Finite Group Databasebeta

← #124869

Subgroups — #124869

Order 512.

566 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 82 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#13961C211yesmaximal, characteristic
2#7910C212yesmaximal
3#7131C212yesmaximal
4#7104C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#432—22no—
7C2×#383C411yescharacteristic
8C2×#376C2211yescharacteristic
9#1446C411yescharacteristic
10#1446C2211yescharacteristic
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1436—22no—
14#1434—22no—
15#1280—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C2×C4×Q8—22no—
19C2×C4×Q8—22no—
20C2×C4×Q8—42no—
21C2×C4×Q8—42no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C42—21no—
25C22×C42C2×C411yescharacteristic
26C22×C42D412yes—
27#432—28no—
28#432—44no—
29C2×C4⋊C8—42no—
30C2×C22⋊C8—24no—
31C2×#97—42no—
32#383—42no—
33#376—42no—
34C2×#95—21no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—21no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95D412yes—
44C2×#95C2311yescharacteristic
45C22×Q8—21no—
46C22×Q8—44no—
47C22×Q8—41no—
48C22×Q8—41no—
49C22×Q8M1612yes—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22×C411yescharacteristic
58C23×C4C2×D412yes—
59C22×C8—44no—
60C4×Q8—48no—
61C4×Q8—48no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4M1612yes—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—24no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—24no—
79C2×C42—42no—
80C2×C42—42no—
81C4⋊C8—48no—
82C22⋊C8—28no—
83C22⋊C8—44no—
84#97—48no—
85#95—41no—
86#95—42no—
87#95—41no—
88#95—41no—
89#95—41no—
90#95—44no—
91#95—44no—
92#95—42no—
93#95—42no—
94#95—44no—
95#95—42no—
96#95—42no—
97C24C2×C22⋊C411yescharacteristic
98C2×Q8—44no—
99C2×Q8—42no—
100C2×Q8—44no—
101C2×Q8—42no—
102C2×Q8—48no—
103C2×Q8—44no—
104C2×Q8—44no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4#9912yes—
145C22×C4#10012yes—
146C22×C4#10414yes—
147C22×C4C2×M1611yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C22×C4AGL(1,ℤ/8)12yes—
150C22×C4#13712yes—
151C2×C8—48no—
152C2×C8—84no—
153C2×C8—84no—
154C4⋊C4—42no—
155C4⋊C4—48no—
156C4⋊C4—42no—
157C42—24no—
158C42—24no—
159C42—24no—
160C42—24no—
161C42—24no—
162C42—24no—
163C42—48no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23#40711yescharacteristic
168C23C2×#9911yescharacteristic
169C23C2×#10011yescharacteristic
170C23#41712yes—
171C23C2×#10412yes—
172Q8—44no—
173Q8—44no—
174Q8—48no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C8—88no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22—21no—
232C22—21no—
233C22#146211yescharacteristic
234C22#148912yes—
235C22#150412yes—
236C22#152511yescharacteristic
237C22#156011yescharacteristic
238C22#212012yes—
239C22#212911yescharacteristic
240C22#213511yescharacteristic
241C4—42no—
242C4—42no—
243C4—42no—
244C4—44no—
245C4—41no—
246C4—41no—
247C4—44no—
248C4—42no—
249C4—82no—
250C4—161no—
251C4—161no—
252C2—21no—
253C2—21no—
254C2—22no—
255C2#763111yescharacteristic
256C2#1150111yescharacteristic
257C2#1151011yescharacteristic
258C2#1175612yes—
259C2#1176312yes—
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×#376 ◁ 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
#139612
#79102
#71312
#71042