Finite Group Databasebeta

← #124995

Subgroups — #124995

Order 512.

478 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 78 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#14170C211yesmaximal, characteristic
2#8699C212yesmaximal
3#7132C212yesmaximal
4#7104C212yesmaximal
5C22×#125C411yescharacteristic
6C2×#376C411yescharacteristic
7C2×#376C2211yescharacteristic
8#1817—22no—
9#1445C411yescharacteristic
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1445C2211yescharacteristic
13#1436—22no—
14#1436—22no—
15#1280—22no—
16#1280C2214yes—
17C2×#125—22no—
18C2×#125—22no—
19C2×#125—42no—
20C2×#125—42no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C42—21no—
26C22×C42C2×C411yescharacteristic
27C22×C42D412yes—
28C2×C22⋊C8—24no—
29#376—42no—
30#376—42no—
31C2×#95—22no—
32C2×#95—22no—
33C2×#95—21no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95C2×C411yescharacteristic
37C2×#95C2×C411yescharacteristic
38C2×#95C2×C411yescharacteristic
39C2×#95D412yes—
40C2×#95C2311yescharacteristic
41#336—44no—
42C23×C4—21no—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4—21no—
46C23×C4C22⋊C412yes—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22×C411yescharacteristic
50C23×C4C2×D412yes—
51C22×C8—44no—
52#125—48no—
53#125—48no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4M1612yes—
67C2×C4⋊C4M1612yes—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—24no—
73C2×C42—24no—
74C2×C42—42no—
75C2×C42—42no—
76C2×C42—42no—
77C2×C42—42no—
78C22⋊C8—44no—
79C22⋊C8—44no—
80#95—41no—
81#95—42no—
82#95—41no—
83#95—41no—
84#95—41no—
85#95—44no—
86#95—44no—
87#95—44no—
88#95—42no—
89#95—44no—
90C24C2×C22⋊C411yescharacteristic
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4#9912yes—
132C22×C4#10012yes—
133C22×C4C2×M1611yescharacteristic
134C22×C4C2×M1611yescharacteristic
135C22×C4#13514yes—
136C2×C8—84no—
137C2×C8—84no—
138C2×C8—84no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—48no—
142C4⋊C4—48no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C42—24no—
146C42—24no—
147C42—48no—
148C42—48no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23#40711yescharacteristic
153C23C2×#9911yescharacteristic
154C23C2×#10011yescharacteristic
155C23#41612yes—
156C23#42112yes—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—82no—
202C2×C4—82no—
203C8—164no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22#146211yescharacteristic
213C22#148812yes—
214C22#151612yes—
215C22#153611yescharacteristic
216C22#154311yescharacteristic
217C22#212312yes—
218C22#212611yescharacteristic
219C22#213211yescharacteristic
220C4—44no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—82no—
226C4—82no—
227C4—82no—
228C4—82no—
229C4—82no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2#764511yescharacteristic
234C2#1150411yescharacteristic
235C2#1150711yescharacteristic
236C2#1184612yes—
237C2#1185312yes—
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
#141702
#86992
#71322
#71042