Finite Group Databasebeta

← #10591630

Subgroups — #10591630

Order 576.

227 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 68 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
1Q8×C3⋊Dic3C211yesmaximal, characteristic
2#63919C211yesmaximal, characteristic
3C3×#835C212yesmaximal
4#63573C212yesmaximal
5#63572C211yesmaximal, characteristic
6#4791—32nomaximal
7C2×C32×Q8C2211yescharacteristic
8C2×#627—21no—
9C4⋊(C3⋊Dic3)—21no—
10C4⋊(C3⋊Dic3)C2211yescharacteristic
11C4×C3⋊Dic3—21no—
12C4×C3⋊Dic3C2211yescharacteristic
13C3×C4⋊Dic3C2212yes—
14C2×C3×C3⋊C8C2212yes—
15#3724—21no—
16Q8×Dic3—32no—
17Q8×Dic3—61no—
18#888—32no—
19C3×#103—32no—
20#835D312yes—
21#817—32no—
22Dic3⋊C8—32no—
23#808—32no—
24C32×Q8—21no—
25C32×Q8Q812yescomplemented
26C2×C32×C4—21no—
27C2×C32×C4C2311yescharacteristic
28C2×C3⋊Dic3—21no—
29C2×C3⋊Dic3D411yescharacteristic
30C2×C3⋊Dic3D411yescharacteristic
31C2×C3×Dic3—22no—
32#627—42no—
33C3×C3⋊C8—22no—
34#475—91noSylow
35C2×C3×Q8—21no—
36C2×C3×Q8C2×D312yes—
37C2×Dic6—62no—
38C2×C3×C8—32no—
39C3×C4⋊C4—32no—
40C4⋊Dic3—32no—
41C4⋊Dic3—61no—
42C4⋊Dic3—62no—
43C4⋊Dic3—62no—
44C4⋊Dic3C2×D312yes—
45Dic3⋊C4—62no—
46C4×Dic3—32no—
47C4×Dic3—61no—
48C4×Dic3—62no—
49C4×Dic3—62no—
50C2×C3⋊C8C2×D312yes—
51C22×C32C2×D411yescharacteristic
52C32×C4—22no—
53C32×C4C2×Q811yescharacteristic
54C32×C4#4111yescharacteristic
55C3⋊Dic3—21no—
56C3⋊Dic3—41no—
57C3⋊Dic3SD1612yes—
58C3×Dic3—42no—
59C4⋊Q8—91no—
60C4×Q8—91no—
61#106—92no—
62C4⋊C8—91no—
63#103—32no—
64C3×Q8—22no—
65C3×Q8—22no—
66C3×Q8—22no—
67C3×Q8Dic614yescomplemented
68C2×C3×C4—22no—
69C2×C3×C4—22no—
70C2×C3×C4—21no—
71C2×C3×C4—62no—
72C2×C3×C4C22×D312yes—
73C2×Dic3—22no—
74C2×Dic3—32no—
75C2×Dic3—32no—
76C2×Dic3—61no—
77C2×Dic3—61no—
78C2×Dic3—62no—
79C2×Dic3—62no—
80Dic6—124no—
81C3×C8—62no—
82C3⋊C8—22no—
83C2×C32C22⋊Q811yescharacteristic
84C2×C32C2×SD1611yescharacteristic
85C2×C32#13711yescharacteristic
86C2×Q8—181no—
87C2×Q8D3211yescharacteristic
88C2×C8—32no—
89C4⋊C4—32no—
90C4⋊C4—91no—
91C4⋊C4—181no—
92C4⋊C4—181no—
93C42—91no—
94C42—181no—
95C22×C3—21no—
96C22×C3D3×D412yes—
97C3×C4—24no—
98C3×C4—24no—
99C3×C4—21no—
100C3×C4—21no—
101C3×C4—122no—
102C3×C4C2×Dic612yes—
103C3×C4#23712yes—
104Dic3—34no—
105Dic3—42no—
106Dic3—62no—
107Dic3—62no—
108Dic3—121no—
109Dic3—122no—
110Dic3—122no—
111C32#47511yescharacteristic, complemented, Sylow
112Q8—21no—
113Q8—362no—
114Q8#62712yescomplemented
115C2×C4—21no—
116C2×C4—62no—
117C2×C4—91no—
118C2×C4—91no—
119C2×C4—181no—
120C2×C4C2×D3211yescharacteristic
121C8—62no—
122C2×C3—21no—
123C2×C3—21no—
124C2×C3—21no—
125C2×C3#87812yes—
126C2×C3D3×SD1612yes—
127C2×C3#91812yes—
128C22#381111yescharacteristic
129C4—22no—
130C4—92no—
131C4—122no—
132C4—181no—
133C4—361no—
134C4#380511yescharacteristic
135C4C2×#62711yescharacteristic
136C3—21no—
137C3#479112yescomplemented
138C2#6394011yescharacteristic
139C2#6394511yescharacteristic
140C2#6398211yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socle, prime radicalC22×C32C2×D4
hypercenterC2×Q8D32
hypocenterC32#475
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
25. C32×Q8Q8semidirect
67. C3×Q8Dic6semidirect
111. C32#475semidirect
114. Q8#627semidirect
137. C3#4791semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic32
#639192
C3×#8352
#635732
#635722
#47913