Finite Group Databasebeta

← #64330

Subgroups — #64330

Order 288.

359 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×#638C211yesmaximal, characteristic, complemented
2C2×C3×#67C212yesmaximal, complemented
3#3811C212yesmaximal, complemented
4D3×#67C214yesmaximal, complemented
5#3805C212yesmaximal, complemented
6#3804C214yesmaximal, complemented
7#1004—32nomaximal
8C23×C32C2211yescharacteristic, complemented
9C22×C3⋊D3C2211yescharacteristic, complemented
10C22×C3×D3C2212yescomplemented
11C2×D32C2212yescomplemented
12#638C2214yescomplemented
13C2×C3⋊Dic3C2211yescharacteristic, complemented
14C3×#67C2214yescomplemented
15C3×#67C2214yescomplemented
16C2×C3×Dic3C2212yescomplemented
17#627C2212yescomplemented
18#626C2214yescomplemented
19#625C2212yescomplemented
20#624C2212yescomplemented
21D3×Dic3C2214yescomplemented
22C2×C3×D4—32no—
23C2×#67—32no—
24C2×#67—34no—
25C2×#67—61no—
26C2×#67D312yescomplemented
27#237—34no—
28#237—34no—
29D3×D4—34no—
30D3×D4—34no—
31#235—34no—
32C22×C32—41no—
33C22×C32C2312yes—
34C22×C32C2311yescharacteristic
35C2×C3⋊D3—41no—
36C2×C3⋊D3C2312yes—
37C2×C3×D3—42no—
38C2×C3×D3C2314yes—
39D32—42no—
40C3⋊Dic3C2312yes—
41C3×Dic3C2314yes—
42H(5,𝔽2)—91noSylow
43C23×C3—21no—
44C23×C3—32no—
45C23×C3C2×D312yescomplemented
46C22×D3—32no—
47C22×D3—34no—
48C22×D3—61no—
49C22×D3C2×D312yescomplemented
50C3×D4—34no—
51C3×D4—34no—
52C2×C3×C4—32no—
53#67—34no—
54#67—34no—
55#67—34no—
56#67—34no—
57#67—64no—
58#67C2×D314yescomplemented
59#67C2×D314yescomplemented
60C2×Dic3—32no—
61C2×Dic3—34no—
62C2×Dic3—61no—
63C2×Dic3C2×D312yescomplemented
64D12—34no—
65C4×D3—34no—
66C4×D3—34no—
67Dic6—34no—
68C2×C32—21no—
69C2×C32—22no—
70C2×C32C2411yescharacteristic
71C3⋊D3—22no—
72C3×D3—24no—
73#41—94no—
74#41—92no—
75C2×D4—32no—
76C2×D4—92no—
77C2×D4—94no—
78C2×D4—91no—
79C22×C3—21no—
80C22×C3—22no—
81C22×C3—34no—
82C22×C3—42no—
83C22×C3—42no—
84C22×C3—122no—
85C22×C3C22×D312yes—
86C22×C3C22×D312yes—
87C22×C3C22×D312yes—
88C2×D3—34no—
89C2×D3—42no—
90C2×D3—62no—
91C2×D3—124no—
92C2×D3—122no—
93C2×D3—122no—
94C2×D3C22×D314yes—
95C3×C4—34no—
96Dic3—34no—
97Dic3—62no—
98Dic3C22×D314yes—
99C32H(5,𝔽2)11yescharacteristic, complemented, Sylow
100C23—32no—
101C23—92no—
102C23—91no—
103C23D3211yescharacteristic, complemented
104Q8—92no—
105D4—34no—
106D4—34no—
107D4—94no—
108D4—92no—
109D4—94no—
110C2×C4—32no—
111C2×C4—92no—
112C2×C4—94no—
113C2×C4—91no—
114C2×C3—22no—
115C2×C3—22no—
116C2×C3—22no—
117C2×C3—22no—
118C2×C3—21no—
119C2×C3—42no—
120C2×C3—64no—
121C2×C3C23×D312yes—
122D3—24no—
123D3—64no—
124D3—122no—
125C22—34no—
126C22—41no—
127C22—92no—
128C22—122no—
129C22—361no—
130C22—362no—
131C22C2×D3212yes—
132C22C2×D3211yescharacteristic
133C4—34no—
134C4—92no—
135C3—21no—
136C3#100412yescomplemented
137C2—22no—
138C2—21no—
139C2—64no—
140C2—182no—
141C2C22×D3211yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32
commutator, socle, cosocleC2×C32C24
FittingC23×C32C22
hypercenterC23D32
hypocenterC32H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#638C2semidirect
2. C2×C3×#67C2semidirect
3. #3811C2semidirect
4. D3×#67C2semidirect
5. #3805C2semidirect
6. #3804C2semidirect
8. C23×C32C22semidirect
9. C22×C3⋊D3C22semidirect
10. C22×C3×D3C22semidirect
11. C2×D32C22semidirect
12. #638C22semidirect
13. C2×C3⋊Dic3C22semidirect
14. C3×#67C22semidirect
15. C3×#67C22semidirect
16. C2×C3×Dic3C22semidirect
17. #627C22semidirect
18. #626C22semidirect
19. #625C22semidirect
20. #624C22semidirect
21. D3×Dic3C22semidirect
26. C2×#67D3semidirect
45. C23×C3C2×D3semidirect
49. C22×D3C2×D3semidirect
58. #67C2×D3semidirect
59. #67C2×D3semidirect
63. C2×Dic3C2×D3semidirect
99. C32H(5,𝔽2)semidirect
103. C23D32semidirect
136. C3#1004semidirect

Maximal subgroups

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

subgroupindex
C2×#6382
C2×C3×#672
#38112
D3×#672
#38052
#38042
#10043