Finite Group Databasebeta

← #128052

Subgroups — #128052

Order 512.

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

normal subgroups only

This group has 622 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#20727C212yesmaximal
2#14346C211yesmaximal, characteristic
3#7308C214yesmaximal
4C22×C4×Q8C2211yescharacteristic
5#2920—22no—
6C2×#383C411yescharacteristic
7C2×#382C411yescharacteristic
8#1821—22no—
9#1801—22no—
10#1801C414yescomplemented
11#1784C2214yes—
12C4×#101—22no—
13C4×#101C414yescomplemented
14#1446C2212yes—
15#1445C412yes—
16#1441—22no—
17#1434—22no—
18C23×Q8C2311yescharacteristic
19C2×C4×Q8—42no—
20C2×C4×Q8—42no—
21C2×C4×Q8C2×C412yes—
22C2×C4×Q8C2×C412yes—
23C2×C4×Q8D411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×Q8Q811yescharacteristic
26C2×C4×Q8Q811yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#432—42no—
33#432—42no—
34#431—42no—
35#431—42no—
36#423—24no—
37C2×#101—24no—
38C2×#101C2×C418yes—
39#407—24no—
40C4×M16—24no—
41#383—42no—
42#382—42no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—24no—
47C2×#95—24no—
48C2×#95C2×C412yes—
49C2×#95C2×C412yes—
50#343—24no—
51#343—24no—
52C22×Q8—41no—
53C22×Q8—41no—
54C22×Q8—44no—
55C22×Q8C22×C412yes—
56C22×Q8C2×D411yescharacteristic
57C22×Q8C2×Q811yescharacteristic
58C23×C4—22no—
59C23×C4—22no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D412yes—
65C2×M16—28no—
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—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—82no—
80C2×C4⋊C4C4212yes—
81C2×C4⋊C4C22⋊C412yes—
82C2×C4⋊C4C22⋊C412yes—
83C2×C4⋊C4C4⋊C411yescharacteristic
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C4⋊C412yes—
86C2×C4⋊C4C4⋊C412yes—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42C4211yescharacteristic
94C2×C42C4211yescharacteristic
95C2×C42C22⋊C412yes—
96C2×C42C22⋊C412yes—
97C2×C42C4⋊C412yes—
98C2×C42C4⋊C412yes—
99C2×C42C4⋊C411yescharacteristic
100C2×C42C4⋊C411yescharacteristic
101C4⋊C8—44no—
102C4⋊C8—44no—
103#101—48no—
104#101—48no—
105C22⋊C8—44no—
106C22⋊C8—44no—
107#97—44no—
108C4×C8—44no—
109#95—42no—
110#95—42no—
111#95—48no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—84no—
116C24C2×C22⋊C411yescharacteristic
117C2×Q8—24no—
118C2×Q8—24no—
119C2×Q8—42no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—48no—
125C2×Q8—48no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4#9511yescharacteristic
152C22×C4#9511yescharacteristic
153C22×C4#9512yes—
154C22×C4#9512yes—
155C22×C4#9511yescharacteristic
156C22×C4#9511yescharacteristic
157C22×C4#9912yes—
158C22×C4#9912yes—
159C22×C4C2×C4211yescharacteristic
160C22×C4C2×C22⋊C412yes—
161C22×C4C2×C4⋊C411yescharacteristic
162C22×C4C2×C4⋊C412yes—
163M16—48no—
164M16—48no—
165C2×C8—48no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—84no—
174C42—22no—
175C42—22no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—22no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23C2×#9511yescharacteristic
191C23C2×#9911yescharacteristic
192C23C2×#9911yescharacteristic
193Q8—44no—
194Q8—48no—
195Q8—44no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—84no—
228C2×C4—88no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#34212yes—
237C2×C4#34212yes—
238C2×C4#41014yes—
239C8—88no—
240C22—24no—
241C22—24no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22C2×#34211yescharacteristic
249C22#174012yes—
250C22#213312yes—
251C22#213411yescharacteristic
252C22#213511yescharacteristic
253C4—22no—
254C4—22no—
255C4—24no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—162no—
264C4—162no—
265C2—21no—
266C2—22no—
267C2—21no—
268C2—22no—
269C2#810011yescharacteristic
270C2#1258611yescharacteristic
271C2#1258711yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #1801C4semidirect
13. C4×#101C4semidirect

Maximal subgroups

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

subgroupindex
#207272
#143462
#73082