Finite Group Databasebeta

← #135539

Subgroups — #135539

Order 512.

574 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 74 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#14161C211yesmaximal, characteristic
2#8551C212yesmaximal
3#7308C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×C4×C4⋊C4C412yes—
6#1784C2214yes—
7#1463—24no—
8#1446C412yes—
9#1446C2212yes—
10C4⋊#95—24no—
11C23×Q8C2×C411yescharacteristic
12C23×Q8C2311yescharacteristic
13C2×C4⋊Q8—24no—
14C2×C4⋊Q8—42no—
15C2×C4⋊Q8—42no—
16C22×C4⋊C4—22no—
17C22×C4⋊C4D414yes—
18C22×C42—22no—
19C22×C42C2×C411yescharacteristic
20C2×#101—28no—
21#407—24no—
22C4×C4⋊C4—42no—
23C4×C4⋊C4—42no—
24C2×#95—24no—
25C2×#95—24no—
26C2×#95C2×C414yes—
27#324—44no—
28#324—44no—
29C22×Q8—22no—
30C22×Q8—42no—
31C22×Q8—42no—
32C22×Q8—82no—
33C22×Q8—82no—
34C22×Q8#4114yes—
35C23×C4—24no—
36C23×C4C22⋊C412yes—
37C23×C4C22⋊C412yes—
38C23×C4C22×C411yescharacteristic
39C23×C4C2×D412yes—
40C2×M16—44no—
41C4⋊Q8—44no—
42C4⋊Q8—48no—
43C4⋊Q8—44no—
44C2×C4⋊C4—28no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—28no—
48C2×C4⋊C4—44no—
49C2×C4⋊C4—44no—
50C2×C4⋊C4—44no—
51C2×C42—22no—
52C2×C42—22no—
53C2×C42—22no—
54C2×C42—48no—
55C2×C42—42no—
56C2×C42—42no—
57#101—88no—
58C22⋊C8—48no—
59#95—44no—
60#95—84no—
61#95—84no—
62C24C2×C22⋊C411yescharacteristic
63C2×Q8—48no—
64C2×Q8—44no—
65C2×Q8—44no—
66C2×Q8—84no—
67C2×Q8—88no—
68C2×Q8—88no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—24no—
72C22×C4—24no—
73C22×C4—22no—
74C22×C4—24no—
75C22×C4—28no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—42no—
79C22×C4—44no—
80C22×C4—48no—
81C22×C4—48no—
82C22×C4—44no—
83C22×C4—44no—
84C22×C4—44no—
85C22×C4#9912yes—
86C22×C4#9912yes—
87C22×C4#11712yes—
88C22×C4#12314yes—
89M16—88no—
90C2×C8—84no—
91C4⋊C4—48no—
92C4⋊C4—48no—
93C4⋊C4—416no—
94C42—22no—
95C42—24no—
96C42—22no—
97C42—84no—
98C42—84no—
99C23—22no—
100C23—22no—
101C23—22no—
102C23#38111yescharacteristic
103C23C2×#9912yes—
104Q8—88no—
105Q8—88no—
106C2×C4—28no—
107C2×C4—28no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—24no—
114C2×C4—28no—
115C2×C4—24no—
116C2×C4—44no—
117C2×C4—48no—
118C2×C4—416no—
119C2×C4—42no—
120C2×C4—44no—
121C2×C4—84no—
122C2×C4—88no—
123C2×C4—84no—
124C2×C4—84no—
125C2×C4—84no—
126C2×C4#35514yes—
127C2×C4#35614yes—
128C8—164no—
129C22—24no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—22no—
134C22—22no—
135C22#181911yescharacteristic
136C22#182812yes—
137C22C2×#35512yes—
138C22C2×#35612yes—
139C4—24no—
140C4—24no—
141C4—44no—
142C4—42no—
143C4—88no—
144C4—82no—
145C2—22no—
146C2—22no—
147C2—22no—
148C2#874811yescharacteristic
149C2#1286012yes—
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#141612
#85512
#73082