Finite Group Databasebeta

← #135474

Subgroups — #135474

Order 512.

524 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 70 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#14709C211yesmaximal, characteristic
2#8554C211yesmaximal, characteristic
3#8554C211yesmaximal, characteristic
4#7215C214yesmaximal
5C22×#125C2211yescharacteristic
6C2×#382C411yescharacteristic
7C2×C43C411yescharacteristic
8#1821—21no—
9#1821C2212yes—
10#1463—21no—
11#1463—21no—
12#1463—21no—
13#1463C2212yes—
14#1436—21no—
15#1436—21no—
16#1436C412yes—
17#1436C2212yes—
18C4⋊#95—21no—
19C4⋊#95—21no—
20C2×#125—24no—
21C2×#125—42no—
22C2×#125—41no—
23C2×#125—41no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42—21no—
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C412yes—
35#407—22no—
36#407—22no—
37#382—41no—
38#382—41no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—21no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95C2×C412yes—
45C43—42no—
46#343—24no—
47#343—42no—
48#324—24no—
49#324—41no—
50#324—41no—
51#324—42no—
52#324—41no—
53#324—41no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C2×M16—42no—
64C2×M16—42no—
65#125—48no—
66#125—48no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4#4114yes—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—24no—
96C2×C42—42no—
97C2×C42—42no—
98C2×C42—42no—
99C2×C42—44no—
100C2×C42—42no—
101C2×C42—44no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#95—42no—
107#95—44no—
108#95—44no—
109#95—44no—
110#95—44no—
111#95—42no—
112C24C2×C22⋊C411yescharacteristic
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4#9911yescharacteristic
152C22×C4#9911yescharacteristic
153C22×C4#9912yes—
154C22×C4#11712yes—
155C22×C4#12312yes—
156C22×C4#12312yes—
157M16—84no—
158M16—84no—
159C2×C8—82no—
160C2×C8—82no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C42—24no—
172C42—24no—
173C42—48no—
174C42—44no—
175C42—48no—
176C42—44no—
177C23—22no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23#38111yescharacteristic
182C23C2×#9911yescharacteristic
183C23C2×#9911yescharacteristic
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—28no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#35512yes—
229C2×C4#35612yes—
230C8—162no—
231C8—162no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22#181911yescharacteristic
243C22#182712yes—
244C22#212012yes—
245C22C2×#35511yescharacteristic
246C22C2×#35611yescharacteristic
247C4—22no—
248C4—22no—
249C4—48no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—81no—
255C4—81no—
256C4—162no—
257C2—22no—
258C2—21no—
259C2—22no—
260C2—21no—
261C2#873811yescharacteristic
262C2#1285311yescharacteristic
263C2#1286011yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#147092
#85542
#72152