Finite Group Databasebeta

← #190543

Subgroups — #190543

Order 512.

1160 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 218 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
1C2×#2333C211yesmaximal, characteristic
2#21906C212yesmaximal, complemented
3#8702C212yesmaximal, complemented
4#8561C214yesmaximal, complemented
5#8307C212yesmaximal
6#8152C214yesmaximal, complemented
7C22×#121C2211yescharacteristic
8#2333—28no—
9C2×#390C412yes—
10C2×#388C2212yes—
11C2×#387C412yes—
12C2×#386C412yes—
13C2×#386C412yes—
14C2×#381C2211yescharacteristic
15C2×C4×C22⋊C4C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1924—28no—
18#1797C2214yes—
19#1779C2214yes—
20#1749C2214yes—
21C2×#323C2218yes—
22#1280C2218yes—
23C23×D4C2×C411yescharacteristic
24C23×D4C2×C412yes—
25C24×C4C2×C412yes—
26C24×C4C2311yescharacteristic
27C2×#121—24no—
28C2×#121—28no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C412yes—
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2×C412yes—
37C22×C22⋊C4C2312yes—
38C22×C42C2×C412yes—
39C22×C42C2311yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C4⋊C8—24no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8C2318yes—
45C2×#97—24no—
46#390—28no—
47#388—28no—
48#387—28no—
49#386—28no—
50#386—28no—
51#381—24no—
52C4×C22⋊C4—28no—
53C4×C22⋊C4D418yes—
54C2×#95C2×C412yes—
55C2×#95C2×C412yes—
56C2×#95C2×C412yes—
57C2×#95C2×C412yes—
58C2×#95C2312yes—
59#336—24no—
60#336—24no—
61#323—216no—
62C25C22×C411yescharacteristic
63C25C22×C412yes—
64C22×D4—24no—
65C22×D4—24no—
66C22×D4—24no—
67C22×D4—22no—
68C22×D4—24no—
69C22×D4—81no—
70C22×D4—81no—
71C22×D4—82no—
72C22×D4—82no—
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—44no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—82no—
79C23×C4—82no—
80C23×C4C22×C412yes—
81C23×C4C22×C412yes—
82C23×C4C22×C412yes—
83C23×C4C22×C412yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C22×C412yes—
86C23×C4C2×D414yes—
87C23×C4C2411yescharacteristic
88C22×C8—28no—
89#121—232no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—82no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—82no—
104C2×C22⋊C4—82no—
105C2×C22⋊C4—81no—
106C2×C22⋊C4—81no—
107C2×C22⋊C4C22⋊C418yes—
108C2×C22⋊C4#4118yes—
109C2×C42—24no—
110C2×C42—81no—
111C2×C42—81no—
112C2×C42—82no—
113C2×C42C2×D414yes—
114C2×C42C2×D414yes—
115C4⋊C8—84no—
116C22⋊C8—216no—
117C22⋊C8—84no—
118C22⋊C8—84no—
119#97—84no—
120#95—24no—
121#95—24no—
122#95—24no—
123#95—24no—
124#95C22⋊C418yes—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—24no—
129C24—24no—
130C24—24no—
131C24—41no—
132C24—41no—
133C24—42no—
134C24—42no—
135C24—42no—
136C24—42no—
137C24—42no—
138C24C2×C22⋊C412yes—
139C24#11714yes—
140C24C23×C411yescharacteristic
141C2×D4—216no—
142C2×D4—28no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—84no—
146C2×D4—84no—
147C2×D4—88no—
148C2×D4—88no—
149C2×D4—88no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—28no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—48no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4—88no—
180C22×C4—816no—
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C22⋊C414yes—
183C22×C4C2×C22⋊C414yes—
184C22×C4#11714yes—
185C22×C4C22×D412yes—
186C22×C4C2×#4114yes—
187C2×C8—48no—
188C2×C8—48no—
189C2×C8—48no—
190C4⋊C4—28no—
191C4⋊C4—84no—
192C22⋊C4—216no—
193C22⋊C4—48no—
194C22⋊C4—48no—
195C22⋊C4—84no—
196C22⋊C4—84no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C42—44no—
200C42—44no—
201C42—84no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23—24no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—22no—
211C23—28no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—44no—
225C23—44no—
226C23—84no—
227C23—84no—
228C23—84no—
229C23—84no—
230C23—84no—
231C23—84no—
232C23C22×C22⋊C411yescharacteristic
233C23C2×#11712yes—
234C23#56814yes—
235D4—88no—
236D4—816no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—88no—
263C2×C4—84no—
264C2×C4—816no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4#52514yes—
268C8—88no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—44no—
274C22—42no—
275C22—44no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—44no—
280C22—44no—
281C22—88no—
282C22—88no—
283C22—88no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22#228011yescharacteristic
289C22#287912yes—
290C22#288311yescharacteristic
291C22#288311yescharacteristic
292C22#288812yes—
293C22#292314yes—
294C4—24no—
295C4—44no—
296C4—42no—
297C4—44no—
298C4—44no—
299C4—82no—
300C4—82no—
301C2—24no—
302C2—82no—
303C2—84no—
304C2#1740512yes—
305C2#1740812yes—
306C2#2071911yescharacteristic
307C2#2072411yescharacteristic
308C2#2072911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21906C2semidirect
3. #8702C2semidirect
4. #8561C2semidirect
6. #8152C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23332
#219062
#87022
#85612
#83072
#81522