Finite Group Databasebeta

← #124990

Subgroups — #124990

Order 512.

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

normal subgroups only

This group has 644 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#13958C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7132C212yesmaximal, complemented
4#7104C212yesmaximal, complemented
5C22×#117C411yescharacteristic
6C2×#381C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1779—22no—
9#1445C411yescharacteristic
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1445C2211yescharacteristic
13#1434—22no—
14C22⋊#95—22no—
15#1280—22no—
16#1280C2214yes—
17C24×C4C2×C411yescharacteristic
18C24×C4D412yes—
19C2×#117—22no—
20C2×#117—22no—
21C2×#117—42no—
22C2×#117—42no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C22⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C42—21no—
28C2×C22⋊C8—24no—
29C2×C22⋊C8—42no—
30C2×C22⋊C8—42no—
31#381—42no—
32C2×#95—22no—
33C2×#95—21no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—42no—
38C2×#95C2×C411yescharacteristic
39C2×#95C2×C411yescharacteristic
40C2×#95C2×C411yescharacteristic
41C2×#95D412yes—
42C2×#95C2311yescharacteristic
43C25C22⋊C411yescharacteristic
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—21no—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—24no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—42no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4—82no—
59C23×C4C22⋊C412yes—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C2×D412yes—
63C22×C8—44no—
64#117—48no—
65#117—48no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4M1612yes—
73C2×C4⋊C4M1612yes—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—44no—
78C2×C22⋊C4—42no—
79C2×C42—24no—
80C2×C42—42no—
81C2×C42—42no—
82C22⋊C8—44no—
83C22⋊C8—44no—
84C22⋊C8—84no—
85C22⋊C8—84no—
86#95—41no—
87#95—42no—
88#95—41no—
89#95—41no—
90#95—41no—
91#95—44no—
92#95—42no—
93#95—44no—
94#95—82no—
95#95—82no—
96C24—24no—
97C24—21no—
98C24—21no—
99C24—22no—
100C24—22no—
101C24—41no—
102C24—41no—
103C24#9912yes—
104C24C2×C22⋊C411yescharacteristic
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—48no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—44no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4—88no—
154C22×C4—88no—
155C22×C4#10012yes—
156C22×C4C2×M1611yescharacteristic
157C22×C4C2×M1611yescharacteristic
158C22×C4#13514yes—
159C2×C8—84no—
160C2×C8—84no—
161C2×C8—84no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C22⋊C4—48no—
167C22⋊C4—84no—
168C42—48no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—24no—
173C23—24no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—44no—
181C23—44no—
182C23—44no—
183C23—44no—
184C23—42no—
185C23—42no—
186C23—41no—
187C23—41no—
188C23—42no—
189C23#35214yes—
190C23#40711yescharacteristic
191C23C2×#9911yescharacteristic
192C23C2×#10011yescharacteristic
193C23#41612yes—
194C23#42112yes—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—48no—
202C2×C4—48no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—88no—
221C2×C4—88no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C8—164no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—44no—
252C22—44no—
253C22—48no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22#146211yescharacteristic
259C22#148812yes—
260C22#151612yes—
261C22#153611yescharacteristic
262C22#154311yescharacteristic
263C22C2×#35212yes—
264C22#212711yescharacteristic
265C22#213311yescharacteristic
266C4—44no—
267C4—42no—
268C4—41no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—161no—
274C4—161no—
275C2—21no—
276C2—21no—
277C2—22no—
278C2—44no—
279C2#764511yescharacteristic
280C2#1150011yescharacteristic
281C2#1150811yescharacteristic
282C2#1184112yes—
283C2#1185012yes—
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. #7132C2semidirect
4. #7104C2semidirect

Maximal subgroups

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

subgroupindex
#139582
#86982
#71322
#71042