Finite Group Databasebeta

← #135459

Subgroups — #135459

Order 512.

427 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 92 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#20526C211yesmaximal, characteristic
2#11418C212yesmaximal
3#7201C214yesmaximal, complemented
4C22×C22⋊C8C411yescharacteristic
5C22×C22⋊C8C2211yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7#2112C2214yes—
8#1752—22no—
9#1752C412yes—
10#1752C2212yes—
11#1749—22no—
12#1314—22no—
13#1314—22no—
14C24×C4C2×C411yescharacteristic
15C23×C8—21no—
16C23×C8C2×C411yescharacteristic
17C23×C8C2311yescharacteristic
18C22×C22⋊C4C811yescharacteristic
19C22×C22⋊C4C811yescharacteristic
20C22×C42—21no—
21C2×M32—24no—
22C2×C22⋊C8—21no—
23C2×C22⋊C8—24no—
24C2×C22⋊C8—24no—
25C2×C22⋊C8C2×C412yes—
26C2×C22⋊C8D412yes—
27C2×C22⋊C8D412yes—
28C2×#97—22no—
29C2×#97—22no—
30C4×C22⋊C4—21no—
31C4×C22⋊C4—21no—
32C4×C22⋊C4C812yes—
33C4×C22⋊C4C2×C412yes—
34C2×#95—21no—
35C22⋊C16—24no—
36C22⋊C16—24no—
37#336—22no—
38#336—22no—
39C25C2×C811yescharacteristic
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—21no—
43C23×C4—21no—
44C23×C4—81no—
45C23×C4—81no—
46C23×C4C22⋊C411yescharacteristic
47C23×C4C22⋊C411yescharacteristic
48C23×C4C2×C811yescharacteristic
49C23×C4M1611yescharacteristic
50C23×C4M1611yescharacteristic
51C23×C4C22×C411yescharacteristic
52C22×C8—22no—
53C22×C8—24no—
54C22×C8—22no—
55C22×C8—81no—
56C22×C8—81no—
57C22×C8—81no—
58C22×C8—81no—
59C22×C8C22⋊C412yes—
60C22×C8C2×D412yes—
61C22×C8#4114yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—42no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4C2×C812yes—
67C2×C22⋊C4C2×C812yes—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—81no—
71C2×C42—81no—
72M32—84no—
73C2×C16—44no—
74C22⋊C8—24no—
75C22⋊C8—22no—
76C22⋊C8—24no—
77C22⋊C8—24no—
78C22⋊C8M1614yes—
79#97—82no—
80#97—82no—
81#95—22no—
82#95—22no—
83C24—22no—
84C24—22no—
85C24—22no—
86C24—81no—
87C24—81no—
88C24C22⋊C811yescharacteristic
89C24C22⋊C811yescharacteristic
90C24C22×C811yescharacteristic
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—82no—
109C22×C4—82no—
110C22×C4—84no—
111C22×C4—81no—
112C22×C4—81no—
113C22×C4—81no—
114C22×C4—81no—
115C22×C4—81no—
116C22×C4—81no—
117C22×C4—81no—
118C22×C4C22⋊C811yescharacteristic
119C22×C4C22⋊C811yescharacteristic
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4#11712yes—
122C22×C4C2×M1612yes—
123C22×C4C2×M1611yescharacteristic
124C2×C8—22no—
125C2×C8—24no—
126C2×C8—42no—
127C2×C8—44no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8—84no—
133C2×C8#12314yes—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C42—42no—
139C42—42no—
140C42—82no—
141C16—84no—
142C23—28no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—82no—
156C23—82no—
157C23—84no—
158C23—84no—
159C23#34912yes—
160C23#34912yes—
161C23C2×C22⋊C811yescharacteristic
162C23#43112yes—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—84no—
189C2×C4—84no—
190C2×C4#38111yescharacteristic
191C2×C4#40711yescharacteristic
192C2×C4#43212yes—
193C8—24no—
194C8—44no—
195C8—42no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—88no—
214C22#181711yescharacteristic
215C22C2×#34911yescharacteristic
216C22C2×#34911yescharacteristic
217C4—21no—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4#182811yescharacteristic
224C4#182911yescharacteristic
225C2—21no—
226C2—21no—
227C2—42no—
228C2—42no—
229C2—42no—
230C2#873411yescharacteristic
231C2#1284611yescharacteristic
232C2#1285111yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1817
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7201C2semidirect

Maximal subgroups

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

subgroupindex
#205262
#114182
#72012