Finite Group Databasebeta

← #10593482

Subgroups — #10593482

Order 576.

160 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 29 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
1#64289C211yesmaximal, characteristic
2#64228C211yesmaximal, characteristic, complemented
3#63945C212yesmaximal
4#63733C211yesmaximal, characteristic, complemented
5#63728C212yesmaximal, complemented
6#3832C2211yescharacteristic
7#3797C2212yes—
8#3788—21no—
9#3788C2211yescharacteristic
10#3787—21no—
11#3787C2211yescharacteristic
12#3776—21no—
13#3776—21no—
14C3×#216—22no—
15#3717—22no—
16#3712C2212yes—
17#918—62no—
18C32×Q8D411yescharacteristic
19C4⋊(C3⋊D3)—21no—
20C4⋊(C3⋊D3)D411yescharacteristic
21C4×C3⋊D3—21no—
22C4×C3⋊D3C2311yescharacteristic
23C3×Dic6—22no—
24#627—22no—
25#624—22no—
26#622—21no—
27#622D411yescharacteristic
28#622D411yescharacteristic
29C3×C3⋊C8—22no—
30#473—91nomaximal, Sylow
31#239—62no—
32D3×Q8—62no—
33C3×Q16—62no—
34#216—22no—
35#215—62no—
36#204—62no—
37ΓL(1,𝔽25)—62no—
38C2×C3⋊D3—21no—
39C2×C3⋊D3#4111yescharacteristic
40C32×C4—21no—
41C32×C4C2×D411yescharacteristic
42C3⋊Dic3C2×D411yescharacteristic
43C3×Dic3—42no—
44#137—92no—
45C2×Q16—91no—
46#131—91no—
47#108—91no—
48#101—92no—
49C3×Q8—22no—
50C3×Q8—62no—
51D12—62no—
52D12—122no—
53C4×D3—62no—
54C4×D3—122no—
55C4×D3—122no—
56Dic6—22no—
57Dic6—122no—
58C3×C8—62no—
59C3⋊C8—22no—
60C2×C32#12111yescharacteristic
61C3⋊D3—21no—
62C3⋊D3—41no—
63#41—91no—
64C2×Q8—92no—
65Q16—62no—
66Q16—181no—
67Q16—181no—
68SD16—182no—
69M16—92no—
70M16—91no—
71M16—181no—
72C2×C8—91no—
73C2×C8—181no—
74C2×D3—62no—
75C2×D3—122no—
76C3×C4—22no—
77C3×C4—42no—
78C3×C4—122no—
79Dic3—42no—
80Dic3—62no—
81C32#47311yescharacteristic, complemented, Sylow
82Q8—62no—
83Q8—182no—
84Q8GO+(4,𝔽2)11yescharacteristic
85D4—91no—
86D4—181no—
87C2×C4—91no—
88C2×C4—181no—
89C2×C4—182no—
90C8—62no—
91C8—91no—
92C8—91no—
93C8—181no—
94C2×C3—22no—
95D3—122no—
96D3—242no—
97C22—91no—
98C22—181no—
99C4—21no—
100C4—91no—
101C4—122no—
102C4C2×GO+(4,𝔽2)11yescharacteristic
103C3—22no—
104C2—181no—
105C2—361no—
106C2#6424211yescharacteristic
subgroupG/H
centerC2#64242
commutator, cosocleC4×C3⋊D3C23
FrattiniC4C2×GO+(4,𝔽2)
FittingC32×Q8D4
socleC2×C32#121
prime radical#3832C22
hypercenterQ8GO+(4,𝔽2)
hypocenterC32#473
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
2. #64228C2semidirect
4. #63733C2semidirect
5. #63728C2semidirect
81. C32#473semidirect

Maximal subgroups

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

subgroupindex
#642892
#642282
#639452
#637332
#637282
#4739