Finite Group Databasebeta

← #127853

Subgroups — #127853

Order 512.

697 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 128 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#21790C211yesmaximal, characteristic
2#17362C212yesmaximal
3#7287C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#432C411yescharacteristic
6C2×#432C411yescharacteristic
7C2×#407—22no—
8#1955—22no—
9#1955C412yes—
10#1955C2212yes—
11#1779—22no—
12#1465—22no—
13#1465C414yes—
14#1463—22no—
15#1462C414yes—
16#1459—22no—
17#1459C2214yes—
18C23×D4Q811yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C2×C4×D4C2×C412yes—
24C2×C4×D4C2×C411yescharacteristic
25C2×C4×D4D412yes—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4Q811yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C22⋊C4C2×C412yes—
31C22×C42C2×C411yescharacteristic
32#432—42no—
33#432—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C412yes—
36#407—24no—
37#407—24no—
38#407—42no—
39#407—42no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—24no—
45C2×C22⋊C8—42no—
46C2×#97—21no—
47C2×#97C2×C412yes—
48#324—24no—
49#324—24no—
50#323—24no—
51#323C2×C418yes—
52C25C4⋊C412yes—
53C22×D4—42no—
54C22×D4—44no—
55C22×D4C4⋊C412yes—
56C22×D4C4⋊C411yescharacteristic
57C22×D4C2×Q811yescharacteristic
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—84no—
63C23×C4C4211yescharacteristic
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C4⋊C411yescharacteristic
69C23×C4C4⋊C412yes—
70C23×C4C2×D412yes—
71C2×M16—44no—
72C2×M16—44no—
73C2×M16—88no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—44no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—41no—
80C4×D4—41no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85C4×D4—41no—
86C4×D4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4C22⋊C412yes—
89C2×C4⋊C4C4⋊C411yescharacteristic
90C2×C4⋊C4C2×D411yescharacteristic
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4C4212yes—
94C2×C22⋊C4C22⋊C412yes—
95C2×C22⋊C4C4⋊C412yes—
96C2×C22⋊C4C22×C412yes—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42C4211yescharacteristic
100C2×C42C22⋊C411yescharacteristic
101C2×C42C4⋊C411yescharacteristic
102C2×C42C22×C411yescharacteristic
103C4⋊C8—42no—
104C4⋊C8—42no—
105C22⋊C8—28no—
106C22⋊C8—44no—
107C22⋊C8—44no—
108C22⋊C8—44no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—84no—
112C22⋊C8—84no—
113#97—42no—
114#97—42no—
115C24—22no—
116C24—22no—
117C24—24no—
118C24—42no—
119C24—82no—
120C24—82no—
121C24#9512yes—
122C24#9511yescharacteristic
123C24C2×C4⋊C412yes—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—42no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—416no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—48no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—84no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4—84no—
154C22×C4—84no—
155C22×C4—81no—
156C22×C4#9511yescharacteristic
157C22×C4#9511yescharacteristic
158C22×C4#9512yes—
159C22×C4#9511yescharacteristic
160C22×C4#9912yes—
161C22×C4#9912yes—
162C22×C4C2×C4211yescharacteristic
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×C22⋊C412yes—
165C22×C4C2×C4⋊C411yescharacteristic
166M16—88no—
167M16—88no—
168C2×C8—48no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—88no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C42—22no—
183C42—22no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—41no—
196C23—44no—
197C23—44no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—44no—
203C23—44no—
204C23—82no—
205C23—82no—
206C23—88no—
207C23—88no—
208C23#34212yes—
209C23C2×#9511yescharacteristic
210C23#43014yes—
211D4—44no—
212D4—48no—
213D4—44no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4#34212yes—
243C2×C4C2×#9911yescharacteristic
244C2×C4C2×#9911yescharacteristic
245C8—82no—
246C8—88no—
247C8—82no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—24no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—82no—
263C22—82no—
264C22—88no—
265C22—88no—
266C22#173812yes—
267C22C2×#34211yescharacteristic
268C4—21no—
269C4—22no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—41no—
276C4#212712yes—
277C2—22no—
278C2—42no—
279C2—44no—
280C2—82no—
281C2#805511yescharacteristic
282C2#1257911yescharacteristic
283C2#1258211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7287C2semidirect

Maximal subgroups

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

subgroupindex
#217902
#173622
#72872