Finite Group Databasebeta

← #64230

Subgroups — #64230

Order 288.

148 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 27 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×GO+(4,𝔽2)C212yesmaximal, complemented
2#3802C211yesmaximal, characteristic, complemented
3C4×D32C211yesmaximal, characteristic, complemented
4C4⋊#157C211yesmaximal, characteristic, complemented
5D32⋊C4C212yesmaximal, complemented
6C2×D32C2212yes—
7C2×D32C2211yescharacteristic
8C2×#157C2212yes—
9GO+(4,𝔽2)—24no—
10C4×C3⋊D3C2211yescharacteristic
11C3×D12—21no—
12C3×C4×D3—21no—
13#625—21no—
14#624C2211yescharacteristic
15D3×Dic3—21no—
16D3×D4—61no—
17C2×C4×D3—61no—
18C2×C3⋊D3C2311yescharacteristic
19C2×C3×D3—21no—
20C2×C3×D3—22no—
21D32—21no—
22D32—22no—
23D32—22no—
24D32D412yescomplemented
25#157—22no—
26C32×C4D411yescharacteristic, complemented
27C3⋊Dic3D411yescharacteristic, complemented
28C3×Dic3—21no—
29#121—91nomaximal, Sylow
30C22×D3—61no—
31C22×D3—62no—
32C3×D4—61no—
33C2×C3×C4—61no—
34#67—62no—
35C2×Dic3—61no—
36D12—21no—
37C4×D3—21no—
38C4×D3—61no—
39C4×D3—61no—
40C4×D3—61no—
41C4×D3—61no—
42C2×C32C2×D411yescharacteristic
43C3⋊D3C2×D411yescharacteristic
44C3⋊D3#4111yescharacteristic
45C3×D3—22no—
46C3×D3—42no—
47C2×D4—91no—
48C2×D4—92no—
49C22×C4—91no—
50C4⋊C4—91no—
51C22⋊C4—92no—
52C22×C3—61no—
53C22×C3—62no—
54C2×D3—21no—
55C2×D3—22no—
56C2×D3—62no—
57C2×D3—61no—
58C2×D3—61no—
59C2×D3—62no—
60C2×D3—122no—
61C2×D3—122no—
62C3×C4—21no—
63C3×C4—21no—
64C3×C4—61no—
65Dic3—21no—
66Dic3—61no—
67Dic3—61no—
68C32#12111yescharacteristic, complemented, Sylow
69C23—91no—
70C23—92no—
71D4—61no—
72D4—181no—
73D4—184no—
74C2×C4—61no—
75C2×C4—91no—
76C2×C4—91no—
77C2×C4—92no—
78C2×C4—181no—
79C2×C3—21no—
80C2×C3—21no—
81C2×C3—62no—
82C2×C3—122no—
83D3—22no—
84D3—42no—
85D3—61no—
86D3—61no—
87D3—61no—
88D3—61no—
89C22—61no—
90C22—62no—
91C22—92no—
92C22—91no—
93C22—181no—
94C22—182no—
95C22—182no—
96C4—61no—
97C4—91no—
98C4—182no—
99C4GO+(4,𝔽2)11yescharacteristic, complemented
100C3—21no—
101C3—21no—
102C2—62no—
103C2—91no—
104C2—91no—
105C2—122no—
106C2C2×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C23
FittingC32×C4D4
socleC2×C32C2×D4
hypercenterC4GO+(4,𝔽2)
hypocenterC32#121
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×GO+(4,𝔽2)C2semidirect
2. #3802C2semidirect
3. C4×D32C2semidirect
4. C4⋊#157C2semidirect
5. D32⋊C4C2semidirect
24. D32D4semidirect
26. C32×C4D4semidirect
27. C3⋊Dic3D4semidirect
68. C32#121semidirect
99. C4GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C2×GO+(4,𝔽2)2
#38022
C4×D322
C4⋊#1572
D32⋊C42
#1219