Finite Group Databasebeta

← #10590856

Subgroups — #10590856

Order 576.

254 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 60 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×#3743C211yesmaximal, characteristic
2#63871C212yesmaximal
3C2×#3718C211yesmaximal, characteristic, complemented
4C3×#816C212yesmaximal, complemented
5#63574C211yesmaximal, characteristic, complemented
6#4816—32nomaximal
7C2×C4×C3⋊D3C2211yescharacteristic
8C2×C3×Dic6C2212yes—
9C2×C32×C8C2211yescharacteristic
10C2×C3⋊(C3⋊C8)C2211yescharacteristic
11#3743—22no—
12C3×C4⋊Dic3C2212yes—
13Dic3⋊Dic3—22no—
14#3722—22no—
15#3718—22no—
16#946—32no—
17C2×#216—32no—
18C2×ΓL(1,𝔽25)—32no—
19C2×ΓL(1,𝔽25)—61no—
20#897—32no—
21C3×#103—32no—
22#816D312yescomplemented
23#808—32no—
24C22×C3⋊D3D411yescharacteristic
25C2×C32×C4C2311yescharacteristic
26C2×C3⋊Dic3D411yescharacteristic
27C4×C3⋊D3—21no—
28C4×C3⋊D3—21no—
29C2×C3×Dic3—22no—
30C2×C3×Dic3—22no—
31C3×Dic6—22no—
32C3×Dic6—22no—
33C32×C8—21no—
34C3⋊(C3⋊C8)D412yes—
35#470—91noSylow
36C2×C3×Q8—32no—
37C2×C4×D3—32no—
38C2×C4×D3—61no—
39C2×Dic6C2×D312yes—
40C2×C3×C8—21no—
41C2×C3×C8C2×D312yes—
42C3×C4⋊C4—32no—
43#216—64no—
44(C2×D3)⋊C4—62no—
45(C2×D3)⋊C4—62no—
46C4⋊Dic3—62no—
47C4⋊Dic3C2×D312yes—
48Dic3⋊C4—62no—
49C2×C3⋊C8—32no—
50C2×C3⋊C8—61no—
51ΓL(1,𝔽25)—64no—
52ΓL(1,𝔽25)—64no—
53C22×C32C2×D411yescharacteristic
54C2×C3⋊D3—21no—
55C2×C3⋊D3—22no—
56C32×C4C2×D411yescharacteristic
57C32×C4#4111yescharacteristic
58C3⋊Dic3—21no—
59C3×Dic3—42no—
60C3×Dic3—42no—
61C2×Q16—91no—
62C2×M16—91no—
63C22⋊Q8—92no—
64#106—91no—
65#103—32no—
66C22×D3—32no—
67C22×D3—61no—
68C3×Q8—62no—
69C3×Q8—62no—
70C2×C3×C4—21no—
71C2×C3×C4—62no—
72C2×C3×C4—62no—
73C2×C3×C4C22×D312yes—
74C2×Dic3—22no—
75C2×Dic3—22no—
76C2×Dic3—32no—
77C2×Dic3—61no—
78C4×D3—62no—
79C4×D3—62no—
80C4×D3—62no—
81C4×D3—62no—
82Dic6—22no—
83Dic6—22no—
84C3×C8—22no—
85C3×C8—22no—
86C3⋊C8—34no—
87C3⋊C8—62no—
88C2×C32#12111yescharacteristic
89C2×C32#13712yes—
90C3⋊D3—41no—
91C2×Q8—32no—
92C22×C4—91no—
93Q16—182no—
94M16—182no—
95C2×C8—91no—
96C2×C8D3211yescharacteristic
97C4⋊C4—32no—
98C4⋊C4—182no—
99C22⋊C4—182no—
100C22×C3—21no—
101C22×C3D3×D412yes—
102C2×D3—62no—
103C2×D3—62no—
104C2×D3—62no—
105C2×D3—62no—
106C2×D3—122no—
107C3×C4—21no—
108C3×C4—21no—
109C3×C4—122no—
110C3×C4—122no—
111C3×C4#23512yes—
112C3×C4D3×D412yes—
113Dic3—42no—
114Dic3—42no—
115Dic3—62no—
116Dic3—62no—
117C32#47011yescharacteristic, complemented, Sylow
118C23—91no—
119Q8—62no—
120Q8—62no—
121C2×C4—62no—
122C2×C4—62no—
123C2×C4—91no—
124C2×C4—181no—
125C2×C4—181no—
126C2×C4C2×D3211yescharacteristic
127C8—21no—
128C8—92no—
129C2×C3—22no—
130C2×C3—21no—
131C2×C3#88412yes—
132C2×C3#91512yes—
133C2×C3#91812yes—
134D3—122no—
135D3—122no—
136C22—182no—
137C22—181no—
138C22#380211yescharacteristic
139C4—122no—
140C4—122no—
141C4—181no—
142C4#379811yescharacteristic
143C4#381111yescharacteristic
144C3—21no—
145C3#481612yescomplemented
146C2—361no—
147C2#6381112yes—
148C2#6391211yescharacteristic
subgroupG/H
centerC22#3802
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenterC2×C8D32
hypocenterC32#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C2×#3718C2semidirect
4. C3×#816C2semidirect
5. #63574C2semidirect
22. #816D3semidirect
117. C32#470semidirect
145. C3#4816semidirect

Maximal subgroups

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

subgroupindex
C2×#37432
#638712
C2×#37182
C3×#8162
#635742
#48163