Finite Group Databasebeta

← #346279

Subgroups — #346279

Order 512.

591 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 96 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#10393C218yesmaximal, complemented
2#10392C214yesmaximal
3#8543C211yesmaximal, characteristic
4#8543C212yesmaximal
5C22×#123C2211yescharacteristic
6#2842C2214yes—
7#1819C2214yes—
8#1819C2214yes—
9#1819C2214yes—
10#1781C2212yes—
11#1781C2214yes—
12#1304C22112yes—
13C23×D4C2311yescharacteristic
14C24×C4Q811yescharacteristic
15C2×#123—44no—
16C2×#123—42no—
17C22×C4⋊C4Q812yes—
18C22×C22⋊C4Q811yescharacteristic
19C22×C22⋊C4C2312yes—
20C2×#99—44no—
21C2×#99—42no—
22#399—44no—
23#396—48no—
24#379C2314yes—
25#379C2318yes—
26#342—44no—
27#342—44no—
28#342—44no—
29C25C2×Q811yescharacteristic
30C25C2411yescharacteristic
31C22×D4—24no—
32C22×D4—22no—
33C22×D4—82no—
34C23×C4—42no—
35C23×C4—41no—
36C23×C4—82no—
37C23×C4C2×Q812yes—
38C23×C4C2×Q812yes—
39C23×C4C2×Q811yescharacteristic
40#123—416no—
41C2×C4⋊C4—42no—
42C2×C4⋊C4—42no—
43C2×C4⋊C4—42no—
44C2×C4⋊C4—44no—
45C2×C4⋊C4—48no—
46C2×C22⋊C4—24no—
47C2×C22⋊C4—24no—
48C2×C22⋊C4—24no—
49C2×C22⋊C4—28no—
50C2×C22⋊C4—28no—
51C2×C22⋊C4—28no—
52C2×C22⋊C4—28no—
53C2×C22⋊C4—24no—
54C2×C22⋊C4—42no—
55C2×C22⋊C4—41no—
56#99—88no—
57#99—84no—
58#95—44no—
59C24—22no—
60C24—21no—
61C24—42no—
62C24—81no—
63C24—82no—
64C24—82no—
65C24C22×Q811yescharacteristic
66C24H(5,𝔽2)14yes—
67C24H(5,𝔽2)12yes—
68C2×D4—48no—
69C2×D4—44no—
70C2×D4—44no—
71C2×D4—84no—
72C2×D4—82no—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—44no—
82C22×C4—44no—
83C22×C4—48no—
84C22×C4—81no—
85C22×C4—84no—
86C22×C4—82no—
87C22×C4—84no—
88C22×C4—82no—
89C22×C4—82no—
90C22×C4—82no—
91C4⋊C4—48no—
92C4⋊C4—164no—
93C4⋊C4—168no—
94C22⋊C4—44no—
95C22⋊C4—48no—
96C22⋊C4—88no—
97C22⋊C4—84no—
98C22⋊C4—88no—
99C22⋊C4—88no—
100C22⋊C4—88no—
101C23—24no—
102C23—22no—
103C23—41no—
104C23—42no—
105C23—44no—
106C23—42no—
107C23—44no—
108C23—44no—
109C23—44no—
110C23—42no—
111C23—41no—
112C23—41no—
113C23—82no—
114C23—82no—
115C23—82no—
116C23—88no—
117C23—164no—
118C23#54312yes—
119C23#54311yescharacteristic
120C23H(3,𝔽4)14yes—
121D4—88no—
122C2×C4—44no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—41no—
127C2×C4—88no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—88no—
135C2×C4—84no—
136C2×C4—88no—
137C2×C4—88no—
138C2×C4—88no—
139C22—22no—
140C22—21no—
141C22—24no—
142C22—28no—
143C22—22no—
144C22—22no—
145C22—22no—
146C22—44no—
147C22—42no—
148C22—84no—
149C22—82no—
150C22—84no—
151C22—82no—
152C22—82no—
153C22—164no—
154C22—162no—
155C22#284011yescharacteristic
156C4—81no—
157C4—82no—
158C4—82no—
159C4—168no—
160C4—164no—
161C2—44no—
162C2—42no—
163C2—41no—
164C2—82no—
165C2#3287311yescharacteristic
166C2#3287412yes—
subgroupG/H
center, socleC22#2840
commutator, Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #10393C2semidirect

Maximal subgroups

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

subgroupindex
#103932
#103922
#85432