Finite Group Databasebeta

← #10593805

Subgroups — #10593805

Order 576.

234 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3792C212yesmaximal
2C2×C4×C3⋊Dic3C211yesmaximal, characteristic
3#63779C212yesmaximal, complemented
4C4⋊#622C212yesmaximal, complemented
5C22×C32×C4C411yescharacteristic
6C22×C3⋊Dic3C411yescharacteristic
7C22×C3⋊Dic3C2211yescharacteristic
8#3792—24no—
9C2×#622C2214yes—
10C4×C3⋊Dic3C411yescharacteristic
11C4×C3⋊Dic3C411yescharacteristic
12C4×C3⋊Dic3C2211yescharacteristic
13C4×C3⋊Dic3C2211yescharacteristic
14C2×C4×Dic3—62no—
15C23×C32C2×C411yescharacteristic
16C2×C32×C4—22no—
17C2×C32×C4C2×C411yescharacteristic
18C2×C32×C4C2×C411yescharacteristic
19C2×C3⋊Dic3—22no—
20C2×C3⋊Dic3C2×C411yescharacteristic
21C2×C3⋊Dic3C2×C411yescharacteristic
22C2×C3⋊Dic3C2×C411yescharacteristic
23C2×C3⋊Dic3D412yes—
24C2×C3⋊Dic3Q812yes—
25C2×C3⋊Dic3C2311yescharacteristic
26#622—24no—
27#423—91nomaximal, Sylow
28C22×C3×C4—22no—
29C22×Dic3—62no—
30C22×Dic3—62no—
31C4×Dic3—62no—
32C4×Dic3—62no—
33C4×Dic3—62no—
34C4×Dic3—62no—
35C22×C32—21no—
36C22×C32—21no—
37C22×C32C4⋊C412yes—
38C22×C32C22×C411yescharacteristic
39C32×C4—21no—
40C32×C4M1612yescomplemented
41C3⋊Dic3—21no—
42C3⋊Dic3C4⋊C412yes—
43C3⋊Dic3M1612yescomplemented
44C3⋊Dic3C2×D411yescharacteristic
45C3⋊Dic3C2×Q811yescharacteristic
46C2×M16—92no—
47C2×C42—91no—
48C4⋊C8—92no—
49C4⋊C8—92no—
50C23×C3—22no—
51C2×C3×C4—28no—
52C2×C3×C4—22no—
53C2×C3×C4—22no—
54C2×Dic3—62no—
55C2×Dic3—62no—
56C2×Dic3—62no—
57C2×Dic3—64no—
58C2×Dic3—64no—
59C2×Dic3—62no—
60C2×Dic3—68no—
61C2×C32—22no—
62C2×C32C2×C4⋊C411yescharacteristic
63C2×C32C2×M1611yescharacteristic
64C2×C32C2×M1611yescharacteristic
65C22×C4—91no—
66C22×C4—91no—
67C22×C4#15711yescharacteristic
68M16—184no—
69C2×C8—94no—
70C42—91no—
71C42—91no—
72C42—91no—
73C42—91no—
74C22×C3—24no—
75C22×C3—24no—
76C22×C3—22no—
77C22×C3—24no—
78C3×C4—24no—
79C3×C4—24no—
80Dic3—64no—
81Dic3—64no—
82Dic3—62no—
83Dic3—62no—
84Dic3—64no—
85C32#42311yescharacteristic, complemented, Sylow
86C23C2×#15711yescharacteristic
87C2×C4—22no—
88C2×C4—91no—
89C2×C4—91no—
90C2×C4—91no—
91C2×C4—92no—
92C2×C4—92no—
93C2×C4—91no—
94C2×C4—182no—
95C2×C4C2×#15711yescharacteristic
96C2×C4C2×#15711yescharacteristic
97C8—184no—
98C2×C3—28no—
99C2×C3—22no—
100C2×C3—22no—
101C2×C3—22no—
102C22—21no—
103C22—21no—
104C22C4⋊#15712yes—
105C22C22×#15711yescharacteristic
106C4—21no—
107C4—92no—
108C4—91no—
109C4—91no—
110C4—92no—
111C4—181no—
112C4#379212yescomplemented
113C3—22no—
114C2—22no—
115C2#6428311yescharacteristic
116C2C2×C4⋊#15711yescharacteristic
117C2C2×#379211yescharacteristic
subgroupG/H
center, FrattiniC22C22×#157
commutator, socleC22×C32C22×C4
FittingC22×C32×C4C4
prime radicalC23×C32C2×C4
cosocleC2×C3⋊Dic3C23
hypercenterC22×C4#157
hypocenterC32#423
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #63779C2semidirect
4. C4⋊#622C2semidirect
40. C32×C4M16semidirect
43. C3⋊Dic3M16semidirect
85. C32#423semidirect
112. C4#3792semidirect

Maximal subgroups

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

subgroupindex
C2×#37922
C2×C4×C3⋊Dic32
#637792
C4⋊#6222
#4239