Finite Group Databasebeta

← #10590840

Subgroups — #10590840

Order 576.

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

normal subgroups only

This group has 294 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, complemented
2#63916C212yesmaximal, complemented
3C2×#3713C211yesmaximal, characteristic, complemented
4C3×#821C212yesmaximal, complemented
5#63574C211yesmaximal, characteristic, complemented
6#4782—32nomaximal
7C2×C4×C3⋊D3C2211yescharacteristic
8C2×C3×D12C2212yes—
9C2×#626—22no—
10C2×C32×C8C2211yescharacteristic
11C2×C3⋊(C3⋊C8)C2211yescharacteristic
12#3743—22no—
13C3×C4⋊Dic3C2212yes—
14(C2×D3)⋊Dic3—22no—
15#3713—22no—
16#938—32no—
17C2×#213—32no—
18C2×ΓL(1,𝔽25)—32no—
19C2×ΓL(1,𝔽25)—61no—
20#895—32no—
21C3×#102—32no—
22#821D312yescomplemented
23#808—32no—
24C22×C3⋊D3D411yescharacteristic
25C22×C3×D3—22no—
26C2×C32×C4C2311yescharacteristic
27C2×C3⋊Dic3D411yescharacteristic
28C4×C3⋊D3—21no—
29C4×C3⋊D3—21no—
30C2×C3×Dic3—22no—
31C3×D12—22no—
32C3×D12—22no—
33#626—44no—
34C32×C8—21no—
35C3⋊(C3⋊C8)D412yescomplemented
36#469—91noSylow
37C2×C3×D4—32no—
38C2×#67—62no—
39C2×D12—62no—
40C2×D12C2×D312yes—
41C2×C4×D3—32no—
42C2×C4×D3—61no—
43C2×C3×C8—21no—
44C2×C3×C8C2×D312yes—
45C3×C4⋊C4—32no—
46#217—62no—
47#213—64no—
48(C2×D3)⋊C4—62no—
49C4⋊Dic3C2×D312yes—
50C2×C3⋊C8—32no—
51C2×C3⋊C8—61no—
52ΓL(1,𝔽25)—64no—
53ΓL(1,𝔽25)—64no—
54C22×C32C2×D411yescharacteristic
55C2×C3⋊D3—21no—
56C2×C3⋊D3—22no—
57C2×C3×D3—42no—
58C2×C3×D3—42no—
59C2×C3×D3—42no—
60C32×C4C2×D411yescharacteristic
61C32×C4#4111yescharacteristic
62C3⋊Dic3—21no—
63C3×Dic3—42no—
64C2×D8—91no—
65C2×M16—91no—
66#121—92no—
67#106—91no—
68#102—32no—
69C23×C3—62no—
70C22×D3—22no—
71C22×D3—32no—
72C22×D3—61no—
73C3×D4—62no—
74C3×D4—62no—
75C2×C3×C4—21no—
76C2×C3×C4—62no—
77C2×C3×C4C22×D312yes—
78#67—124no—
79C2×Dic3—22no—
80C2×Dic3—32no—
81C2×Dic3—61no—
82D12—22no—
83D12—22no—
84D12—124no—
85C4×D3—62no—
86C4×D3—62no—
87C4×D3—62no—
88C4×D3—62no—
89C3×C8—22no—
90C3×C8—22no—
91C3⋊C8—34no—
92C3⋊C8—62no—
93C2×C32#12111yescharacteristic
94C2×C32AGL(1,ℤ/8)12yes—
95C3⋊D3—41no—
96C3×D3—82no—
97C2×D4—32no—
98C2×D4—182no—
99C22×C4—91no—
100D8—182no—
101M16—182no—
102C2×C8—91no—
103C2×C8D3211yescharacteristic
104C4⋊C4—32no—
105C22⋊C4—182no—
106C22×C3—21no—
107C22×C3—122no—
108C22×C3—122no—
109C22×C3—122no—
110C22×C3D3×D412yes—
111C2×D3—42no—
112C2×D3—42no—
113C2×D3—42no—
114C2×D3—62no—
115C2×D3—62no—
116C2×D3—62no—
117C2×D3—62no—
118C2×D3—122no—
119C3×C4—21no—
120C3×C4—21no—
121C3×C4—122no—
122C3×C4#23512yes—
123C3×C4D3×D412yes—
124Dic3—42no—
125Dic3—62no—
126Dic3—62no—
127C32#46911yescharacteristic, complemented, Sylow
128C23—62no—
129C23—91no—
130D4—62no—
131D4—62no—
132D4—364no—
133C2×C4—62no—
134C2×C4—91no—
135C2×C4—181no—
136C2×C4—181no—
137C2×C4C2×D3211yescharacteristic
138C8—21no—
139C8—92no—
140C2×C3—22no—
141C2×C3—21no—
142C2×C3—242no—
143C2×C3#88412yes—
144C2×C3#91112yes—
145C2×C3#91412yes—
146D3—82no—
147D3—122no—
148D3—122no—
149C22—122no—
150C22—122no—
151C22—122no—
152C22—182no—
153C22—181no—
154C22#380211yescharacteristic
155C4—122no—
156C4—181no—
157C4#379811yescharacteristic
158C4#381111yescharacteristic
159C3—21no—
160C3#478212yescomplemented
161C2—242no—
162C2—361no—
163C2#6379812yes—
164C2#6391211yescharacteristic
subgroupG/H
centerC22#3802
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
hypercenterC2×C8D32
hypocenterC32#469
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
1. C2×#3743C2semidirect
2. #63916C2semidirect
3. C2×#3713C2semidirect
4. C3×#821C2semidirect
5. #63574C2semidirect
22. #821D3semidirect
35. C3⋊(C3⋊C8)D4semidirect
127. C32#469semidirect
160. C3#4782semidirect

Maximal subgroups

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

subgroupindex
C2×#37432
#639162
C2×#37132
C3×#8212
#635742
#47823