Finite Group Databasebeta

← #135162

Subgroups — #135162

Order 512.

900 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 118 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#13756C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7187C214yesmaximal
4C22×#117C411yescharacteristic
5C22×#117C2211yescharacteristic
6C23×C42C411yescharacteristic
7#1817—22no—
8C2×#324—22no—
9C2×#323C2214yes—
10C4⋊#95—22no—
11C22⋊#95C412yes—
12C22⋊#95C2212yes—
13C4×#95—22no—
14C24×C4C2×C411yescharacteristic
15C24×C4C2×C412yes—
16C2×#117—24no—
17C2×#117—24no—
18C2×#117—42no—
19C2×#117—42no—
20C2×#117—42no—
21C2×#117—42no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4D412yes—
24C22×C22⋊C4C2×C411yescharacteristic
25C22×C22⋊C4C2×C412yes—
26C22×C22⋊C4C2311yescharacteristic
27C22×C42—21no—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42—22no—
31C22×C42—42no—
32C22×C42—42no—
33C22×C42C814yescomplemented
34C22×C42D412yes—
35C2×C22⋊C8—24no—
36C2×#95—24no—
37C2×#95—24no—
38#336—44no—
39#324—44no—
40#323—28no—
41C25C22×C411yescharacteristic
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—48no—
52C23×C4—42no—
53C23×C4—42no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C2×C812yes—
58C23×C4C2×C814yes—
59C23×C4M1612yes—
60C23×C4C2×D412yes—
61C22×C8—44no—
62#117—416no—
63#117—416no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—41no—
73C2×C22⋊C4—41no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4M1614yes—
79C2×C22⋊C4#4114yes—
80C2×C42—24no—
81C2×C42—24no—
82C2×C42—24no—
83C2×C42—24no—
84C2×C42—24no—
85C2×C42—42no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42—42no—
89C2×C42—48no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—48no—
94C22⋊C8—44no—
95C22⋊C8—44no—
96#95—48no—
97#95—48no—
98C24—21no—
99C24—21no—
100C24—21no—
101C24—21no—
102C24—44no—
103C24C2×C22⋊C411yescharacteristic
104C24#11712yes—
105C24C22×C811yescharacteristic
106C24C2×M1611yescharacteristic
107C24C2×M1612yes—
108C22×C4—28no—
109C22×C4—28no—
110C22×C4—28no—
111C22×C4—28no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—28no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—48no—
135C22×C4—48no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—416no—
139C22×C4—48no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—48no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C22⋊C812yes—
151C22×C4C22⋊C812yes—
152C22×C4#9912yes—
153C22×C4#9912yes—
154C22×C4#10012yes—
155C22×C4#10112yes—
156C22×C4#12314yes—
157C2×C8—84no—
158C2×C8—88no—
159C4⋊C4—48no—
160C4⋊C4—48no—
161C22⋊C4—44no—
162C22⋊C4—44no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C42—48no—
166C42—48no—
167C42—48no—
168C42—48no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23—44no—
181C23—44no—
182C23—44no—
183C23—44no—
184C23—44no—
185C23—44no—
186C23#38111yescharacteristic
187C23C2×C22⋊C811yescharacteristic
188C23#40711yescharacteristic
189C23C2×#9911yescharacteristic
190C23C2×#9911yescharacteristic
191C23C2×#10011yescharacteristic
192C23C2×#10111yescharacteristic
193C23#43112yes—
194C23#43212yes—
195C2×C4—216no—
196C2×C4—216no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—28no—
210C2×C4—24no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—416no—
214C2×C4—48no—
215C2×C4—44no—
216C2×C4—48no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—82no—
226C2×C4—84no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#32314yes—
231C2×C4#32414yes—
232C8—164no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—42no—
242C22—42no—
243C22—44no—
244C22—44no—
245C22—42no—
246C22—42no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—44no—
251C22C2×#32311yescharacteristic
252C22C2×#32411yescharacteristic
253C22#146211yescharacteristic
254C22#146311yescharacteristic
255C22#181711yescharacteristic
256C22#181911yescharacteristic
257C22#182111yescharacteristic
258C22#212014yes—
259C4—24no—
260C4—24no—
261C4—44no—
262C4—48no—
263C4—84no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—21no—
268C2—21no—
269C2—42no—
270C2—42no—
271C2#869811yescharacteristic
272C2#1142512yes—
273C2#1151812yes—
274C2#1285312yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
33. C22×C42C8semidirect

Maximal subgroups

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

subgroupindex
#137562
#83002
#71872