Finite Group Databasebeta

← #10590057

Subgroups — #10590057

Order 576.

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

normal subgroups only

This group has 182 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#63923C211yesmaximal, characteristic
2#63774C211yesmaximal, characteristic, complemented
3#63559C211yesmaximal, characteristic, complemented
4C22×D32C411yescharacteristic, complemented
5C2×C4⋊(C3⋊D3)C2211yescharacteristic
6C2×#626—21no—
7C22⋊#157—21no—
8C3×(C2×D3)⋊C4—21no—
9C3×#208—21no—
10#3722C411yescharacteristic, complemented
11#882—61no—
12#823—61no—
13C22×C3⋊D3C2×C411yescharacteristic
14C22×C3⋊D3D411yescharacteristic
15C22×C3×D3—21no—
16C2×D32—41no—
17C2×D32—42no—
18C2×#157—41no—
19C2×C32×C4D411yescharacteristic
20C4⋊(C3⋊D3)—41no—
21C2×C3×Dic3—21no—
22#626—42no—
23C3×C3⋊C8—41no—
24AΣL(1,𝔽16)—91nomaximal, Sylow
25C23×D3—61no—
26C2×#67—61no—
27C2×D12—61no—
28C2×D12—61no—
29C2×D12—61no—
30C3×M16—61no—
31C3×C22⋊C4—61no—
32(C2×D3)⋊C4—21no—
33(C2×D3)⋊C4—61no—
34#208—21no—
35C22×C32C22⋊C411yescharacteristic
36C2×C3⋊D3—21no—
37C2×C3⋊D3—21no—
38C2×C3⋊D3—41no—
39C2×C3⋊D3—41no—
40C2×C3×D3—22no—
41C2×C3×D3—41no—
42C2×C3×D3—41no—
43D32—42no—
44D32—42no—
45#157—81no—
46C32×C4—21no—
47C3×Dic3—41no—
48#120—91no—
49#100—91no—
50#99—91no—
51C23×C3—61no—
52C22×D3—21no—
53C22×D3—61no—
54C22×D3—61no—
55C22×D3—61no—
56C22×D3—61no—
57C22×D3—122no—
58C22×D3—122no—
59C22×D3—122no—
60C2×C3×C4—21no—
61C2×C3×C4—21no—
62C2×C3×C4—61no—
63#67—122no—
64C2×Dic3—21no—
65D12—121no—
66D12—121no—
67D12—122no—
68D12—122no—
69C3×C8—121no—
70C3⋊C8—41no—
71C2×C32—21no—
72C2×C32#9911yescharacteristic
73C3⋊D3—41no—
74C3⋊D3—41no—
75C3×D3—42no—
76C24—91no—
77C2×D4—91no—
78C2×D4—181no—
79M16—61no—
80C22⋊C4—61no—
81C22⋊C4—91no—
82C22⋊C4—181no—
83C22×C3—21no—
84C22×C3—21no—
85C22×C3—62no—
86C22×C3—121no—
87C22×C3—121no—
88C2×D3—22no—
89C2×D3—41no—
90C2×D3—41no—
91C2×D3—62no—
92C2×D3—121no—
93C2×D3—121no—
94C2×D3—121no—
95C2×D3—124no—
96C2×D3—124no—
97C2×D3—121no—
98C2×D3—121no—
99C2×D3—121no—
100C2×D3—121no—
101C2×D3—241no—
102C2×D3—241no—
103C3×C4—21no—
104C3×C4—21no—
105C3×C4—41no—
106C3×C4—121no—
107Dic3—41no—
108C32AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
109C23—61no—
110C23—91no—
111C23—91no—
112C23—362no—
113C23—361no—
114D4—361no—
115D4—362no—
116C2×C4—61no—
117C2×C4—361no—
118C2×C4GO+(4,𝔽2)11yescharacteristic
119C8—121no—
120C2×C3—21no—
121C2×C3—21no—
122C2×C3—21no—
123C2×C3—21no—
124C2×C3—41no—
125C2×C3—122no—
126D3—42no—
127D3—122no—
128D3—241no—
129D3—241no—
130D3—241no—
131C22—62no—
132C22—121no—
133C22—121no—
134C22—181no—
135C22—181no—
136C22—361no—
137C22—361no—
138C22—362no—
139C22—362no—
140C22D32⋊C411yescharacteristic
141C4—21no—
142C4—121no—
143C4—721no—
144C3—21no—
145C3—21no—
146C2—21no—
147C2—122no—
148C2—361no—
149C2—361no—
150C2#6373811yescharacteristic
subgroupG/H
centerC2#63738
commutatorC22×C3⋊D3C2×C4
Frattini, hypercenterC2×C4GO+(4,𝔽2)
FittingC2×C32×C4D4
socleC2×C32#99
prime radical#63923C2
cosocleC2×C4⋊(C3⋊D3)C22
hypocenterC32AΣL(1,𝔽16)
LCSG ▷ C22×C3⋊D3 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #63774C2semidirect
3. #63559C2semidirect
4. C22×D32C4semidirect
10. #3722C4semidirect
108. C32AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
#639232
#637742
#635592
AΣL(1,𝔽16)9