Finite Group Databasebeta

← #128158

Subgroups — #128158

Order 512.

658 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 134 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#17357C212yesmaximal
2#14343C211yesmaximal, characteristic
3#7324C214yesmaximal
4C22×#117C2211yescharacteristic
5#2920—22no—
6C2×#380C412yes—
7#1464—22no—
8#1464C414yescomplemented
9#1463—22no—
10#1462C2214yes—
11#1458—22no—
12#1458C414yescomplemented
13#1445—22no—
14#1440C412yes—
15#1440C2212yes—
16C4⋊#95—22no—
17C24×C4C2×C411yescharacteristic
18C2×#117—42no—
19C2×#117—42no—
20C2×#117C2×C412yes—
21C2×#117C2×C412yes—
22C2×#117D411yescharacteristic
23C2×#117D411yescharacteristic
24C2×#117Q811yescharacteristic
25C2×#117Q811yescharacteristic
26C22×C4⋊C4—22no—
27C22×C4⋊C4D412yes—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4C2×C414yes—
30C22×C22⋊C4C2311yescharacteristic
31C22×C42D412yes—
32#432—24no—
33#432—42no—
34#431—24no—
35#431—42no—
36#423—42no—
37#407—24no—
38C4×M16—42no—
39#380—42no—
40#380—42no—
41C2×#95—22no—
42C2×#95—24no—
43C2×#95—24no—
44#324—24no—
45#324—24no—
46#323—24no—
47#323C2×C418yes—
48C25C22×C411yescharacteristic
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—24no—
52C23×C4—44no—
53C23×C4C4211yescharacteristic
54C23×C4C4211yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C4⋊C411yescharacteristic
59C23×C4C4⋊C411yescharacteristic
60C23×C4C2×D412yes—
61C2×M16—44no—
62#117—42no—
63#117—42no—
64#117—42no—
65#117—42no—
66#117—42no—
67#117—41no—
68#117—41no—
69#117—41no—
70#117—41no—
71#117—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4—44no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4C22⋊C412yes—
79C2×C4⋊C4C22⋊C412yes—
80C2×C4⋊C4C4⋊C412yes—
81C2×C4⋊C4C4⋊C412yes—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4—84no—
86C2×C22⋊C4C4212yes—
87C2×C22⋊C4C4⋊C411yescharacteristic
88C2×C22⋊C4C4⋊C411yescharacteristic
89C2×C22⋊C4C22×C412yes—
90C2×C22⋊C4C2×D411yescharacteristic
91C2×C22⋊C4C2×Q811yescharacteristic
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42C22⋊C412yes—
97C2×C42C22⋊C412yes—
98C2×C42C4⋊C412yes—
99C2×C42C4⋊C412yes—
100C4⋊C8—44no—
101C4⋊C8—44no—
102C22⋊C8—28no—
103C22⋊C8—44no—
104#97—44no—
105C4×C8—44no—
106#95—44no—
107#95—84no—
108#95—84no—
109C24—22no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24C2×C4211yescharacteristic
114C24C2×C22⋊C411yescharacteristic
115C24C2×C4⋊C411yescharacteristic
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—48no—
134C22×C4—48no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4#9511yescharacteristic
147C22×C4#9511yescharacteristic
148C22×C4#9511yescharacteristic
149C22×C4#9512yes—
150C22×C4#9512yes—
151C22×C4#9511yescharacteristic
152C22×C4#9912yes—
153C22×C4#9912yes—
154C22×C4C2×C22⋊C412yes—
155C22×C4C2×C4⋊C412yes—
156M16—84no—
157M16—84no—
158C2×C8—48no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C22⋊C4—42no—
166C22⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—42no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—88no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—22no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—44no—
188C23—44no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—82no—
196C23—82no—
197C23C2×#9511yescharacteristic
198C23C2×#9912yes—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4#34211yescharacteristic
235C2×C4#34211yescharacteristic
236C2×C4#34212yes—
237C2×C4#34414yes—
238C8—88no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—82no—
251C22—82no—
252C22—84no—
253C22—84no—
254C22C2×#34211yescharacteristic
255C22C2×#34412yes—
256C22#212212yes—
257C22#212312yes—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—41no—
265C4—41no—
266C4—164no—
267C4#130614yes—
268C2—22no—
269C2—22no—
270C2—22no—
271C2—81no—
272C2—81no—
273C2C2×#130611yescharacteristic
274C2#1257512yes—
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
8. #1464C4semidirect
12. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#173572
#143432
#73242