Finite Group Databasebeta

← #10593744

Subgroups — #10593744

Order 576.

172 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 42 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#64086C211yesmaximal, characteristic
2C22⋊#622C211yesmaximal, characteristic
3C22⋊#622C211yesmaximal, characteristic
4#63779C211yesmaximal, characteristic, complemented
5#63777C211yesmaximal, characteristic, complemented
6C4⋊#622C211yesmaximal, characteristic, complemented
7C4×#622C211yesmaximal, characteristic, complemented
8C22×C3⋊Dic3C2211yescharacteristic
9C2×#622C2211yescharacteristic
10C2×#622C2211yescharacteristic
11C2×#622C2211yescharacteristic
12C2×#622C2211yescharacteristic
13C32×C22⋊C4C411yescharacteristic
14#3757C411yescharacteristic
15#3750C411yescharacteristic
16#3750C411yescharacteristic
17C4×C3⋊Dic3C2211yescharacteristic
18C4×C3⋊Dic3C2211yescharacteristic
19#877—62no—
20C23×C32C2×C411yescharacteristic
21C2×C32×C4C2×C411yescharacteristic
22C2×C32×C4C2×C411yescharacteristic
23C2×C3⋊Dic3—21no—
24C2×C3⋊Dic3—21no—
25C2×C3⋊Dic3C2×C411yescharacteristic
26C2×C3⋊Dic3C2×C411yescharacteristic
27C2×C3⋊Dic3C2×C411yescharacteristic
28C2×C3⋊Dic3C2311yescharacteristic
29#622—21no—
30#622—21no—
31#622—21no—
32#622—21no—
33#433—91nomaximal, Sylow
34C22×Dic3—62no—
35C3×C22⋊C4—22no—
36#217—62no—
37Dic3⋊C4—62no—
38Dic3⋊C4—62no—
39C4×Dic3—62no—
40C4×Dic3—62no—
41C22×C32—21no—
42C22×C32—21no—
43C22×C32—21no—
44C22×C32C22×C411yescharacteristic
45C32×C4—21no—
46C32×C4—21no—
47C3⋊Dic3—21no—
48C3⋊Dic3—21no—
49C3⋊Dic3—21no—
50C3⋊Dic3#4111yescharacteristic
51C3⋊Dic3#4111yescharacteristic
52#117—91no—
53C4⋊C8—91no—
54C4⋊C8—91no—
55C22⋊C8—91no—
56C22⋊C8—91no—
57#97—91no—
58C4×C8—91no—
59C23×C3—22no—
60C2×C3×C4—22no—
61C2×C3×C4—22no—
62C2×Dic3—62no—
63C2×Dic3—62no—
64C2×Dic3—62no—
65C2×Dic3—62no—
66C2×Dic3—64no—
67C2×Dic3—64no—
68C2×C32—41no—
69C2×C32#11711yescharacteristic
70C2×C32#13111yescharacteristic
71C2×C32#13111yescharacteristic
72C22×C4—91no—
73C2×C8—91no—
74C2×C8—91no—
75C2×C8—91no—
76C2×C8—91no—
77C4⋊C4—91no—
78C4⋊C4—91no—
79C22⋊C4—91no—
80C22⋊C4#15711yescharacteristic
81C42—91no—
82C42—91no—
83C22×C3—24no—
84C22×C3—22no—
85C22×C3—42no—
86C22×C3—42no—
87C3×C4—42no—
88C3×C4—42no—
89Dic3—62no—
90Dic3—62no—
91Dic3—64no—
92Dic3—122no—
93Dic3—122no—
94C32#43311yescharacteristic, complemented, Sylow
95C23C2×#15711yescharacteristic
96C2×C4—91no—
97C2×C4—91no—
98C2×C4—91no—
99C2×C4—91no—
100C2×C4—181no—
101C2×C4—181no—
102C2×C4C2×#15711yescharacteristic
103C2×C4C2×#15711yescharacteristic
104C8—181no—
105C8—181no—
106C8—181no—
107C8—181no—
108C2×C3—22no—
109C2×C3—22no—
110C2×C3—22no—
111C2×C3—44no—
112C22—21no—
113C22—21no—
114C22—21no—
115C22C22×#15711yescharacteristic
116C4—21no—
117C4—21no—
118C4—91no—
119C4—91no—
120C4—181no—
121C4—181no—
122C4—181no—
123C3—22no—
124C2—41no—
125C2#6428611yescharacteristic
126C2#6428711yescharacteristic
127C2#6428711yescharacteristic
subgroupG/H
center, FrattiniC22C22×#157
commutator, socleC22×C32C22×C4
FittingC32×C22⋊C4C4
prime radicalC23×C32C2×C4
cosocleC2×C3⋊Dic3C23
hypercenterC22⋊C4#157
hypocenterC32#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #63779C2semidirect
5. #63777C2semidirect
6. C4⋊#622C2semidirect
7. C4×#622C2semidirect
94. C32#433semidirect

Maximal subgroups

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

subgroupindex
#640862
C22⋊#6222
#637792
#637772
C4⋊#6222
C4×#6222
#4339