Finite Group Databasebeta

← #93202

Subgroups — #93202

Order 512.

282 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 26 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#8462C211yesmaximal, characteristic
2#7058C212yesmaximal, complemented
3#3485C411yescharacteristic
4#1782C411yescharacteristic
5#1780C2211yescharacteristic
6#1516—22no—
7#1314—22no—
8C2×H(5,𝔽2)C811yescharacteristic
9C22×#41C2×C411yescharacteristic
10#547—42no—
11#546—42no—
12#536—21no—
13#534—41no—
14#525—21no—
15C2×#123—41no—
16#518C811yescharacteristic
17C2×#99—41no—
18C2×C22⋊C8—41no—
19C2×C22⋊C8D412yes—
20#381—21no—
21C22⋊C16—42no—
22#328—42no—
23#327—42no—
24H(5,𝔽2)—82no—
25C2×#41—21no—
26C2×#41—42no—
27C2×#41—41no—
28C2×#41—82no—
29C22×Q8M1611yescharacteristic
30C22×D4—21no—
31C22×D4—41no—
32C22×D4—41no—
33C22×D4C2×C811yescharacteristic
34C23×C4—21no—
35C23×C4C22⋊C411yescharacteristic
36C22×C8—22no—
37C4⋊Q8—41no—
38#124—41no—
39#123—42no—
40#123—44no—
41#123—42no—
42#123—41no—
43C22⋊Q8—22no—
44C22⋊Q8—42no—
45#121—22no—
46#121—44no—
47#120—42no—
48#120—41no—
49C4×D4—42no—
50C4×D4—42no—
51#117—21no—
52C2×C4⋊C4—42no—
53C2×C22⋊C4—41no—
54C2×C22⋊C4—42no—
55C2×C22⋊C4—41no—
56C2×C16—82no—
57#99—82no—
58C22⋊C8—42no—
59C22⋊C8—82no—
60#95—42no—
61C24—21no—
62C24—41no—
63C24#10011yescharacteristic
64#41—82no—
65#41—84no—
66#41—82no—
67#41—84no—
68#41—82no—
69C2×Q8—21no—
70C2×Q8—41no—
71C2×Q8—81no—
72C2×D4—21no—
73C2×D4—41no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—41no—
79C2×D4—42no—
80C2×D4—81no—
81C2×D4—82no—
82C2×D4—82no—
83C2×D4—82no—
84C2×D4—82no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—41no—
91C22×C4—42no—
92C22×C4—81no—
93C22×C4—81no—
94C22×C4—81no—
95C22×C4C22⋊C811yescharacteristic
96C22×C4#9911yescharacteristic
97C2×C8—42no—
98C2×C8—82no—
99C4⋊C4—22no—
100C4⋊C4—44no—
101C4⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—44no—
105C22⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—82no—
108C42—41no—
109C16—162no—
110C23—22no—
111C23—21no—
112C23—41no—
113C23—41no—
114C23—42no—
115C23—41no—
116C23—41no—
117C23—82no—
118C23—82no—
119C23—162no—
120C23#32311yescharacteristic
121Q8—81no—
122Q8—82no—
123D4—42no—
124D4—42no—
125D4—81no—
126D4—82no—
127D4—84no—
128D4—84no—
129D4—162no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—82no—
139C2×C4—84no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—82no—
143C2×C4—162no—
144C2×C4—161no—
145C2×C4—162no—
146C2×C4AΣL(1,𝔽16)12yes—
147C8—82no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—42no—
153C22—82no—
154C22—82no—
155C22—81no—
156C22—82no—
157C22—161no—
158C22—161no—
159C22—322no—
160C22#131611yescharacteristic
161C4—22no—
162C4—41no—
163C4—81no—
164C4—81no—
165C4—82no—
166C4—321no—
167C2—41no—
168C2—81no—
169C2—82no—
170C2—161no—
171C2#733811yescharacteristic
172C2#734612yes—
subgroupG/H
center, socleC22#1316
commutatorC22×D4C2×C8
Frattini, cosocle#1780C22
prime radical#3485C4
LCSG ▷ C22×D4 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1780 ◁ G
LECSG ▷ #1780 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7058C2semidirect

Maximal subgroups

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

subgroupindex
#84622
#70582