Finite Group Databasebeta

← #64313

Subgroups — #64313

Order 288.

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

normal subgroups only

This group has 352 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
1#3830C211yesmaximal, characteristic, complemented
2C3×D3×D4C212yesmaximal, complemented
3D3×#67C214yesmaximal, complemented
4C2×#625C212yesmaximal, complemented
5#3805C212yesmaximal, complemented
6#3802C211yesmaximal, characteristic, complemented
7#3798C211yesmaximal, characteristic, complemented
8#3795C212yesmaximal, complemented
9#1004—32nomaximal
10C22×C3×D3C2214yescomplemented
11C2×D32C2212yescomplemented
12C32×D4C2211yescharacteristic, complemented
13#638C2212yescomplemented
14C2×C3⋊Dic3C2212yescomplemented
15C4×C3⋊D3C2211yescharacteristic, complemented
16#634C2211yescharacteristic, complemented
17C3×#67C2214yescomplemented
18C3×D12C2212yescomplemented
19C3×C4×D3C2212yescomplemented
20#627C2211yescharacteristic, complemented
21#626C2212yescomplemented
22#625C2214yescomplemented
23#625C2211yescharacteristic, complemented
24#625C2212yescomplemented
25D3×Dic3C2214yescomplemented
26C2×C3×D4—32no—
27C2×#67—34no—
28C2×#67—34no—
29#237—32no—
30#237—34no—
31#237—32no—
32#237—61no—
33D3×D4—34no—
34D3×D4—32no—
35D3×D4D312yescomplemented
36#235—32no—
37#235—32no—
38C22×C32C2312yes—
39C2×C3⋊D3C2311yescharacteristic
40C2×C3×D3—44no—
41C2×C3×D3C2314yes—
42C2×C3×D3C2312yes—
43D32—42no—
44C32×C4C2311yescharacteristic
45C3⋊Dic3C2312yes—
46C3⋊Dic3C2311yescharacteristic
47C3×Dic3C2312yes—
48H(5,𝔽2)—91noSylow
49C23×C3—34no—
50C22×D3—34no—
51C22×D3C2×D314yescomplemented
52C3×D4—21no—
53C3×D4—34no—
54C3×D4—32no—
55C3×D4C2×D312yescomplemented
56C2×C3×C4—32no—
57#67—34no—
58#67—34no—
59#67—32no—
60#67—34no—
61#67—34no—
62#67—32no—
63#67—62no—
64#67C2×D314yescomplemented
65C2×Dic3—34no—
66C2×Dic3—34no—
67C2×Dic3—62no—
68D12—32no—
69D12C2×D312yescomplemented
70C4×D3—34no—
71C4×D3—32no—
72C4×D3—61no—
73C4×D3C2×D312yescomplemented
74Dic6—32no—
75Dic6—32no—
76Dic6—61no—
77C2×C32—22no—
78C2×C32C2411yescharacteristic
79C3⋊D3—21no—
80C3×D3—24no—
81C3×D3—22no—
82#41—91no—
83#41—92no—
84#41—92no—
85#41—91no—
86C2×D4—32no—
87C2×D4—91no—
88C2×D4—94no—
89C2×D4—92no—
90C22×C3—22no—
91C22×C3—34no—
92C22×C3—32no—
93C22×C3—124no—
94C22×C3C22×D314yes—
95C2×D3—32no—
96C2×D3—44no—
97C2×D3—61no—
98C2×D3—124no—
99C2×D3C22×D314yes—
100C2×D3C22×D312yes—
101C3×C4—21no—
102C3×C4—32no—
103C3×C4C22×D312yes—
104Dic3—34no—
105Dic3—32no—
106Dic3—62no—
107Dic3—61no—
108Dic3C22×D312yes—
109C32H(5,𝔽2)11yescharacteristic, complemented, Sylow
110C23—34no—
111C23—92no—
112Q8—91no—
113Q8—91no—
114D4—34no—
115D4—32no—
116D4—92no—
117D4—91no—
118D4—94no—
119D4—92no—
120D4—92no—
121D4D3211yescharacteristic, complemented
122C2×C4—32no—
123C2×C4—94no—
124C2×C4—92no—
125C2×C4—91no—
126C2×C3—24no—
127C2×C3—21no—
128C2×C3—42no—
129C2×C3—64no—
130C2×C3—62no—
131C2×C3C23×D312yes—
132D3—22no—
133D3—24no—
134D3—62no—
135D3—121no—
136C22—32no—
137C22—34no—
138C22—91no—
139C22—124no—
140C22—362no—
141C22C2×D3212yes—
142C4—32no—
143C4—92no—
144C4—91no—
145C4C2×D3211yescharacteristic
146C3—21no—
147C3#100412yescomplemented
148C2—22no—
149C2—62no—
150C2—64no—
151C2—181no—
152C2C22×D3211yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32
commutator, socle, cosocleC2×C32C24
FittingC32×D4C22
hypercenterD4D32
hypocenterC32H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #3830C2semidirect
2. C3×D3×D4C2semidirect
3. D3×#67C2semidirect
4. C2×#625C2semidirect
5. #3805C2semidirect
6. #3802C2semidirect
7. #3798C2semidirect
8. #3795C2semidirect
10. C22×C3×D3C22semidirect
11. C2×D32C22semidirect
12. C32×D4C22semidirect
13. #638C22semidirect
14. C2×C3⋊Dic3C22semidirect
15. C4×C3⋊D3C22semidirect
16. #634C22semidirect
17. C3×#67C22semidirect
18. C3×D12C22semidirect
19. C3×C4×D3C22semidirect
20. #627C22semidirect
21. #626C22semidirect
22. #625C22semidirect
23. #625C22semidirect
24. #625C22semidirect
25. D3×Dic3C22semidirect
35. D3×D4D3semidirect
51. C22×D3C2×D3semidirect
55. C3×D4C2×D3semidirect
64. #67C2×D3semidirect
69. D12C2×D3semidirect
73. C4×D3C2×D3semidirect
109. C32H(5,𝔽2)semidirect
121. D4D32semidirect
147. C3#1004semidirect

Maximal subgroups

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

subgroupindex
#38302
C3×D3×D42
D3×#672
C2×#6252
#38052
#38022
#37982
#37952
#10043