Finite Group Databasebeta

← #108628

Subgroups — #108628

Order 512.

850 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 158 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#13776C211yesmaximal, characteristic
2#10727C212yesmaximal
3C2×#1306C212yesmaximal
4C2×#1306C212yesmaximal
5C23×C4⋊C4C2211yescharacteristic
6C2×#407C2214yes—
7C2×#381C2211yescharacteristic
8C2×#381C2211yescharacteristic
9C2×C4×C22⋊C4C412yes—
10C2×#343—24no—
11#1436—24no—
12C22⋊#95C412yes—
13#1306—24no—
14#1306C418yescomplemented
15C24×C4C2×C411yescharacteristic
16C24×C4C2×C411yescharacteristic
17C24×C4C2311yescharacteristic
18C22×M16—24no—
19C22×C4⋊C4—22no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4—44no—
22C22×C4⋊C4D414yes—
23C22×C22⋊C4C2×C412yes—
24C22×C22⋊C4C2×C412yes—
25C22×C42—22no—
26#407—24no—
27#407—24no—
28#407—24no—
29#407C2×C418yes—
30C2×C22⋊C8—24no—
31C2×C22⋊C8—24no—
32#381—22no—
33#381—22no—
34#381C2×C414yes—
35#381D412yes—
36#381Q812yes—
37C4×C22⋊C4—44no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—24no—
42#343—48no—
43C25C22×C411yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—24no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4—42no—
52C23×C4—42no—
53C23×C4—42no—
54C23×C4—82no—
55C23×C4—82no—
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C412yes—
58C23×C4C22×C412yes—
59C23×C4C2×D412yes—
60C23×C4C2×D411yescharacteristic
61C23×C4C2×Q811yescharacteristic
62C23×C4#4112yes—
63C23×C4#4112yes—
64C23×C4#4112yes—
65C23×C4#4112yes—
66C2×M16—28no—
67C2×M16—44no—
68C2×M16—48no—
69C2×M16—48no—
70C22×C8—44no—
71C2×C4⋊C4—28no—
72C2×C4⋊C4—28no—
73C2×C4⋊C4—44no—
74C2×C4⋊C4—48no—
75C2×C4⋊C4—48no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4C4214yes—
83C2×C22⋊C4C4⋊C414yes—
84C2×C42—44no—
85C2×C42—84no—
86C22⋊C8—28no—
87C22⋊C8—28no—
88C22⋊C8—44no—
89C22⋊C8—44no—
90#95—28no—
91#95—24no—
92#95—24no—
93#95—44no—
94#95—84no—
95C24—22no—
96C24—22no—
97C24—42no—
98C24—42no—
99C24C2×C4211yescharacteristic
100C24C2×C22⋊C411yescharacteristic
101C24C2×C4⋊C411yescharacteristic
102C24#11711yescharacteristic
103C24#11711yescharacteristic
104C24#11711yescharacteristic
105C24#11711yescharacteristic
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—88no—
142C22×C4—88no—
143C22×C4#9914yes—
144C22×C4#11714yes—
145C22×C4C4×D414yes—
146C22×C4C4×D412yes—
147C22×C4C4×Q812yes—
148C22×C4#12112yes—
149C22×C4C22⋊Q812yes—
150C22×C4#12314yes—
151C22×C4#12412yes—
152C22×C4#12512yes—
153C22×C4#12614yes—
154M16—416no—
155M16—88no—
156C2×C8—48no—
157C2×C8—84no—
158C2×C8—84no—
159C4⋊C4—416no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—44no—
163C22⋊C4—44no—
164C42—88no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—44no—
184C23—44no—
185C23—48no—
186C23#37611yescharacteristic
187C23C4×C22⋊C411yescharacteristic
188C23C4×C4⋊C411yescharacteristic
189C23#38011yescharacteristic
190C23#38111yescharacteristic
191C23#38211yescharacteristic
192C23#38311yescharacteristic
193C23C2×#9912yes—
194C23#41012yes—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—28no—
200C2×C4—24no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4#38714yes—
228C2×C4#38814yes—
229C8—88no—
230C22—22no—
231C22—22no—
232C22—24no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—44no—
244C22—44no—
245C22—44no—
246C22—44no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22#143411yescharacteristic
254C22C4×#9912yes—
255C22#179912yes—
256C22#181911yescharacteristic
257C22#182011yescharacteristic
258C22#182912yes—
259C22#183012yes—
260C4—24no—
261C4—24no—
262C4—44no—
263C4—84no—
264C4—84no—
265C4—82no—
266C2—22no—
267C2—22no—
268C2—41no—
269C2—41no—
270C2—41no—
271C2—41no—
272C2#875111yescharacteristic
273C2#875311yescharacteristic
274C2#875512yes—
275C2#1080111yescharacteristic
276C2#1083012yes—
subgroupG/H
center, socleC23#381
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1306C4semidirect

Maximal subgroups

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

subgroupindex
#137762
#107272
C2×#13062