Finite Group Databasebeta

← #64241

Subgroups — #64241

Order 288.

224 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 29 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
1C22×D32C211yesmaximal, characteristic, complemented
2C2×GO+(4,𝔽2)C212yesmaximal, complemented
3#3811C211yesmaximal, characteristic, complemented
4C22⋊#157C211yesmaximal, characteristic, complemented
5D32⋊C4C212yesmaximal, complemented
6C22×C3⋊D3C2211yescharacteristic
7C22×C3×D3—21no—
8C2×D32—21no—
9C2×D32—24no—
10C2×D32C2212yes—
11C2×D32C2211yescharacteristic, complemented
12C2×#157C2212yescomplemented
13GO+(4,𝔽2)—24no—
14C3×#67—21no—
15#626—21no—
16#624C2211yescharacteristic, complemented
17C23×D3—61no—
18D3×D4—61no—
19C22×C32D411yescharacteristic, complemented
20C2×C3⋊D3—22no—
21C2×C3⋊D3D411yescharacteristic, complemented
22C2×C3⋊D3C2311yescharacteristic
23C2×C3×D3—24no—
24C2×C3×D3—22no—
25C2×C3×D3—21no—
26D32—24no—
27D32—22no—
28D32—22no—
29D32D414yescomplemented
30#157—22no—
31C3×Dic3—21no—
32#120—91nomaximal, Sylow
33C23×C3—61no—
34C22×D3—21no—
35C22×D3—62no—
36C22×D3—61no—
37C22×D3—61no—
38C22×D3—61no—
39C22×D3—68no—
40C22×D3—62no—
41C3×D4—61no—
42#67—21no—
43#67—61no—
44D12—61no—
45C4×D3—61no—
46C2×C32—21no—
47C2×C32C2×D411yescharacteristic
48C3⋊D3—21no—
49C3⋊D3C2×D412yes—
50C3×D3—24no—
51C3×D3—41no—
52C24—91no—
53C2×D4—91no—
54C2×D4—92no—
55C22⋊C4—92no—
56C22⋊C4—91no—
57C22×C3—21no—
58C22×C3—21no—
59C22×C3—64no—
60C22×C3—62no—
61C22×C3—61no—
62C2×D3—24no—
63C2×D3—22no—
64C2×D3—21no—
65C2×D3—64no—
66C2×D3—61no—
67C2×D3—61no—
68C2×D3—61no—
69C2×D3—61no—
70C2×D3—64no—
71C2×D3—68no—
72C2×D3—64no—
73C2×D3—122no—
74C2×D3—122no—
75C3×C4—61no—
76Dic3—21no—
77C32#12011yescharacteristic, complemented, Sylow
78C23—61no—
79C23—92no—
80C23—91no—
81C23—91no—
82C23—184no—
83C23—181no—
84D4—61no—
85D4—181no—
86D4—184no—
87C2×C4—91no—
88C2×C4—92no—
89C2×C3—22no—
90C2×C3—21no—
91C2×C3—21no—
92C2×C3—41no—
93C2×C3—64no—
94C2×C3—121no—
95D3—24no—
96D3—41no—
97D3—62no—
98D3—62no—
99D3—62no—
100D3—121no—
101C22—64no—
102C22—62no—
103C22—61no—
104C22—94no—
105C22—91no—
106C22—91no—
107C22—182no—
108C22—182no—
109C22—184no—
110C22—182no—
111C22GO+(4,𝔽2)11yescharacteristic, complemented
112C4—61no—
113C4—182no—
114C3—21no—
115C3—21no—
116C2—21no—
117C2—64no—
118C2—92no—
119C2—121no—
120C2—181no—
121C2C2×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C23
FittingC22×C32D4
socleC2×C32C2×D4
hypercenterC22GO+(4,𝔽2)
hypocenterC32#120
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×D32C2semidirect
2. C2×GO+(4,𝔽2)C2semidirect
3. #3811C2semidirect
4. C22⋊#157C2semidirect
5. D32⋊C4C2semidirect
11. C2×D32C22semidirect
12. C2×#157C22semidirect
16. #624C22semidirect
19. C22×C32D4semidirect
21. C2×C3⋊D3D4semidirect
29. D32D4semidirect
77. C32#120semidirect
111. C22GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C22×D322
C2×GO+(4,𝔽2)2
#38112
C22⋊#1572
D32⋊C42
#1209