Finite Group Databasebeta

← #10593634

Subgroups — #10593634

Order 576.

184 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 37 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
1C4×C3⋊S4C211yesmaximal, characteristic
2C3×C8×A4C211yesmaximal, characteristic, complemented
3C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#5400—33nomaximal
5#5111—31nomaximal
6C2×C3⋊S4C411yescharacteristic
7C3×C4×A4C2211yescharacteristic
8C3⋊SL(2,ℤ/4)C411yescharacteristic
9#3743—41nomaximal
10C4×S4—33no—
11C22×C3×C8D311yescharacteristic, complemented
12C4×#67—31no—
13C2×ΓL(1,𝔽25)—31no—
14C8×A4D313yescomplemented
15A4⋊C8—33no—
16#830—31no—
17(C2×D3)⋊C8—31no—
18#815—31no—
19Dic3⋊C8—31no—
20C2×C3×A4C2×C411yescharacteristic
21C3⋊S4—21no—
22C4×C3⋊D3—41no—
23C32×C8—41no—
24C3⋊(C3⋊C8)—41no—
25#435—91noSylow
26C2×S4—33no—
27C22×C3×C4C2×D311yescharacteristic
28C2×#67—31no—
29C2×C4×D3—31no—
30C4×A4C2×D313yes—
31SL(2,ℤ/4)—33no—
32C2×C3×C8—31no—
33C2×C3×C8—31no—
34#217—31no—
35(C2×D3)⋊C4—31no—
36Dic3⋊C4—31no—
37C4×Dic3—31no—
38C2×C3⋊C8—31no—
39C2×C3⋊C8—31no—
40ΓL(1,𝔽25)—61no—
41ΓL(1,𝔽25)—61no—
42ΓL(1,𝔽25)—123no—
43C2×C3⋊D3—41no—
44C3×A4M1611yescharacteristic, complemented
45C32×C4—41no—
46C3⋊Dic3—41no—
47C2×M16—91no—
48C22×C8C3⋊D311yescharacteristic, complemented
49C4×D4—91no—
50C4⋊C8—91no—
51C22⋊C8—91no—
52C22⋊C8—91no—
53#97—91no—
54C23×C3C4×D311yescharacteristic
55C22×D3—31no—
56C2×A4C4×D313yes—
57S4—63no—
58C2×C3×C4—31no—
59C2×C3×C4—31no—
60#67—61no—
61#67—61no—
62C2×Dic3—31no—
63C2×Dic3—31no—
64C2×Dic3—31no—
65C4×D3—61no—
66C4×D3—61no—
67C4×D3—123no—
68C3×C8—31no—
69C3×C8—43no—
70C3×C8S411yescharacteristic, complemented
71C3⋊C8—61no—
72C3⋊C8—61no—
73C3⋊C8—123no—
74C2×C32—41no—
75C3⋊D3—81no—
76C2×D4—91no—
77C22×C4—91no—
78C22×C4C2×C3⋊D311yescharacteristic
79M16—181no—
80M16—181no—
81C2×C8—31no—
82C2×C8—31no—
83C2×C8—91no—
84C2×C8—91no—
85C4⋊C4—91no—
86C22⋊C4—91no—
87C22⋊C4—91no—
88C42—91no—
89C22×C3—31no—
90C22×C3—31no—
91C22×C3ΓL(1,𝔽25)11yescharacteristic, complemented
92C2×D3—61no—
93C2×D3—61no—
94C2×D3—61no—
95C2×D3—123no—
96A4ΓL(1,𝔽25)13yescomplemented
97C3×C4—31no—
98C3×C4—43no—
99C3×C4C2×S411yescharacteristic
100Dic3—61no—
101Dic3—61no—
102Dic3—61no—
103Dic3—123no—
104C32—41noSylow
105C23—91no—
106C23C4×C3⋊D311yescharacteristic
107D4—181no—
108D4—181no—
109C2×C4—31no—
110C2×C4—31no—
111C2×C4—91no—
112C2×C4—91no—
113C2×C4—91no—
114C2×C4—181no—
115C2×C4—181no—
116C8—31no—
117C8—181no—
118C8—181no—
119C8C3⋊S411yescharacteristic, complemented
120C2×C3—31no—
121C2×C3—31no—
122C2×C3—43no—
123C2×C3C4×S411yescharacteristic
124D3—121no—
125D3—243no—
126C22—31no—
127C22—31no—
128C22—181no—
129C22—181no—
130C22—181no—
131C22#374311yescharacteristic, complemented
132C4—31no—
133C4—181no—
134C4—181no—
135C4—181no—
136C4C2×C3⋊S411yescharacteristic
137C3—43no—
138C3#540011yescharacteristic, complemented
139C2—31no—
140C2—31no—
141C2—361no—
142C2C4×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC4C2×C3⋊S4
commutatorC2×C3×A4C2×C4
FittingC22×C3×C8D3
socleC23×C3C4×D3
prime radicalC2×C3⋊S4C4
cosocleC3×C4×A4C22
hypercenterC8C3⋊S4
hypocenterC3×A4M16
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C3×C8×A4C2semidirect
3. C3⋊(A4⋊C8)C2semidirect
11. C22×C3×C8D3semidirect
14. C8×A4D3semidirect
44. C3×A4M16semidirect
48. C22×C8C3⋊D3semidirect
70. C3×C8S4semidirect
91. C22×C3ΓL(1,𝔽25)semidirect
96. A4ΓL(1,𝔽25)semidirect
119. C8C3⋊S4semidirect
131. C22#3743semidirect
138. C3#5400semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊S42
C3×C8×A42
C3⋊(A4⋊C8)2
#54003
#51113
#37434