Finite Group Databasebeta

← #10596202

Subgroups — #10596202

Order 576.

576 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64155C212yesmaximal
3Q8×C3⋊Dic3C212yesmaximal, complemented
4#64107C214yesmaximal
5#64098C214yesmaximal, complemented
6#64081C211yesmaximal, characteristic
7#64079C211yesmaximal, characteristic, complemented
8#5731—34nomaximal
9C2×C32×Q8C2212yes—
10C2×C4×C3⋊D3C2211yescharacteristic
11C32×C4⋊C4C2214yes—
12C32×C42C2211yescharacteristic
13#3752C2212yes—
14#3752C2214yes—
15C4⋊(C3⋊Dic3)C2212yes—
16C4⋊(C3⋊Dic3)C2214yes—
17#3750C2214yes—
18#3750C2212yes—
19#3750C2214yes—
20C4×C3⋊Dic3C2211yescharacteristic
21C4×C3⋊Dic3C2214yes—
22C3×C4⋊Q8D314yescomplemented
23#946—38no—
24Q8×Dic3—38no—
25#898—316no—
26#889—316no—
27#872—34no—
28#870—34no—
29C22×C3⋊D3C2311yescharacteristic
30C32×Q8—24no—
31C2×C32×C4C2314yes—
32C2×C32×C4C2312yes—
33C2×C32×C4C2311yescharacteristic
34C2×C3⋊Dic3C2312yes—
35C2×C3⋊Dic3C2314yes—
36C2×C3⋊Dic3C2311yescharacteristic
37C4×C3⋊D3—22no—
38#541—91noSylow
39C2×C3×Q8C2×D318yes—
40C2×C4×D3—34no—
41C3×C4⋊C4C2×D3116yes—
42C3×C42C2×D314yes—
43(C2×D3)⋊C4—38no—
44(C2×D3)⋊C4—316no—
45C4⋊Dic3—38no—
46C4⋊Dic3—316no—
47Dic3⋊C4—316no—
48Dic3⋊C4—38no—
49Dic3⋊C4—316no—
50C4×Dic3—34no—
51C4×Dic3—316no—
52C22×C32C2411yescharacteristic
53C2×C3⋊D3—21no—
54C2×C3⋊D3—22no—
55C32×C4—24no—
56C32×C4—22no—
57C32×C4#4112yes—
58C3⋊Dic3—21no—
59C3⋊Dic3—22no—
60C3⋊Dic3—24no—
61C4⋊Q8C3⋊D311yescharacteristic, complemented
62#126—94no—
63#125—94no—
64#125—91no—
65C22⋊Q8—92no—
66C4×Q8—92no—
67#117—91no—
68C22×D3—34no—
69C3×Q8—216no—
70C2×C3×C4C22×D3116yes—
71C2×C3×C4C22×D318yes—
72C2×C3×C4C22×D314yes—
73C2×Dic3—38no—
74C2×Dic3—316no—
75C2×Dic3—34no—
76C4×D3—68no—
77C2×C32C2×#4111yescharacteristic
78C2×C322−1+412yes—
79C3⋊D3—41no—
80C2×Q8C2×C3⋊D312yes—
81C22×C4—91no—
82C4⋊C4—92no—
83C4⋊C4—92no—
84C4⋊C4—94no—
85C4⋊C4—94no—
86C4⋊C4—94no—
87C4⋊C4C2×C3⋊D314yes—
88C22⋊C4—92no—
89C22⋊C4—94no—
90C42—91no—
91C42—94no—
92C42C2×C3⋊D311yescharacteristic
93C22×C3C23×D314yes—
94C2×D3—64no—
95C2×D3—68no—
96C3×C4—216no—
97C3×C4—28no—
98C3×C4#23718yes—
99Dic3—64no—
100Dic3—68no—
101Dic3—616no—
102C32#54111yescharacteristic, complemented, Sylow
103C23—91no—
104Q8—24no—
105C2×C4—92no—
106C2×C4—94no—
107C2×C4—91no—
108C2×C4—182no—
109C2×C4C22×C3⋊D314yes—
110C2×C4C22×C3⋊D312yes—
111C2×C4C22×C3⋊D311yescharacteristic
112C2×C3C2×#23714yes—
113C2×C3#100718yes—
114D3—124no—
115C22—181no—
116C22—182no—
117C22C23×C3⋊D311yescharacteristic
118C4—24no—
119C4—22no—
120C4—181no—
121C4—182no—
122C4—184no—
123C4#383012yes—
124C3#573114yescomplemented
125C2—361no—
126C2C2×#383011yescharacteristic
127C2#6436412yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊Q8C2
prime radicalC22×C3⋊D3C23
hypercenterC4⋊Q8C3⋊D3
hypocenterC32#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C4⋊Q8C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. #64098C2semidirect
7. #64079C2semidirect
22. C3×C4⋊Q8D3semidirect
61. C4⋊Q8C3⋊D3semidirect
102. C32#541semidirect
124. C3#5731semidirect

Maximal subgroups

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

subgroupindex
C32×C4⋊Q82
#641552
Q8×C3⋊Dic32
#641072
#640982
#640812
#640792
#57313