Finite Group Databasebeta

← #10593448

Subgroups — #10593448

Order 576.

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

normal subgroups only

This group has 177 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
1C2×#3776C212yesmaximal
2#63919C211yesmaximal, characteristic
3#63838C211yesmaximal, characteristic
4#63779C211yesmaximal, characteristic
5#63740C212yesmaximal
6C2×#627C2212yes—
7C2×#627C2211yescharacteristic
8C2×#622C2212yes—
9#3776—24no—
10C4×C3⋊Dic3C2211yescharacteristic
11C3×C4⋊Dic3—21no—
12C3×Dic3⋊C4—21no—
13#3724—21no—
14#3724C2211yescharacteristic
15Dic32—21no—
16#888—61no—
17#887—61no—
18C2×C32×C4D411yescharacteristic
19C2×C3⋊Dic3D411yescharacteristic
20C2×C3⋊Dic3C2311yescharacteristic
21C2×C3×Dic3—21no—
22C2×C3×Dic3—21no—
23C2×C3×Dic3—22no—
24#627—21no—
25#627—22no—
26#627—22no—
27#627D412yes—
28#622—22no—
29#462—91nomaximal, Sylow
30C2×Dic6—61no—
31C2×Dic6—62no—
32C3×C4⋊C4—61no—
33C3×C4⋊C4—61no—
34C4⋊Dic3—21no—
35Dic3⋊C4—21no—
36Dic3⋊C4—61no—
37Dic3⋊C4—62no—
38C4×Dic3—61no—
39C4×Dic3—61no—
40C4×Dic3—61no—
41C4×Dic3—61no—
42C22×C32C2×D411yescharacteristic
43C32×C4—21no—
44C3⋊Dic3Q1612yescomplemented
45C3⋊Dic3C2×D411yescharacteristic
46C3⋊Dic3#4111yescharacteristic
47C3×Dic3—22no—
48C3×Dic3—42no—
49C3×Dic3—41no—
50C2×Q16—92no—
51C4⋊Q8—91no—
52C4×Q8—91no—
53C4⋊C8—91no—
54#103—92no—
55C2×C3×C4—21no—
56C2×C3×C4—21no—
57C2×C3×C4—61no—
58C2×C3×C4—61no—
59C2×C3×C4—62no—
60C2×Dic3—21no—
61C2×Dic3—21no—
62C2×Dic3—22no—
63C2×Dic3—61no—
64C2×Dic3—61no—
65C2×Dic3—61no—
66C2×Dic3—61no—
67Dic6—62no—
68Dic6—62no—
69Dic6—122no—
70Dic6—122no—
71C2×C32#12111yescharacteristic
72C2×C32C2×Q1611yescharacteristic
73C2×C32#13711yescharacteristic
74C2×Q8—91no—
75C2×Q8—92no—
76Q16—184no—
77C2×C8—92no—
78C4⋊C4—61no—
79C4⋊C4—61no—
80C4⋊C4—91no—
81C4⋊C4—181no—
82C42—91no—
83C42—181no—
84C22×C3—21no—
85C22×C3—21no—
86C3×C4—22no—
87C3×C4—41no—
88C3×C4—62no—
89C3×C4—122no—
90C3×C4—121no—
91Dic3—22no—
92Dic3—42no—
93Dic3—41no—
94Dic3—61no—
95Dic3—61no—
96Dic3—61no—
97Dic3—61no—
98Dic3—62no—
99Dic3—62no—
100C32#46211yescharacteristic, complemented, Sylow
101Q8—92no—
102Q8—181no—
103Q8—182no—
104Q8—182no—
105C2×C4—61no—
106C2×C4—61no—
107C2×C4—62no—
108C2×C4—91no—
109C2×C4—91no—
110C2×C4GO+(4,𝔽2)11yescharacteristic
111C8—182no—
112C2×C3—21no—
113C2×C3—21no—
114C2×C3—21no—
115C2×C3—21no—
116C2×C3—21no—
117C2×C3—21no—
118C22C2×GO+(4,𝔽2)11yescharacteristic
119C4—21no—
120C4—62no—
121C4—92no—
122C4—91no—
123C4—91no—
124C4—122no—
125C4—121no—
126C3—21no—
127C3—21no—
128C2#6422811yescharacteristic
129C2#6423011yescharacteristic
130C2#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#462
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
44. C3⋊Dic3Q16semidirect
100. C32#462semidirect

Maximal subgroups

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

subgroupindex
C2×#37762
#639192
#638382
#637792
#637402
#4629