescharacteristic
84C32×C4C2411yescharacteristic
85C3⋊Dic3C2×D412yes—
86H(5,𝔽2)—92no—
87AGL(1,ℤ/8)—92no—
88AGL(1,ℤ/8)—94no—
89#135—92no—
90#135C3⋊D311yescharacteristic, complemented
91C2×D8—92no—
92C2×D8—91no—
93#131—91no—
94C22×D3—616no—
95C22×D3—68no—
96C3×Q8C22×D318yes—
97C3×D4—28no—
98C3×D4C22×D318yes—
99C2×C3×C4—28no—
100C2×C3×C4C22×D314yes—
101#67—38no—
102#67—616no—
103D12—34no—
104D12—316no—
105D12—616no—
106D12—68no—
107C4×D3—38no—
108C4×D3—616no—
109Dic6—34no—
110C3×C8C22×D314yes—
111C3×C8C22×D314yes—
112C3⋊C8—38no—
113C2×C32—21no—
114C2×C32—42no—
115C2×C32C22×D411yescharacteristic
116C3⋊D3—22no—
117C3⋊D3—44no—
118#41—91no—
119#41—94no—
120#41—182no—
121#41C2×C3⋊D312yescomplemented
122C2×D4—94no—
123C2×D4—92no—
124C2×D4—182no—
125C2×D4—184no—
126Q16C2×C3⋊D311yescharacteristic, complemented
127SD16—94no—
128SD16C2×C3⋊D312yescomplemented
129D8—94no—
130D8—92no—
131D8—92no—
132D8C2×C3⋊D311yescharacteristic, complemented
133M16—91no—
134M16—92no—
135C2×C8—92no—
136C2×C8C2×C3⋊D311yescharacteristic, complemented
137C22×C3—28no—
138C22×C3D3×D414yes—
139C2×D3—38no—
140C2×D3—616no—
141C2×D3—2416no—
142C2×D3—248no—
143C3×C4—28no—
144C3×C4D3×D414yes—
145C3×C4C23×D314yes—
146Dic3—38no—
147C32#57611yescharacteristic, complemented, Sylow
148C23—184no—
149C23—182no—
150Q8—91no—
151Q8C22×C3⋊D312yes—
152D4—22no—
153D4—92no—
154D4—94no—
155D4—91no—
156D4—184no—
157D4—184no—
158D4—182no—
159D4C22×C3⋊D312yes—
160C2×C4—22no—
161C2×C4—92no—
162C2×C4—184no—
163C2×C4C22×C3⋊D311yescharacteristic
164C8—92no—
165C8C22×C3⋊D311yescharacteristic
166C8C22×C3⋊D311yescharacteristic
167C2×C3—24no—
168C2×C3—48no—
169C2×C3C2×D3×D414yes—
170D3—68no—
171D3—1216no—
172C22—22no—
173C22—92no—
174C22—184no—
175C22—722no—
176C22—724no—
177C22D4×C3⋊D311yescharacteristic
178C4—22no—
179C4—92no—
180C4D4×C3⋊D311yescharacteristic
181C4C23×C3⋊D311yescharacteristic
182C3#576914yescomplemented
183C2—21no—
184C2—42no—
185C2—182no—
186C2—364no—
187C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC32×#135C2
socleC2×C32C22×D4
hypercenter#135C3⋊D3
hypocenterC32#576
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #64366C2semidirect
2. C32×#135C2semidirect
3. #64158C2semidirect
4. #64128C2semidirect
5. #64123C2semidirect
6. D8×C3⋊D3C2semidirect
7. C2×#3745C2semidirect
8. #64110C2semidirect
10. C32×#41C22semidirect
11. #3832C22semidirect
13. D4×C3⋊D3C22semidirect
17. C2×C4⋊(C3⋊D3)C22semidirect
18. C32×Q16C22semidirect
19. C32×SD16C22semidirect
20. C32×D8C22semidirect
21. C2×C32×C8C22semidirect
22. #3755C22semidirect
23. #3753C22semidirect
24. #3748C22semidirect
25. #3745C22semidirect
26. #3745C22semidirect
27. #3743C22semidirect
28. C8×C3⋊D3C22semidirect
30. C3×#135D3semidirect
57. C3×#41C2×D3semidirect
65. C3×Q16C2×D3semidirect
66. C3×SD16C2×D3semidirect
67. C3×D8C2×D3semidirect
68. C2×C3×C8C2×D3semidirect
90. #135C3⋊D3semidirect
121. #41C2×C3⋊D3semidirect
126. Q16C2×C3⋊D3semidirect
128. SD16C2×C3⋊D3semidirect
132. D8C2×C3⋊D3semidirect
136. C2×C8C2×C3⋊D3semidirect
147. C32#576semidirect
182. C3#5769semidirect

Maximal subgroups

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

subgroupindex
#643662
C32×#1352
#641582
#641282
#641232
D8×C3⋊D32
C2×#37452
#641102
#57693