Finite Group Databasebeta

← #128197

Subgroups — #128197

Order 512.

618 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 130 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#17365C212yesmaximal
2#13961C211yesmaximal, characteristic
3#7326C214yesmaximal
4C22×C4×Q8C2211yescharacteristic
5#2920—22no—
6C2×#383C411yescharacteristic
7C2×#376C411yescharacteristic
8#1467—22no—
9#1467C414yescomplemented
10#1463—22no—
11#1463C2214yes—
12#1461—22no—
13#1461C414yescomplemented
14#1446C412yes—
15#1445C2212yes—
16#1436—22no—
17#1434—22no—
18C23×Q8C2×C411yescharacteristic
19C2×C4×Q8—42no—
20C2×C4×Q8—41no—
21C2×C4×Q8—41no—
22C2×C4×Q8C2×C412yes—
23C2×C4×Q8C2×C412yes—
24C2×C4×Q8D412yes—
25C2×C4×Q8Q812yes—
26C22×C4⋊C4D412yes—
27C22×C4⋊C4C2311yescharacteristic
28C22×C42—21no—
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31#432—24no—
32#432—42no—
33#431—24no—
34#431—42no—
35#423—42no—
36#407—24no—
37C4×M16—42no—
38#383—41no—
39#383—41no—
40#376—42no—
41C2×#95—24no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—24no—
46C2×#95C2×C412yes—
47C2×#95C2×C412yes—
48#324—24no—
49#324—24no—
50#324—24no—
51#324C2×C418yes—
52C22×Q8—41no—
53C22×Q8—41no—
54C22×Q8—44no—
55C22×Q8C4211yescharacteristic
56C22×Q8C4211yescharacteristic
57C22×Q8C4⋊C412yes—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D412yes—
65C2×M16—44no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×Q8—42no—
69C4×Q8—42no—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4C22⋊C412yes—
79C2×C4⋊C4C22⋊C412yes—
80C2×C4⋊C4C4⋊C412yes—
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4C22×C412yes—
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4C2×Q811yescharacteristic
85C2×C42—22no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—42no—
93C2×C42—44no—
94C2×C42C4212yes—
95C2×C42C22⋊C412yes—
96C2×C42C22⋊C412yes—
97C2×C42C4⋊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—42no—
107#95—42no—
108#95—48no—
109#95—48no—
110#95—42no—
111#95—42no—
112#95—41no—
113#95—41no—
114C24C2×C22⋊C411yescharacteristic
115C2×Q8—24no—
116C2×Q8—24no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—48no—
123C2×Q8—48no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4#9512yes—
149C22×C4#9512yes—
150C22×C4#9512yes—
151C22×C4#9512yes—
152C22×C4#9912yes—
153C22×C4#9911yescharacteristic
154C22×C4#9911yescharacteristic
155C22×C4C2×C4211yescharacteristic
156C22×C4C2×C22⋊C412yes—
157C22×C4C2×C4⋊C411yescharacteristic
158C22×C4C2×C4⋊C412yes—
159M16—88no—
160C2×C8—48no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—22no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—48no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×#9511yescharacteristic
185C23C2×#9911yescharacteristic
186C23C2×#9911yescharacteristic
187Q8—44no—
188Q8—48no—
189Q8—44no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—48no—
218C2×C4—82no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4#34212yes—
228C2×C4#34212yes—
229C2×C4#34414yes—
230C8—88no—
231C22—24no—
232C22—24no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22C2×#34211yescharacteristic
240C22C2×#34412yes—
241C22#212612yes—
242C22#213312yes—
243C4—22no—
244C4—22no—
245C4—24no—
246C4—42no—
247C4—42no—
248C4—42no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—84no—
253C4—162no—
254C2—22no—
255C2—22no—
256C2—22no—
257C2#811511yescharacteristic
258C2#1258111yescharacteristic
259C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1467C4semidirect
13. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#173652
#139612
#73262