Finite Group Databasebeta

← #124859

Subgroups — #124859

Order 512.

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

normal subgroups only

This group has 582 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#14161C211yesmaximal, characteristic
2#8153C212yesmaximal
3#7131C214yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C2×C4×C4⋊C4C411yescharacteristic
6C2×C4×C4⋊C4C2211yescharacteristic
7#1749—22no—
8C2×#324—22no—
9#1446C412yes—
10#1446C2212yes—
11C4⋊#95—22no—
12C4⋊#95—22no—
13#1280C2214yes—
14C23×Q8C2×C412yes—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—41no—
17C2×C4⋊Q8—41no—
18C2×C4⋊Q8—42no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4D412yes—
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C22×C42D412yes—
26C2×C22⋊C8—24no—
27C2×#97—42no—
28C4×C4⋊C4—41no—
29C4×C4⋊C4—41no—
30C4×C4⋊C4—42no—
31C2×#95—24no—
32C2×#95—24no—
33C2×#95C2×C412yes—
34C2×#95C2×C411yescharacteristic
35C2×#95C2311yescharacteristic
36#336—42no—
37#324—28no—
38C22×Q8—22no—
39C22×Q8—42no—
40C22×Q8—42no—
41C22×Q8—84no—
42C22×Q8M1614yes—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4C22⋊C411yescharacteristic
46C23×C4C22⋊C412yes—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C22×C411yescharacteristic
49C23×C4C2×D412yes—
50C22×C8—44no—
51C4⋊Q8—48no—
52C4⋊Q8—48no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—42no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—44no—
73C2×C42—42no—
74C2×C42—44no—
75C22⋊C8—28no—
76C22⋊C8—44no—
77#97—84no—
78#95—41no—
79#95—41no—
80#95—42no—
81#95—84no—
82#95—84no—
83C24C2×C22⋊C411yescharacteristic
84C2×Q8—44no—
85C2×Q8—44no—
86C2×Q8—84no—
87C2×Q8—84no—
88C2×Q8—88no—
89C2×Q8—88no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4#9911yescharacteristic
120C22×C4#9911yescharacteristic
121C22×C4#10012yes—
122C22×C4#10414yes—
123C22×C4C2×M1612yes—
124C22×C4#13714yes—
125C2×C8—48no—
126C2×C8—84no—
127C2×C8—84no—
128C4⋊C4—48no—
129C4⋊C4—48no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—48no—
133C42—24no—
134C42—24no—
135C42—44no—
136C42—82no—
137C42—84no—
138C23—22no—
139C23—21no—
140C23—21no—
141C23#40711yescharacteristic
142C23C2×#9911yescharacteristic
143C23C2×#10011yescharacteristic
144C23#41912yes—
145C23C2×#10412yes—
146Q8—88no—
147Q8—88no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—48no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—82no—
176C2×C4—84no—
177C2×C4—82no—
178C2×C4—84no—
179C2×C4—84no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4#32914yes—
183C2×C4#35412yes—
184C2×C4#35612yes—
185C8—88no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22#146211yescharacteristic
195C22#149112yes—
196C22#150812yes—
197C22C2×#32911yescharacteristic
198C22#155811yescharacteristic
199C22#212212yes—
200C22C2×#35411yescharacteristic
201C22C2×#35611yescharacteristic
202C4—24no—
203C4—22no—
204C4—22no—
205C4—44no—
206C4—42no—
207C4—84no—
208C4—82no—
209C4—84no—
210C2—21no—
211C2—21no—
212C2—22no—
213C2#762911yescharacteristic
214C2#1150311yescharacteristic
215C2#1150611yescharacteristic
216C2#1178712yes—
217C2#1179512yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#141612
#81532
#71312