Finite Group Databasebeta

← #124876

Subgroups — #124876

Order 512.

670 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 86 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#13966C211yesmaximal, characteristic
2#7909C212yesmaximal
3#7122C214yesmaximal, complemented
4C2×#407—22no—
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1442C412yes—
9#1442C2212yes—
10#1434—22no—
11C22⋊#95—22no—
12#1280—22no—
13#1280C2214yes—
14C24×C4C2×C411yescharacteristic
15C24×C4D412yes—
16C22×M16—42no—
17C22×C4⋊C4C2×C412yes—
18C22×C22⋊C4—21no—
19C22×C22⋊C4—21no—
20C22×C42—21no—
21#407—28no—
22#407—44no—
23C2×C22⋊C8—24no—
24C2×C22⋊C8—42no—
25#380—41no—
26#380—41no—
27C4×C22⋊C4—24no—
28C2×#95—22no—
29C2×#95—21no—
30C2×#95—22no—
31C2×#95—22no—
32C2×#95—22no—
33C2×#95—41no—
34C2×#95—41no—
35C2×#95C2×C412yes—
36C2×#95C2×C411yescharacteristic
37C2×#95D412yes—
38C2×#95C2311yescharacteristic
39C25C22⋊C411yescharacteristic
40C23×C4—22no—
41C23×C4—21no—
42C23×C4—21no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—24no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—44no—
52C23×C4—82no—
53C23×C4—82no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22×C411yescharacteristic
57C23×C4C2×D412yes—
58C2×M16—48no—
59C2×M16—84no—
60C2×M16—84no—
61C22×C8—44no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—82no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—44no—
68C2×C22⋊C4—41no—
69C2×C22⋊C4—41no—
70C2×C42—24no—
71C2×C42—82no—
72C22⋊C8—28no—
73C22⋊C8—48no—
74C22⋊C8—44no—
75#95—22no—
76#95—41no—
77#95—42no—
78#95—41no—
79#95—42no—
80#95—44no—
81#95—42no—
82#95—42no—
83#95—82no—
84#95—82no—
85#95M1614yescomplemented
86C24—22no—
87C24—22no—
88C24—21no—
89C24—21no—
90C24—22no—
91C24—22no—
92C24—42no—
93C24#9911yescharacteristic
94C24#9911yescharacteristic
95C24C2×C22⋊C411yescharacteristic
96C22×C4—24no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—84no—
133C22×C4—88no—
134C22×C4—84no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—84no—
140C22×C4—84no—
141C22×C4#10112yes—
142C22×C4#10414yes—
143C22×C4C2×M1612yes—
144C22×C4AGL(1,ℤ/8)12yes—
145C22×C4#13712yes—
146M16—816no—
147C2×C8—48no—
148C2×C8—84no—
149C2×C8—84no—
150C4⋊C4—84no—
151C22⋊C4—48no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C42—84no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23—24no—
159C23—24no—
160C23—22no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23—44no—
169C23—44no—
170C23—44no—
171C23—44no—
172C23—42no—
173C23—42no—
174C23—41no—
175C23—42no—
176C23—41no—
177C23#35412yes—
178C23#35512yes—
179C23#40711yescharacteristic
180C23C2×#9911yescharacteristic
181C23C2×#10111yescharacteristic
182C23#41712yes—
183C23C2×#10412yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—48no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C8—88no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22—48no—
238C22—44no—
239C22—44no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22#146311yescharacteristic
247C22#148912yes—
248C22#150412yes—
249C22#152611yescharacteristic
250C22#156111yescharacteristic
251C22#212112yes—
252C22C2×#35411yescharacteristic
253C22C2×#35511yescharacteristic
254C4—42no—
255C4—44no—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—84no—
260C4—162no—
261C2—22no—
262C2—21no—
263C2—21no—
264C2—42no—
265C2—42no—
266C2#763211yescharacteristic
267C2#1151111yescharacteristic
268C2#1151911yescharacteristic
269C2#1175912yes—
270C2#1176612yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7122C2semidirect
85. #95M16semidirect

Maximal subgroups

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

subgroupindex
#139662
#79092
#71222