Finite Group Databasebeta

← #10593522

Subgroups — #10593522

Order 576.

224 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 33 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×#3792C211yesmaximal, characteristic, complemented
2C2×#3774C211yesmaximal, characteristic, complemented
3#63980C212yesmaximal, complemented
4#63742C211yesmaximal, characteristic, complemented
5#63739C212yesmaximal, complemented
6C22×C3⋊Dic3C2211yescharacteristic
7C2×C3×#67—22no—
8C2×#625C2212yes—
9#3792—22no—
10C2×#622C2211yescharacteristic
11C2×#622C2211yescharacteristic
12#3774—22no—
13#3724C2212yes—
14(C2×D3)⋊Dic3—22no—
15#939—62no—
16C23×C32D411yescharacteristic
17C22×C3×D3—22no—
18C2×C3⋊Dic3—21no—
19C2×C3⋊Dic3—21no—
20C2×C3⋊Dic3D411yescharacteristic
21C2×C3⋊Dic3C2311yescharacteristic
22C3×#67—44no—
23C2×C3×Dic3—22no—
24#625—22no—
25#625—22no—
26#622—21no—
27#622D412yescomplemented
28#469—91nomaximal, Sylow
29C2×C3×D4—62no—
30C2×#67—22no—
31C2×#67—62no—
32C22×Dic3—62no—
33#217—62no—
34(C2×D3)⋊C4—62no—
35Dic3⋊C4—62no—
36C22×C32—22no—
37C22×C32—21no—
38C22×C32C2×D411yescharacteristic
39C2×C3×D3—42no—
40C2×C3×D3—42no—
41C2×C3×D3—42no—
42C3⋊Dic3—21no—
43C3⋊Dic3C2×D411yescharacteristic
44C3⋊Dic3#4111yescharacteristic
45C3×Dic3—42no—
46C2×D8—91no—
47C2×M16—91no—
48#121—92no—
49#106—91no—
50#102—92no—
51C23×C3—22no—
52C23×C3—62no—
53C22×D3—22no—
54C3×D4—124no—
55C2×C3×C4—62no—
56#67—44no—
57#67—122no—
58#67—122no—
59C2×Dic3—22no—
60C2×Dic3—62no—
61C2×Dic3—62no—
62C2×Dic3—122no—
63C2×Dic3—122no—
64C2×C32—41no—
65C2×C32#12111yescharacteristic
66C2×C32AGL(1,ℤ/8)12yes—
67C3×D3—82no—
68C2×D4—62no—
69C2×D4—92no—
70C22×C4—91no—
71D8—182no—
72M16—182no—
73C2×C8—91no—
74C2×C8—91no—
75C4⋊C4—92no—
76C22⋊C4—182no—
77C22×C3—24no—
78C22×C3—22no—
79C22×C3—42no—
80C22×C3—42no—
81C22×C3—122no—
82C22×C3—122no—
83C22×C3—122no—
84C2×D3—42no—
85C2×D3—42no—
86C2×D3—42no—
87C3×C4—122no—
88Dic3—42no—
89Dic3—62no—
90Dic3—62no—
91Dic3—122no—
92C32#46911yescharacteristic, complemented, Sylow
93C23—62no—
94C23GO+(4,𝔽2)11yescharacteristic
95D4—124no—
96D4—182no—
97D4—182no—
98C2×C4—62no—
99C2×C4—91no—
100C2×C4—91no—
101C2×C4—181no—
102C2×C4—181no—
103C8—92no—
104C8—181no—
105C2×C3—22no—
106C2×C3—22no—
107C2×C3—22no—
108C2×C3—44no—
109C2×C3—242no—
110D3—82no—
111C22—22no—
112C22—21no—
113C22—122no—
114C22—122no—
115C22—122no—
116C22C2×GO+(4,𝔽2)11yescharacteristic
117C4—91no—
118C4—91no—
119C4—122no—
120C4—181no—
121C3—22no—
122C2—41no—
123C2—242no—
124C2#6423612yes—
125C2#6424211yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
FittingC23×C32D4
socleC22×C32C2×D4
hypercenterC23GO+(4,𝔽2)
hypocenterC32#469
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#3792C2semidirect
2. C2×#3774C2semidirect
3. #63980C2semidirect
4. #63742C2semidirect
5. #63739C2semidirect
27. #622D4semidirect
92. C32#469semidirect

Maximal subgroups

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

subgroupindex
C2×#37922
C2×#37742
#639802
#637422
#637392
#4699