Finite Group Databasebeta

← #10593385

Subgroups — #10593385

Order 576.

165 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 35 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#63919C211yesmaximal, characteristic
2#63838C211yesmaximal, characteristic
3#63779C211yesmaximal, characteristic
4#63742C212yesmaximal
5#63740C212yesmaximal
6C2×#627—21no—
7C2×#627C2211yescharacteristic
8C2×#622C2212yes—
9C4×C3⋊Dic3C2211yescharacteristic
10C3×C4⋊Dic3—21no—
11C3×Dic3⋊C4—21no—
12#3724C2211yescharacteristic
13#3724C2212yes—
14Dic32—21no—
15#888—61no—
16#887—61no—
17C2×C32×C4D411yescharacteristic
18C2×C3⋊Dic3D411yescharacteristic
19C2×C3⋊Dic3C2311yescharacteristic
20C2×C3×Dic3—21no—
21C2×C3×Dic3—21no—
22C2×C3×Dic3—22no—
23#627—21no—
24#627—42no—
25#627Q812yescomplemented
26#622—22no—
27#475—91nomaximal, Sylow
28C2×Dic6—61no—
29C2×Dic6—62no—
30C3×C4⋊C4—61no—
31C3×C4⋊C4—61no—
32C4⋊Dic3—21no—
33Dic3⋊C4—21no—
34Dic3⋊C4—61no—
35Dic3⋊C4—62no—
36C4×Dic3—61no—
37C4×Dic3—61no—
38C4×Dic3—61no—
39C4×Dic3—61no—
40C22×C32C2×D411yescharacteristic
41C32×C4—21no—
42C3⋊Dic3SD1612yes—
43C3⋊Dic3C2×Q811yescharacteristic
44C3⋊Dic3#4111yescharacteristic
45C3×Dic3—22no—
46C3×Dic3—42no—
47C3×Dic3—41no—
48C4⋊Q8—91no—
49C4×Q8—91no—
50#106—92no—
51C4⋊C8—91no—
52#103—92no—
53C2×C3×C4—21no—
54C2×C3×C4—21no—
55C2×C3×C4—61no—
56C2×C3×C4—61no—
57C2×C3×C4—62no—
58C2×Dic3—21no—
59C2×Dic3—21no—
60C2×Dic3—22no—
61C2×Dic3—61no—
62C2×Dic3—61no—
63C2×Dic3—61no—
64C2×Dic3—61no—
65Dic6—62no—
66Dic6—62no—
67Dic6—124no—
68C2×C32C22⋊Q811yescharacteristic
69C2×C32C2×SD1611yescharacteristic
70C2×C32#13711yescharacteristic
71C2×Q8—91no—
72C2×Q8—181no—
73C2×C8—92no—
74C4⋊C4—61no—
75C4⋊C4—61no—
76C4⋊C4—92no—
77C4⋊C4—91no—
78C42—91no—
79C42—181no—
80C22×C3—21no—
81C22×C3—21no—
82C3×C4—22no—
83C3×C4—41no—
84C3×C4—62no—
85C3×C4—122no—
86C3×C4—121no—
87Dic3—22no—
88Dic3—42no—
89Dic3—41no—
90Dic3—61no—
91Dic3—61no—
92Dic3—62no—
93Dic3—62no—
94Dic3—61no—
95Dic3—61no—
96C32#47511yescharacteristic, complemented, Sylow
97Q8—92no—
98Q8—181no—
99Q8—362no—
100C2×C4—61no—
101C2×C4—61no—
102C2×C4—62no—
103C2×C4—91no—
104C2×C4—91no—
105C2×C4GO+(4,𝔽2)11yescharacteristic
106C8—182no—
107C2×C3—21no—
108C2×C3—21no—
109C2×C3—21no—
110C2×C3—21no—
111C2×C3—21no—
112C2×C3—21no—
113C22C2×GO+(4,𝔽2)11yescharacteristic
114C4—21no—
115C4—62no—
116C4—91no—
117C4—91no—
118C4—92no—
119C4—122no—
120C4—121no—
121C3—21no—
122C3—21no—
123C2D32⋊Q811yescharacteristic
124C2#6422711yescharacteristic
125C2#6423911yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
FittingC2×C32×C4D4
socle, prime radicalC22×C32C2×D4
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#475
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
25. #627Q8semidirect
96. C32#475semidirect

Maximal subgroups

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

subgroupindex
#639192
#638382
#637792
#637422
#637402
#4759