Finite Group Databasebeta

← #124873

Subgroups — #124873

Order 512.

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

normal subgroups only

This group has 534 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#14170C211yesmaximal, characteristic
2#7910C212yesmaximal
3#7132C212yesmaximal
4#7104C212yesmaximal
5C22×#125C411yescharacteristic
6C2×#432—22no—
7C2×#376C411yescharacteristic
8C2×#376C2211yescharacteristic
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—41no—
20C2×#125—41no—
21C2×#125—42no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C42—21no—
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29#432—28no—
30#432—44no—
31C2×C4⋊C8—42no—
32C2×C22⋊C8—24no—
33C2×#97—42no—
34#376—41no—
35#376—41no—
36#376—42no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95C2×C411yescharacteristic
43C2×#95C2×C411yescharacteristic
44C2×#95C2×C411yescharacteristic
45C2×#95D412yes—
46C2×#95C2311yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22⋊C412yes—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22×C411yescharacteristic
55C23×C4C2×D412yes—
56C22×C8—44no—
57#125—44no—
58#125—44no—
59#125—44no—
60#125—44no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4M1612yes—
75C2×C4⋊C4M1612yes—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—24no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42—42no—
89C4⋊C8—48no—
90C22⋊C8—28no—
91C22⋊C8—44no—
92#97—48no—
93#95—41no—
94#95—42no—
95#95—41no—
96#95—41no—
97#95—41no—
98#95—44no—
99#95—44no—
100#95—42no—
101#95—44no—
102#95—42no—
103#95—42no—
104C24C2×C22⋊C411yescharacteristic
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—42no—
124C22×C4—42no—
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—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4#9911yescharacteristic
145C22×C4#9911yescharacteristic
146C22×C4#10012yes—
147C22×C4#10414yes—
148C22×C4C2×M1611yescharacteristic
149C22×C4C2×M1611yescharacteristic
150C22×C4AGL(1,ℤ/8)12yes—
151C22×C4#13712yes—
152C2×C8—48no—
153C2×C8—84no—
154C2×C8—84no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—48no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C42—24no—
163C42—24no—
164C42—24no—
165C42—24no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—48no—
171C23—22no—
172C23—21no—
173C23—21no—
174C23#40711yescharacteristic
175C23C2×#9911yescharacteristic
176C23C2×#10011yescharacteristic
177C23#41712yes—
178C23C2×#10412yes—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
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—42no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—81no—
229C8—88no—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—22no—
234C22—21no—
235C22—21no—
236C22—22no—
237C22—22no—
238C22#146211yescharacteristic
239C22#148912yes—
240C22#150412yes—
241C22#152511yescharacteristic
242C22#156011yescharacteristic
243C22#212312yes—
244C22#212611yescharacteristic
245C22#213211yescharacteristic
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—44no—
254C4—82no—
255C4—82no—
256C4—82no—
257C4—82no—
258C2—21no—
259C2—21no—
260C2—22no—
261C2#763111yescharacteristic
262C2#1150411yescharacteristic
263C2#1150711yescharacteristic
264C2#1176012yes—
265C2#1176712yes—
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
#79102
#71322
#71042