Finite Group Databasebeta

← #129206

Subgroups — #129206

Order 512.

1940 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1940 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C22×#407C211yesmaximal, characteristic
2#12681C212yesmaximal, complemented
3#11418C214yesmaximal, complemented
4C25×C4C411yescharacteristic
5C23×M16C411yescharacteristic
6C23×M16C2211yescharacteristic
7C2×#407—28no—
8C2×#407—24no—
9C22×C22⋊C8C412yes—
10C22×C22⋊C8C2212yes—
11C2×#349—24no—
12#2112C2214yescomplemented
13#1314—28no—
14C26C811yescharacteristic
15C24×C4—24no—
16C24×C4—22no—
17C24×C4—24no—
18C24×C4—22no—
19C24×C4—44no—
20C24×C4—44no—
21C24×C4C812yes—
22C24×C4C811yescharacteristic
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C412yes—
25C22×M16—22no—
26C22×M16—24no—
27C22×M16—24no—
28C22×M16—44no—
29C22×M16—44no—
30C22×M16D414yes—
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C412yes—
33C23×C8C2311yescharacteristic
34C2×M32—24no—
35#407—232no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—24no—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8D418yes—
42#349—48no—
43C22⋊C16—28no—
44C25—24no—
45C25—22no—
46C25—44no—
47C25—48no—
48C25C2×C811yescharacteristic
49C25C2×C812yes—
50C23×C4—24no—
51C23×C4—24no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—28no—
55C23×C4—216no—
56C23×C4—28no—
57C23×C4—24no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—416no—
67C23×C4—416no—
68C23×C4—432no—
69C23×C4—48no—
70C23×C4—48no—
71C23×C4—44no—
72C23×C4—44no—
73C23×C4—48no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—44no—
77C23×C4—44no—
78C23×C4—44no—
79C23×C4C22⋊C412yes—
80C23×C4C22⋊C414yes—
81C23×C4C2×C811yescharacteristic
82C23×C4C2×C812yes—
83C23×C4M1612yes—
84C23×C4M1614yes—
85C23×C4C22×C411yescharacteristic
86C2×M16—28no—
87C2×M16—28no—
88C2×M16—28no—
89C2×M16—44no—
90C2×M16—416no—
91C2×M16—416no—
92C2×M16—44no—
93C2×M16—44no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—44no—
100C22×C8C22⋊C412yes—
101C22×C8C22⋊C414yes—
102C22×C8C2×D412yes—
103C22×C8C2×D414yes—
104M32—48no—
105C2×C16—44no—
106C22⋊C8—216no—
107C22⋊C8—216no—
108C24—24no—
109C24—28no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—28no—
114C24—44no—
115C24—44no—
116C24—44no—
117C24—416no—
118C24—44no—
119C24—416no—
120C24—416no—
121C24—48no—
122C24—464no—
123C24—44no—
124C24—48no—
125C24C22⋊C812yes—
126C24C22⋊C814yes—
127C24C22×C811yescharacteristic
128C22×C4—28no—
129C22×C4—28no—
130C22×C4—216no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—216no—
143C22×C4—28no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—416no—
151C22×C4—464no—
152C22×C4—416no—
153C22×C4—48no—
154C22×C4—48no—
155C22×C4—416no—
156C22×C4—416no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—416no—
160C22×C4—44no—
161C22×C4—416no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—48no—
165C22×C4—416no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4C22⋊C812yes—
173C22×C4C22⋊C814yes—
174C22×C4#9914yes—
175C22×C4#10014yes—
176C22×C4C2×C22⋊C412yes—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×M1611yescharacteristic
179C22×C4C2×M1612yes—
180M16—416no—
181M16—416no—
182C2×C8—24no—
183C2×C8—22no—
184C2×C8—24no—
185C2×C8—24no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8#12014yes—
192C16—84no—
193C23—24no—
194C23—22no—
195C23—28no—
196C23—28no—
197C23—28no—
198C23—24no—
199C23—22no—
200C23—22no—
201C23—24no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—44no—
206C23—416no—
207C23—416no—
208C23—432no—
209C23—44no—
210C23—44no—
211C23—432no—
212C23—432no—
213C23—464no—
214C23—464no—
215C23—432no—
216C23—44no—
217C23—44no—
218C23—44no—
219C23#32314yes—
220C23#34918yescomplemented
221C23C2×C22⋊C812yes—
222C23C2×C22⋊C811yescharacteristic
223C2×C4—28no—
224C2×C4—216no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—432no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—416no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4#32314yes—
247C2×C4#37911yescharacteristic
248C2×C4#40712yes—
249C2×C4#40711yescharacteristic
250C2×C4C2×#9912yes—
251C2×C4C2×#10012yes—
252C8—24no—
253C8—42no—
254C8—44no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—24no—
260C22—28no—
261C22—44no—
262C22—48no—
263C22—416no—
264C22—416no—
265C22—48no—
266C22—416no—
267C22—44no—
268C22—464no—
269C22—44no—
270C22—44no—
271C22—44no—
272C22C2×#32312yes—
273C22#177911yescharacteristic
274C22C2×#34914yes—
275C4—24no—
276C4—22no—
277C4—44no—
278C4#146212yes—
279C4#178111yescharacteristic
280C4#178311yescharacteristic
281C2—24no—
282C2—22no—
283C2—44no—
284C2—48no—
285C2#829811yescharacteristic
286C2#1268112yes—
subgroupG/H
centerC2×C4#379
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1779
prime radicalC26C8
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12681C2semidirect
3. #11418C2semidirect
12. #2112C22semidirect
220. C23#349semidirect

Maximal subgroups

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

subgroupindex
C22×#4072
#126812
#114182