Finite Group Databasebeta

← #10593363

Subgroups — #10593363

Order 576.

346 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 49 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×#3802C211yesmaximal, characteristic
2C2×#3788C211yesmaximal, characteristic, complemented
3#63874C211yesmaximal, characteristic, complemented
4#63739C214yesmaximal, complemented
5C22×D32—21no—
6C2×C4×C3⋊D3C2211yescharacteristic
7C2×C3×D12—21no—
8C2×#625C2212yes—
9#3802—22no—
10#3802C414yescomplemented
11C2×#622C2212yescomplemented
12#3788—22no—
13C3×C4×Dic3—21no—
14#3724C2212yescomplemented
15#3722—21no—
16C2×D3×D4—61no—
17#872—61no—
18C22×C3⋊D3D411yescharacteristic
19C22×C3×D3—22no—
20C2×D32—24no—
21C2×D32—44no—
22C2×C32×C4D411yescharacteristic
23C2×C3⋊Dic3C2311yescharacteristic
24C4×C3⋊D3C2×C412yes—
25C4×C3⋊D3D412yes—
26C2×C3×Dic3—22no—
27C3×D12—24no—
28#625—22no—
29#625C2×C414yes—
30#622—22no—
31#418—91nomaximal, Sylow
32C23×D3—62no—
33C2×C3×D4—61no—
34C2×#67—62no—
35D3×D4—64no—
36D3×D4—64no—
37C2×D12—21no—
38C2×C4×D3—61no—
39C2×C4×D3—61no—
40C3×C42—61no—
41(C2×D3)⋊C4—62no—
42Dic3⋊C4—62no—
43C4×Dic3—21no—
44C22×C32C2×D411yescharacteristic
45C2×C3⋊D3—22no—
46C2×C3⋊D3C22⋊C412yes—
47C2×C3×D3—24no—
48C2×C3×D3—44no—
49D32—48no—
50C32×C4C22⋊C412yes—
51C3⋊Dic3C22×C411yescharacteristic
52C3⋊Dic3C2×D411yescharacteristic
53C3×Dic3—42no—
54C22×D4—91no—
55C2×M16—91no—
56#117—91no—
57#102—94no—
58C23×C3—62no—
59C22×D3—22no—
60C22×D3—61no—
61C22×D3—61no—
62C22×D3—68no—
63C22×D3—128no—
64C3×D4—64no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—62no—
68#67—64no—
69#67—64no—
70C2×Dic3—22no—
71C2×Dic3—61no—
72C2×Dic3—61no—
73D12—24no—
74C4×D3—62no—
75C4×D3—62no—
76C4×D3—62no—
77C4×D3—62no—
78C2×C32C2×C22⋊C411yescharacteristic
79C2×C32AGL(1,ℤ/8)12yes—
80C3⋊D3—22no—
81C3×D3—44no—
82C24—181no—
83C2×D4—61no—
84C2×D4—94no—
85C2×D4—92no—
86C2×D4—182no—
87C22×C4—91no—
88M16—182no—
89C2×C8—92no—
90C4⋊C4—92no—
91C22⋊C4—181no—
92C42—61no—
93C22×C3—21no—
94C22×C3—21no—
95C22×C3—64no—
96C22×C3—124no—
97C2×D3—24no—
98C2×D3—44no—
99C2×D3—62no—
100C2×D3—62no—
101C2×D3—64no—
102C2×D3—122no—
103C2×D3—1216no—
104C3×C4—22no—
105C3×C4—22no—
106C3×C4—122no—
107Dic3—42no—
108Dic3—61no—
109Dic3—61no—
110Dic3—61no—
111Dic3—61no—
112C32#41811yescharacteristic, complemented, Sylow
113C23—62no—
114C23—91no—
115C23—184no—
116C23—364no—
117D4—64no—
118D4—94no—
119D4—182no—
120C2×C4—62no—
121C2×C4—91no—
122C2×C4—92no—
123C2×C4—92no—
124C2×C4GO+(4,𝔽2)11yescharacteristic
125C8—182no—
126C2×C3—22no—
127C2×C3—22no—
128C2×C3—21no—
129C2×C3—21no—
130C2×C3—124no—
131D3—44no—
132D3—64no—
133D3—122no—
134C22—64no—
135C22—92no—
136C22—124no—
137C22—182no—
138C22—368no—
139C22C2×GO+(4,𝔽2)11yescharacteristic
140C4—91no—
141C4—91no—
142C4—122no—
143C4D32⋊C412yes—
144C3—21no—
145C3—21no—
146C2—124no—
147C2—182no—
148C2#6422412yes—
149C2C2×D32⋊C411yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C4
FittingC2×C32×C4D4
socleC22×C32C2×D4
prime radicalC2×#3802C2
cosocleC2×C3⋊Dic3C23
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#418
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3788C2semidirect
3. #63874C2semidirect
4. #63739C2semidirect
10. #3802C4semidirect
11. C2×#622C22semidirect
14. #3724C22semidirect
112. C32#418semidirect

Maximal subgroups

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

subgroupindex
C2×#38022
C2×#37882
#638742
#637392
#4189