Finite Group Databasebeta

← #10590078

Subgroups — #10590078

Order 576.

185 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 17 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#63983C211yesmaximal, characteristic
2#63788C211yesmaximal, characteristic, complemented
3#63581C211yesmaximal, characteristic, complemented
4C22×D32C411yescharacteristic, complemented
5C2×#638C2211yescharacteristic
6C2×C3×#67—21no—
7#3792—21no—
8C3×#217—21no—
9#3722C411yescharacteristic, complemented
10(C2×D3)⋊Dic3—21no—
11#937—61no—
12#806—61no—
13C23×C32D411yescharacteristic
14C22×C3⋊D3C2×C411yescharacteristic
15C22×C3×D3—21no—
16C2×D32—41no—
17C2×D32—42no—
18#638—41no—
19C2×C3⋊Dic3D411yescharacteristic
20C3×#67—42no—
21C2×C3×Dic3—21no—
22C2×C3×Dic3—41no—
23#622—41no—
24AΣL(1,𝔽16)—91nomaximal, Sylow
25C23×D3—61no—
26C2×C3×D4—61no—
27C2×#67—21no—
28C2×#67—61no—
29C2×#67—61no—
30C3×C22⋊C4—61no—
31#217—21no—
32#217—61no—
33(C2×D3)⋊C4—61no—
34(C2×D3)⋊C4—61no—
35C22×C32—21no—
36C22×C32—41no—
37C22×C32C22⋊C411yescharacteristic
38C2×C3⋊D3—21no—
39C2×C3⋊D3—41no—
40C2×C3×D3—22no—
41C2×C3×D3—41no—
42C2×C3×D3—41no—
43D32—42no—
44D32—42no—
45C3⋊Dic3—21no—
46C3×Dic3—41no—
47C3×Dic3—81no—
48#120—91no—
49#100—91no—
50#99—91no—
51C23×C3—21no—
52C23×C3—21no—
53C23×C3—61no—
54C22×D3—21no—
55C22×D3—61no—
56C22×D3—61no—
57C22×D3—122no—
58C22×D3—122no—
59C22×D3—122no—
60C3×D4—122no—
61C2×C3×C4—61no—
62C2×C3×C4—121no—
63#67—42no—
64#67—122no—
65#67—121no—
66#67—121no—
67C2×Dic3—21no—
68C2×Dic3—41no—
69C2×Dic3—61no—
70C2×Dic3—61no—
71C2×C32—21no—
72C2×C32—41no—
73C2×C32#9911yescharacteristic
74C3⋊D3—41no—
75C3×D3—42no—
76C24—91no—
77C2×D4—61no—
78C2×D4—91no—
79M16—181no—
80C22⋊C4—61no—
81C22⋊C4—91no—
82C22⋊C4—181no—
83C22×C3—21no—
84C22×C3—21no—
85C22×C3—21no—
86C22×C3—21no—
87C22×C3—41no—
88C22×C3—41no—
89C22×C3—41no—
90C22×C3—62no—
91C22×C3—81no—
92C22×C3—121no—
93C22×C3—121no—
94C2×D3—22no—
95C2×D3—41no—
96C2×D3—41no—
97C2×D3—62no—
98C2×D3—121no—
99C2×D3—124no—
100C2×D3—124no—
101C2×D3—121no—
102C2×D3—121no—
103C2×D3—241no—
104C3×C4—121no—
105C3×C4—241no—
106Dic3—41no—
107Dic3—81no—
108Dic3—121no—
109Dic3—121no—
110C32AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
111C23—61no—
112C23—91no—
113C23—362no—
114C23—361no—
115C23GO+(4,𝔽2)11yescharacteristic
116D4—122no—
117D4—361no—
118C2×C4—61no—
119C2×C4—91no—
120C2×C4—121no—
121C8—361no—
122C2×C3—21no—
123C2×C3—21no—
124C2×C3—21no—
125C2×C3—21no—
126C2×C3—41no—
127C2×C3—41no—
128C2×C3—41no—
129C2×C3—81no—
130C2×C3—122no—
131D3—42no—
132D3—122no—
133D3—241no—
134C22—21no—
135C22—41no—
136C22—62no—
137C22—121no—
138C22—121no—
139C22—181no—
140C22—361no—
141C22—362no—
142C22—362no—
143C22D32⋊C411yescharacteristic
144C4—121no—
145C4—181no—
146C4—241no—
147C3—21no—
148C3—21no—
149C2—21no—
150C2—41no—
151C2—122no—
152C2—361no—
153C2#6373811yescharacteristic
subgroupG/H
centerC2#63738
commutatorC22×C3⋊D3C2×C4
Frattini, hypercenterC23GO+(4,𝔽2)
FittingC23×C32D4
socleC2×C32#99
prime radical#63983C2
cosocleC2×#638C22
hypocenterC32AΣL(1,𝔽16)
LCSG ▷ C22×C3⋊D3 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #63788C2semidirect
3. #63581C2semidirect
4. C22×D32C4semidirect
9. #3722C4semidirect
110. C32AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
#639832
#637882
#635812
AΣL(1,𝔽16)9