Finite Group Databasebeta

← #99159

Subgroups — #99159

Order 512.

686 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 90 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#14339C211yesmaximal, characteristic
2#8299C212yesmaximal
3#7032C214yesmaximal
4C23×C4⋊C4C2211yescharacteristic
5C2×#381C412yes—
6C2×#380C412yes—
7#1779C2214yes—
8C2×#324—22no—
9C2×#324—22no—
10#1441—22no—
11#1441—22no—
12C22⋊#95C2212yes—
13C24×C4C2×C411yescharacteristic
14C24×C4C2×C411yescharacteristic
15C24×C4C2311yescharacteristic
16C22×C4⋊C4—24no—
17C22×C4⋊C4—22no—
18C22×C4⋊C4—44no—
19C22×C4⋊C4D412yes—
20C22×C4⋊C4D412yes—
21C22×C22⋊C4C2×C414yes—
22C2×C22⋊C8—24no—
23C2×C22⋊C8—28no—
24#381—44no—
25#380—44no—
26C2×#95—22no—
27C2×#95—22no—
28C2×#95—22no—
29C2×#95—24no—
30#324—44no—
31#324—44no—
32C25C22×C411yescharacteristic
33C23×C4—22no—
34C23×C4—22no—
35C23×C4—22no—
36C23×C4—22no—
37C23×C4—24no—
38C23×C4—24no—
39C23×C4—24no—
40C23×C4—42no—
41C23×C4—42no—
42C23×C4—82no—
43C23×C4—82no—
44C23×C4C22⋊C411yescharacteristic
45C23×C4C22⋊C411yescharacteristic
46C23×C4C22⋊C412yes—
47C23×C4C2×D412yes—
48C23×C4#4114yes—
49C22×C8—44no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—416no—
57C2×C4⋊C4—44no—
58C2×C4⋊C4—84no—
59C2×C22⋊C4—28no—
60C2×C22⋊C4—44no—
61C2×C22⋊C4—44no—
62C22⋊C8—44no—
63C22⋊C8—44no—
64C22⋊C8—88no—
65#95—44no—
66#95—44no—
67#95—44no—
68#95—84no—
69C24—22no—
70C24—21no—
71C24—21no—
72C24—42no—
73C24—42no—
74C24C2×C22⋊C411yescharacteristic
75C24#11712yes—
76C24#13114yes—
77C22×C4—28no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—28no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—44no—
92C22×C4—48no—
93C22×C4—44no—
94C22×C4—48no—
95C22×C4—48no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4—48no—
100C22×C4—48no—
101C22×C4—44no—
102C22×C4—44no—
103C22×C4—44no—
104C22×C4—84no—
105C22×C4—88no—
106C22×C4—88no—
107C22×C4#9914yes—
108C22×C4#10012yes—
109C22×C4#10112yes—
110C22×C4#12314yes—
111C2×C8—84no—
112C2×C8—88no—
113C4⋊C4—48no—
114C4⋊C4—48no—
115C4⋊C4—88no—
116C22⋊C4—88no—
117C23—22no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—24no—
122C23—24no—
123C23—24no—
124C23—24no—
125C23—48no—
126C23—44no—
127C23—44no—
128C23—44no—
129C23—44no—
130C23#38111yescharacteristic
131C23#40812yes—
132C23C2×#9912yes—
133C23C2×#10011yescharacteristic
134C23C2×#10111yescharacteristic
135C23#43314yes—
136C2×C4—24no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—28no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—416no—
144C2×C4—44no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—44no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—88no—
156C2×C4—88no—
157C2×C4—84no—
158C2×C4—88no—
159C2×C4—84no—
160C2×C4—88no—
161C2×C4—84no—
162C8—164no—
163C22—24no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—24no—
169C22—28no—
170C22—48no—
171C22—48no—
172C22—42no—
173C22—42no—
174C22—44no—
175C22#147112yes—
176C22#147212yes—
177C22#181811yescharacteristic
178C22#181911yescharacteristic
179C22#182111yescharacteristic
180C22#182812yes—
181C22#183012yes—
182C4—42no—
183C4—42no—
184C4—44no—
185C4—44no—
186C4—164no—
187C2—22no—
188C2—22no—
189C2—44no—
190C2#870411yescharacteristic
191C2#873712yes—
192C2#874611yescharacteristic
193C2#875411yescharacteristic
194C2#875712yes—
subgroupG/H
center, socleC23#381
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#143392
#82992
#70322