Finite Group Databasebeta

← #64092

Subgroups — #64092

Order 288.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C2×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×C22⋊C4C211yesmaximal, characteristic, complemented
4#3757C211yesmaximal, characteristic, complemented
5#3752C211yesmaximal, characteristic, complemented
6C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7#3750C211yesmaximal, characteristic, complemented
8#883—34nomaximal
9C23×C32C2211yescharacteristic
10C22×C3⋊D3C2211yescharacteristic
11C2×C32×C4C2211yescharacteristic
12C2×C32×C4C2211yescharacteristic
13#638—22no—
14C2×C3⋊Dic3C2211yescharacteristic
15C2×C3⋊Dic3C2211yescharacteristic
16C2×C3⋊Dic3C2211yescharacteristic
17C4×C3⋊D3—22no—
18C2×#67—34no—
19C2×C4×D3—34no—
20C3×C22⋊C4D314yescomplemented
21#217—34no—
22(C2×D3)⋊C4—34no—
23C4⋊Dic3—34no—
24Dic3⋊C4—34no—
25C22×C32—21no—
26C22×C32—21no—
27C22×C32—21no—
28C22×C32C2311yescharacteristic
29C2×C3⋊D3—21no—
30C2×C3⋊D3—21no—
31C2×C3⋊D3D412yes—
32C32×C4—21no—
33C32×C4—21no—
34C3⋊Dic3—21no—
35C3⋊Dic3—21no—
36C3⋊Dic3—21no—
37#123—91noSylow
38C23×C3C2×D314yes—
39C22×D3—34no—
40C2×C3×C4C2×D314yes—
41C2×C3×C4C2×D314yes—
42#67—68no—
43C2×Dic3—34no—
44C2×Dic3—34no—
45C2×Dic3—34no—
46C4×D3—68no—
47C2×C32—41no—
48C2×C32C2×D411yescharacteristic
49C2×C32#4111yescharacteristic
50C2×C32#4111yescharacteristic
51C3⋊D3—22no—
52C2×D4—91no—
53C22×C4—91no—
54C4⋊C4—91no—
55C4⋊C4—91no—
56C22⋊C4—91no—
57C22⋊C4—91no—
58C22⋊C4C3⋊D311yescharacteristic, complemented
59C22×C3—24no—
60C22×C3—24no—
61C22×C3—24no—
62C22×C3C22×D314yes—
63C2×D3—38no—
64C2×D3—64no—
65C2×D3—64no—
66C3×C4—24no—
67C3×C4—24no—
68Dic3—64no—
69Dic3—64no—
70Dic3—64no—
71C32#12311yescharacteristic, complemented, Sylow
72C23—91no—
73C23C2×C3⋊D311yescharacteristic
74D4—182no—
75C2×C4—91no—
76C2×C4—91no—
77C2×C4—91no—
78C2×C4—182no—
79C2×C4C2×C3⋊D311yescharacteristic
80C2×C4C2×C3⋊D311yescharacteristic
81C2×C3—44no—
82C2×C3#23514yes—
83C2×C3D3×D414yes—
84C2×C3#23714yes—
85D3—68no—
86C22—21no—
87C22—21no—
88C22—21no—
89C22—92no—
90C22—181no—
91C22—181no—
92C22C22×C3⋊D311yescharacteristic
93C4—21no—
94C4—21no—
95C4—181no—
96C4—181no—
97C4—181no—
98C3#88314yescomplemented
99C2—41no—
100C2—182no—
101C2#382811yescharacteristic
102C2D4×C3⋊D311yescharacteristic
103C2#383011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊D3
commutator, socle, cosocleC22×C32C23
FittingC32×C22⋊C4C2
prime radicalC2×#638C2
hypercenterC22⋊C4C3⋊D3
hypocenterC32#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊D3C2semidirect
3. C32×C22⋊C4C2semidirect
4. #3757C2semidirect
5. #3752C2semidirect
6. C4⋊(C3⋊Dic3)C2semidirect
7. #3750C2semidirect
20. C3×C22⋊C4D3semidirect
58. C22⋊C4C3⋊D3semidirect
71. C32#123semidirect
98. C3#883semidirect

Maximal subgroups

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

subgroupindex
C2×#6382
C2×C4×C3⋊D32
C32×C22⋊C42
#37572
#37522
C4⋊(C3⋊Dic3)2
#37502
#8833