Finite Group Databasebeta

← #10596514

Subgroups — #10596514

Order 576.

434 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 85 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#64313C212yesmaximal, complemented
2#64287C211yesmaximal, characteristic, complemented
3#64237C212yesmaximal, complemented
4#64236C214yesmaximal, complemented
5C2×#3774C212yesmaximal, complemented
6#64225C211yesmaximal, characteristic, complemented
7#64224C212yesmaximal, complemented
8#64223C211yesmaximal, characteristic, complemented
9#3830C2211yescharacteristic
10C3×D3×D4—22no—
11D3×#67—24no—
12C2×#625C2214yescomplemented
13#3805C2214yescomplemented
14#3802C2212yescomplemented
15#3798C2212yescomplemented
16#3795—22no—
17#3792C2212yescomplemented
18C2×#622C2212yescomplemented
19#3788C2211yescharacteristic, complemented
20#3787C2211yescharacteristic, complemented
21#3776C2211yescharacteristic, complemented
22#3775C2212yescomplemented
23#3775C2214yescomplemented
24#3774C2211yescharacteristic, complemented
25#3774C2214yescomplemented
26#3774C2212yescomplemented
27#3774C2212yescomplemented
28#1004—62no—
29C22×C3×D3—24no—
30C2×D32—22no—
31C32×D4D411yescharacteristic
32#638D412yes—
33C2×C3⋊Dic3C2312yes—
34C4×C3⋊D3C2311yescharacteristic
35#634D411yescharacteristic
36C3×#67—24no—
37C3×D12—22no—
38C3×C4×D3—22no—
39#627C2312yes—
40#626—22no—
41#625—24no—
42#625C2312yes—
43#625C2314yes—
44D3×Dic3—24no—
45#622C2311yescharacteristic
46#622C2311yescharacteristic
47#622C2312yes—
48#576—91nomaximal, Sylow
49C2×C3×D4—62no—
50C2×#67—64no—
51C2×#67—64no—
52#237—64no—
53#237—62no—
54#237—62no—
55D3×D4—22no—
56D3×D4—62no—
57D3×D4—64no—
58#235—62no—
59#235—62no—
60C22×C32C2×D412yes—
61C2×C3⋊D3C2×D411yescharacteristic
62C2×C3×D3—24no—
63C2×C3×D3—22no—
64C2×C3×D3—84no—
65D32—82no—
66C32×C4C2×D411yescharacteristic
67C3⋊Dic3C2×D412yes—
68C3⋊Dic3C2411yescharacteristic
69C3×Dic3—22no—
70H(5,𝔽2)—92no—
71AGL(1,ℤ/8)—92no—
72AGL(1,ℤ/8)—94no—
73#135—91no—
74#135—92no—
75C2×D8—91no—
76C2×D8—92no—
77#131—91no—
78C23×C3—64no—
79C22×D3—24no—
80C22×D3—64no—
81C3×D4—22no—
82C3×D4—62no—
83C3×D4—64no—
84C2×C3×C4—62no—
85#67—24no—
86#67—62no—
87#67—64no—
88#67—64no—
89#67—62no—
90#67—64no—
91#67—64no—
92C2×Dic3—64no—
93C2×Dic3—64no—
94D12—22no—
95D12—62no—
96C4×D3—22no—
97C4×D3—62no—
98C4×D3—64no—
99Dic6—62no—
100Dic6—62no—
101C2×C32—22no—
102C2×C32C22×D411yescharacteristic
103C3⋊D3—21no—
104C3×D3—44no—
105C3×D3—42no—
106#41—92no—
107#41—91no—
108#41—94no—
109#41—182no—
110C2×D4—62no—
111C2×D4—92no—
112C2×D4—94no—
113C2×D4—184no—
114Q16—91no—
115SD16—92no—
116SD16—94no—
117D8—92no—
118D8—91no—
119D8—92no—
120D8—94no—
121M16—91no—
122M16—92no—
123C2×C8—91no—
124C2×C8—92no—
125C22×C3—24no—
126C22×C3—64no—
127C22×C3—62no—
128C22×C3—244no—
129C2×D3—22no—
130C2×D3—24no—
131C2×D3—62no—
132C2×D3—84no—
133C2×D3—244no—
134C3×C4—22no—
135C3×C4—62no—
136Dic3—22no—
137Dic3—64no—
138Dic3—62no—
139C32#57611yescharacteristic, complemented, Sylow
140C23—64no—
141C23—182no—
142Q8—92no—
143Q8—91no—
144D4—64no—
145D4—62no—
146D4—92no—
147D4—94no—
148D4—92no—
149D4—182no—
150D4—184no—
151D4GO+(4,𝔽2)11yescharacteristic
152C2×C4—62no—
153C2×C4—92no—
154C2×C4—91no—
155C2×C4—184no—
156C8—91no—
157C8—91no—
158C8—92no—
159C2×C3—22no—
160C2×C3—44no—
161C2×C3—124no—
162C2×C3—122no—
163D3—42no—
164D3—44no—
165D3—122no—
166C22—62no—
167C22—64no—
168C22—91no—
169C22—244no—
170C22—722no—
171C22C2×GO+(4,𝔽2)12yes—
172C4—62no—
173C4—91no—
174C4—92no—
175C4C2×GO+(4,𝔽2)11yescharacteristic
176C3—22no—
177C2—22no—
178C2—122no—
179C2—124no—
180C2—181no—
181C2C22×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24
FittingC32×D4D4
socleC2×C32C22×D4
hypercenterD4GO+(4,𝔽2)
hypocenterC32#576
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #64313C2semidirect
2. #64287C2semidirect
3. #64237C2semidirect
4. #64236C2semidirect
5. C2×#3774C2semidirect
6. #64225C2semidirect
7. #64224C2semidirect
8. #64223C2semidirect
12. C2×#625C22semidirect
13. #3805C22semidirect
14. #3802C22semidirect
15. #3798C22semidirect
17. #3792C22semidirect
18. C2×#622C22semidirect
19. #3788C22semidirect
20. #3787C22semidirect
21. #3776C22semidirect
22. #3775C22semidirect
23. #3775C22semidirect
24. #3774C22semidirect
25. #3774C22semidirect
26. #3774C22semidirect
27. #3774C22semidirect
139. C32#576semidirect

Maximal subgroups

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

subgroupindex
#643132
#642872
#642372
#642362
C2×#37742
#642252
#642242
#642232
#5769