Finite Group Databasebeta

← #10593477

Subgroups — #10593477

Order 576.

168 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 33 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×#157C211yesmaximal, characteristic
2#64222C211yesmaximal, characteristic
3#63944C212yesmaximal
4#63732C211yesmaximal, characteristic
5#63730C212yesmaximal
6Q8×C3⋊D3C2211yescharacteristic
7#3797C2212yes—
8C4⋊#157—21no—
9C4⋊#157C2211yescharacteristic
10C4×#157—21no—
11C4×#157C2211yescharacteristic
12#3773—21no—
13#3773—21no—
14C3×#216—22no—
15#3719—22no—
16#3710C2212yes—
17D3×Q16—62no—
18C2×#157—21no—
19C2×#157D411yescharacteristic
20C2×#157D411yescharacteristic
21C32×Q8D411yescharacteristic
22C4×C3⋊D3—21no—
23C4×C3⋊D3C2311yescharacteristic
24#634—21no—
25#634D411yescharacteristic
26C3×Dic6—22no—
27#627—22no—
28#624—22no—
29C3×C3⋊C8—22no—
30#462—91nomaximal, Sylow
31D3×Q8—62no—
32D3×Q8—62no—
33C3×Q16—62no—
34#216—22no—
35#216—62no—
36Dic12—62no—
37D3×C8—62no—
38C2×C3⋊D3C2×D411yescharacteristic
39#157—21no—
40#157—41no—
41#157Q1612yescomplemented
42C32×C4—21no—
43C32×C4C2×D411yescharacteristic
44C3⋊Dic3—21no—
45C3⋊Dic3#4111yescharacteristic
46C3×Dic3—42no—
47C2×Q16—92no—
48C4⋊Q8—91no—
49C4×Q8—91no—
50C4⋊C8—91no—
51#103—92no—
52C3×Q8—22no—
53C3×Q8—62no—
54C4×D3—62no—
55C4×D3—122no—
56C4×D3—122no—
57Dic6—22no—
58Dic6—62no—
59Dic6—122no—
60Dic6—122no—
61C3×C8—62no—
62C3⋊C8—22no—
63C2×C32#12111yescharacteristic
64C3⋊D3C2×Q1611yescharacteristic
65C3⋊D3#13711yescharacteristic
66C2×Q8—92no—
67C2×Q8—91no—
68Q16—62no—
69Q16—182no—
70C2×C8—92no—
71C4⋊C4—91no—
72C4⋊C4—181no—
73C4⋊C4—181no—
74C4⋊C4—181no—
75C42—91no—
76C42—181no—
77C2×D3—62no—
78C3×C4—22no—
79C3×C4—42no—
80C3×C4—122no—
81Dic3—42no—
82Dic3—62no—
83Dic3—122no—
84C32#46211yescharacteristic, complemented, Sylow
85Q8—62no—
86Q8—91no—
87Q8—181no—
88Q8—182no—
89Q8GO+(4,𝔽2)11yescharacteristic
90C2×C4—91no—
91C2×C4—91no—
92C2×C4—91no—
93C2×C4—181no—
94C2×C4—182no—
95C2×C4—181no—
96C8—62no—
97C2×C3—22no—
98D3—62no—
99D3—62no—
100C22—91no—
101C4—21no—
102C4—92no—
103C4—91no—
104C4—122no—
105C4—181no—
106C4—181no—
107C4—361no—
108C4C2×GO+(4,𝔽2)11yescharacteristic
109C3—22no—
110C2—91no—
111C2—91no—
112C2#6424211yescharacteristic
subgroupG/H
centerC2#64242
commutator, cosocleC4×C3⋊D3C23
FrattiniC4C2×GO+(4,𝔽2)
FittingC32×Q8D4
socleC2×C32#121
prime radicalC2×C3⋊D3C2×D4
hypercenterQ8GO+(4,𝔽2)
hypocenterC32#462
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
41. #157Q16semidirect
84. C32#462semidirect

Maximal subgroups

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

subgroupindex
Q8×#1572
#642222
#639442
#637322
#637302
#4629