Finite Group Databasebeta

← #207011

Subgroups — #207011

Order 512.

934 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 298 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#35181C211yesmaximal, characteristic
2#20536C211yesmaximal, characteristic
3#13954C211yesmaximal, characteristic
4#9887C218yesmaximal
5#8026C212yesmaximal
6#8014C212yesmaximal
7C2×#431C2212yes—
8#2851C2214yes—
9#2646—22no—
10#2571C414yescomplemented
11#2570—24no—
12#2552C414yescomplemented
13#2317—22no—
14C2×#389—22no—
15#2285C2212yes—
16C2×#382C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1770—24no—
20#1749C2214yes—
21#1444—22no—
22C4⋊#95C2214yes—
23#1294C2218yes—
24#1294C2218yes—
25C2×#125C2×C414yes—
26C2×#117—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4D412yes—
31C22×C42—21no—
32C22×C42C2312yes—
33C22×C42C2311yescharacteristic
34#431—24no—
35#431Q818yes—
36C2×C4⋊C8—22no—
37C2×C22⋊C8C2314yes—
38C2×#97—24no—
39C2×C4×C8—22no—
40#400—28no—
41#398—28no—
42#389—24no—
43#389—24no—
44#389—48no—
45#389C2×C418yes—
46#387—28no—
47#385—28no—
48#384—24no—
49#384C2×C412yes—
50#384C2×C412yes—
51#382—24no—
52#382—22no—
53#382—24no—
54#382C2×C414yes—
55#380—22no—
56#380C2314yes—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—24no—
59C4×C4⋊C4C2×C414yes—
60C4×C4⋊C4C2×C414yes—
61C4×C4⋊C4D414yes—
62C2×#95—24no—
63C2×#95—21no—
64C2×#95D412yes—
65C2×#95C2312yes—
66C2×#95C2312yes—
67#336—24no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4C2×D414yes—
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2411yescharacteristic
74C22×C8—24no—
75#125—84no—
76#117—44no—
77#117—44no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—28no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—28no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—48no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—84no—
93C2×C4⋊C4C22⋊C414yes—
94C2×C4⋊C4C22⋊C414yes—
95C2×C4⋊C4C22×C414yes—
96C2×C4⋊C4C22×C414yes—
97C2×C22⋊C4—24no—
98C2×C42—24no—
99C2×C42—42no—
100C2×C42—44no—
101C2×C42—82no—
102C2×C42C22×C414yes—
103C2×C42C22×C412yes—
104C2×C42C2×D414yes—
105C2×C42C2×D412yes—
106C2×C42C2×Q814yes—
107C2×C42#4114yes—
108C4⋊C8—28no—
109C22⋊C8—24no—
110C22⋊C8C2×Q818yes—
111#97—84no—
112C4×C8—28no—
113#95—24no—
114#95—28no—
115#95—28no—
116#95—24no—
117#95—24no—
118#95—24no—
119#95—48no—
120#95—42no—
121#95C22⋊C418yes—
122C24C22×D411yescharacteristic
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—41no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4C2×C22⋊C418yes—
145C22×C4C2×C22⋊C412yes—
146C22×C4C2×C22⋊C412yes—
147C22×C4#12412yes—
148C22×C4#12412yes—
149C22×C4C4⋊Q812yes—
150C22×C4C4⋊Q812yes—
151C22×C4AGL(1,ℤ/8)14yes—
152C22×C4#13714yes—
153C22×C4C23×C411yescharacteristic
154C22×C4C22×D411yescharacteristic
155C22×C4C22×Q812yes—
156C22×C4C2×#4112yes—
157C2×C8—28no—
158C2×C8—44no—
159C2×C8—44no—
160C4⋊C4—28no—
161C4⋊C4—28no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—88no—
170C4⋊C4—88no—
171C4⋊C4—816no—
172C4⋊C4—84no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C42—44no—
176C42—44no—
177C42—48no—
178C42—84no—
179C42C4×Q818yes—
180C42C22⋊Q814yes—
181C42C22⋊Q814yes—
182C23—21no—
183C23—21no—
184C23—22no—
185C23C22×C22⋊C411yescharacteristic
186C23C2×#12411yescharacteristic
187C23C2×C4⋊Q811yescharacteristic
188C23C2×AGL(1,ℤ/8)12yes—
189C23C2×#13712yes—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—28no—
198C2×C4—28no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—48no—
204C2×C4—48no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—84no—
222C2×C4—82no—
223C2×C4#39114yes—
224C2×C4#39114yes—
225C2×C4#41714yes—
226C2×C4#41714yes—
227C2×C4#49014yes—
228C2×C4#49114yes—
229C2×C4#50118yes—
230C2×C4C2×C4×Q812yes—
231C2×C4C2×C22⋊Q812yes—
232C8—48no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22C2×#39111yescharacteristic
242C22#288311yescharacteristic
243C22#288411yescharacteristic
244C22C2×#41712yes—
245C22#290112yes—
246C22C2×#49011yescharacteristic
247C22C2×#49111yescharacteristic
248C22C2×#50112yes—
249C4—24no—
250C4—22no—
251C4—22no—
252C4—48no—
253C4—84no—
254C4—84no—
255C4—84no—
256C4—162no—
257C4#199218yes—
258C2—21no—
259C2—21no—
260C2—22no—
261C2#1762812yes—
262C2#1763012yes—
263C2#2130711yescharacteristic
264C2C2×#199211yescharacteristic
265C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2571C4semidirect
12. #2552C4semidirect

Maximal subgroups

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

subgroupindex
#351812
#205362
#139542
#98872
#80262
#80142