Finite Group Databasebeta

← #135564

Subgroups — #135564

Order 512.

650 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 70 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#14159C211yesmaximal, characteristic
2#8555C212yesmaximal
3#7326C214yesmaximal, complemented
4C22×C22⋊Q8C2211yescharacteristic
5C2×C4×C22⋊C4C411yescharacteristic
6C22×#95C411yescharacteristic
7#1821—22no—
8#1463C2214yes—
9#1462—22no—
10#1446C412yes—
11#1445C2212yes—
12#1436—22no—
13C22⋊#95—22no—
14C23×Q8C2×C411yescharacteristic
15C24×C4—21no—
16C24×C4C2×C411yescharacteristic
17C2×C22⋊Q8—24no—
18C2×C22⋊Q8—41no—
19C2×C22⋊Q8—41no—
20C2×C22⋊Q8—42no—
21C22×C4⋊C4D412yes—
22C22×C4⋊C4C2311yescharacteristic
23C22×C22⋊C4—21no—
24C22×C22⋊C4D412yes—
25C22×C42—21no—
26#407—24no—
27C4×C22⋊C4—42no—
28C2×#95—21no—
29C2×#95—24no—
30C2×#95—24no—
31C2×#95—41no—
32C2×#95—41no—
33C2×#95C2×C412yes—
34C2×#95C2×C412yes—
35#343—44no—
36#324—28no—
37#323—42no—
38#323—42no—
39C25C22⋊C411yescharacteristic
40C22×Q8—22no—
41C22×Q8—41no—
42C22×Q8—41no—
43C22×Q8—82no—
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—44no—
51C23×C4—41no—
52C23×C4—41no—
53C23×C4—44no—
54C23×C4—82no—
55C23×C4—82no—
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D412yes—
60C2×M16—44no—
61C22⋊Q8—48no—
62C22⋊Q8—48no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4#4114yes—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C42—42no—
77C2×C42—82no—
78C22⋊C8—44no—
79C22⋊C8—44no—
80#95—42no—
81#95—42no—
82#95—48no—
83#95—42no—
84#95—84no—
85C24—22no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—42no—
90C24—42no—
91C24#9911yescharacteristic
92C24#9911yescharacteristic
93C24C2×C22⋊C411yescharacteristic
94C2×Q8—44no—
95C2×Q8—84no—
96C2×Q8—84no—
97C2×Q8—84no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—84no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4—84no—
129C22×C4—816no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—82no—
134C22×C4#9912yes—
135C22×C4#11712yes—
136C22×C4#12314yes—
137M16—88no—
138C2×C8—84no—
139C4⋊C4—44no—
140C4⋊C4—48no—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C22⋊C4—48no—
144C42—84no—
145C23—22no—
146C23—22no—
147C23—21no—
148C23—21no—
149C23—24no—
150C23—24no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—48no—
160C23—42no—
161C23—48no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23#35512yes—
166C23#35612yes—
167C23#38111yescharacteristic
168C23C2×#9911yescharacteristic
169C23C2×#9911yescharacteristic
170Q8—88no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—48no—
184C2×C4—44no—
185C2×C4—48no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—82no—
189C2×C4—84no—
190C2×C4—84no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—88no—
196C2×C4—88no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C8—164no—
202C22—24no—
203C22—24no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—48no—
216C22—48no—
217C22—42no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22#181911yescharacteristic
223C22#182912yes—
224C22#212612yes—
225C22C2×#35511yescharacteristic
226C22C2×#35611yescharacteristic
227C4—42no—
228C4—42no—
229C4—44no—
230C4—84no—
231C4—84no—
232C4—82no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—42no—
239C2#875111yescharacteristic
240C2#1285711yescharacteristic
241C2#1286011yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect

Maximal subgroups

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

subgroupindex
#141592
#85552
#73262