Finite Group Databasebeta

← #10593808

Subgroups — #10593808

Order 576.

314 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 48 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×#3828C211yesmaximal, characteristic
2#64283C212yesmaximal, complemented
3#63788C212yesmaximal, complemented
4#63780C212yesmaximal, complemented
5C22×C32×C4C411yescharacteristic
6C2×#638C411yescharacteristic
7C2×#638C2211yescharacteristic
8#3828—22no—
9#3828—22no—
10C2×C4⋊(C3⋊D3)C411yescharacteristic
11C2×C4×C3⋊D3C411yescharacteristic
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×#634C2211yescharacteristic
14#3792C2214yes—
15#3788—22no—
16#3787—22no—
17C2×#235—62no—
18C23×C32C2×C411yescharacteristic
19C22×C3⋊D3C2×C411yescharacteristic
20C22×C3⋊D3C2×C411yescharacteristic
21C2×C32×C4—21no—
22C2×C32×C4—21no—
23C2×C32×C4C2×C411yescharacteristic
24C2×C32×C4C2×C411yescharacteristic
25#638—22no—
26#638—22no—
27C2×C3⋊Dic3C2×C411yescharacteristic
28C2×C3⋊Dic3C2311yescharacteristic
29C4⋊(C3⋊D3)—22no—
30C4×C3⋊D3—21no—
31C4×C3⋊D3—21no—
32C4×C3⋊D3D412yes—
33C4×C3⋊D3D412yes—
34#634—22no—
35#622—24no—
36#413—91nomaximal, Sylow
37C22×C3×C4—22no—
38C2×#67—62no—
39C2×#67—62no—
40#235—68no—
41#235—68no—
42C2×D12—62no—
43C2×C4×D3—62no—
44C2×C4×D3—62no—
45C2×Dic6—62no—
46C22×C32—21no—
47C22×C32—41no—
48C22×C32C22×C411yescharacteristic
49C2×C3⋊D3—21no—
50C2×C3⋊D3—41no—
51C2×C3⋊D3—41no—
52C2×C3⋊D3C22⋊C412yes—
53C32×C4—21no—
54C32×C4C22⋊C411yescharacteristic
55C32×C4C22⋊C411yescharacteristic
56C3⋊Dic3—21no—
57C3⋊Dic3C2×D412yes—
58C2×#41—91no—
59C2×M16—92no—
60#101—92no—
61#100—92no—
62C23×C3—22no—
63C22×D3—62no—
64C22×D3—62no—
65C2×C3×C4—24no—
66C2×C3×C4—24no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69#67—68no—
70#67—68no—
71C2×Dic3—62no—
72C2×Dic3—62no—
73D12—68no—
74C4×D3—64no—
75C4×D3—64no—
76C4×D3—64no—
77C4×D3—64no—
78Dic6—68no—
79C2×C32—21no—
80C2×C32—41no—
81C2×C32C2×C22⋊C411yescharacteristic
82C3⋊D3—22no—
83C3⋊D3—41no—
84#41—182no—
85#41—182no—
86C2×Q8—91no—
87C2×D4—91no—
88C2×D4—91no—
89C2×D4—91no—
90C22×C4—91no—
91C22×C4—91no—
92C22×C4#15711yescharacteristic
93M16—94no—
94M16—182no—
95C2×C8—182no—
96C22×C3—24no—
97C22×C3—22no—
98C22×C3—44no—
99C2×D3—64no—
100C2×D3—64no—
101C2×D3—124no—
102C2×D3—124no—
103C3×C4—24no—
104C3×C4—22no—
105C3×C4—22no—
106Dic3—64no—
107Dic3—64no—
108C32#41311yescharacteristic, complemented, Sylow
109C23—91no—
110C23—91no—
111C23C2×#15711yescharacteristic
112Q8—182no—
113D4—182no—
114D4—182no—
115D4—182no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—91no—
119C2×C4—92no—
120C2×C4—92no—
121C2×C4—91no—
122C2×C4—181no—
123C2×C4—181no—
124C2×C4C2×#15711yescharacteristic
125C2×C4C2×#15711yescharacteristic
126C8—184no—
127C2×C3—24no—
128C2×C3—22no—
129C2×C3—44no—
130D3—124no—
131D3—124no—
132C22—21no—
133C22—41no—
134C22—92no—
135C22—181no—
136C22—361no—
137C22—361no—
138C22C22×#15711yescharacteristic
139C4—21no—
140C4—92no—
141C4—181no—
142C4C22⋊#15711yescharacteristic
143C4C22⋊#15711yescharacteristic
144C3—22no—
145C2—21no—
146C2—41no—
147C2—182no—
148C2—361no—
149C2C2×C22⋊#15711yescharacteristic
subgroupG/H
centerC4C22⋊#157
commutatorC22×C32C22×C4
FrattiniC22C22×#157
FittingC22×C32×C4C4
socleC2×C32C2×C22⋊C4
prime radicalC2×#3828C2
cosocleC2×C3⋊Dic3C23
hypercenterC22×C4#157
hypocenterC32#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #64283C2semidirect
3. #63788C2semidirect
4. #63780C2semidirect
108. C32#413semidirect

Maximal subgroups

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

subgroupindex
C2×#38282
#642832
#637882
#637802
#4139