Finite Group Databasebeta

← #124877

Subgroups — #124877

Order 512.

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

normal subgroups only

This group has 742 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#13965C211yesmaximal, characteristic
2#7909C212yesmaximal
3#7123C214yesmaximal, complemented
4C2×#407—22no—
5C2×#385—22no—
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1444C412yes—
10#1444C2212yes—
11C4⋊#95—22no—
12#1280—22no—
13#1280C2214yes—
14C24×C4C2×C411yescharacteristic
15C24×C4D412yes—
16C22×M16—42no—
17C22×C4⋊C4—22no—
18C22×C4⋊C4—22no—
19C22×C4⋊C4C2×C412yes—
20C22×C22⋊C4—21no—
21C22×C22⋊C4—21no—
22C22×C42—21no—
23#407—28no—
24#407—44no—
25C2×C22⋊C8—24no—
26C2×C22⋊C8—42no—
27#385—48no—
28#380—41no—
29#380—41no—
30C4×C22⋊C4—24no—
31C2×#95—21no—
32C2×#95—22no—
33C2×#95—22no—
34C2×#95—41no—
35C2×#95—41no—
36C2×#95C2×C412yes—
37C2×#95C2×C411yescharacteristic
38C2×#95D412yes—
39C2×#95C2311yescharacteristic
40C25C22⋊C411yescharacteristic
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—24no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—44no—
53C23×C4—82no—
54C23×C4—82no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C412yes—
57C23×C4C22×C411yescharacteristic
58C23×C4C2×D412yes—
59C2×M16—48no—
60C2×M16—84no—
61C2×M16—84no—
62C22×C8—44no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4—44no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—82no—
70C2×C4⋊C4—82no—
71C2×C4⋊C4—82no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—44no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C42—24no—
79C2×C42—82no—
80C22⋊C8—28no—
81C22⋊C8—48no—
82C22⋊C8—44no—
83#95—22no—
84#95—41no—
85#95—42no—
86#95—41no—
87#95—42no—
88#95—44no—
89#95—44no—
90#95M1614yescomplemented
91C24—22no—
92C24—22no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—22no—
97C24—42no—
98C24#9911yescharacteristic
99C24#9911yescharacteristic
100C24C2×C22⋊C411yescharacteristic
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—44no—
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—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—84no—
138C22×C4—88no—
139C22×C4—84no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4—84no—
145C22×C4—84no—
146C22×C4#10112yes—
147C22×C4#10414yes—
148C22×C4C2×M1612yes—
149C22×C4AGL(1,ℤ/8)12yes—
150C22×C4#13712yes—
151M16—816no—
152C2×C8—48no—
153C2×C8—84no—
154C2×C8—84no—
155C4⋊C4—88no—
156C4⋊C4—84no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C22⋊C4—216no—
160C22⋊C4—82no—
161C22⋊C4—82no—
162C42—84no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23—24no—
167C23—24no—
168C23—22no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—22no—
175C23—22no—
176C23—44no—
177C23—44no—
178C23—44no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23#35312yes—
186C23#35612yes—
187C23#40711yescharacteristic
188C23C2×#9911yescharacteristic
189C23C2×#10111yescharacteristic
190C23#41712yes—
191C23C2×#10412yes—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—48no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—48no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C8—88no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—22no—
239C22—21no—
240C22—21no—
241C22—24no—
242C22—22no—
243C22—22no—
244C22—24no—
245C22—48no—
246C22—44no—
247C22—44no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22#146311yescharacteristic
255C22#148912yes—
256C22#150412yes—
257C22#152611yescharacteristic
258C22#156111yescharacteristic
259C22#212112yes—
260C22C2×#35311yescharacteristic
261C22C2×#35611yescharacteristic
262C4—48no—
263C4—42no—
264C4—44no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—162no—
269C2—22no—
270C2—21no—
271C2—21no—
272C2—42no—
273C2—42no—
274C2#763211yescharacteristic
275C2#1151211yescharacteristic
276C2#1151911yescharacteristic
277C2#1175712yes—
278C2#1176812yes—
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. #7123C2semidirect
90. #95M16semidirect

Maximal subgroups

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

subgroupindex
#139652
#79092
#71232