Finite Group Databasebeta

← #135187

Subgroups — #135187

Order 512.

374 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 106 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#20366C211yesmaximal, characteristic
2#12844C212yesmaximal
3#7186C212yesmaximal
4#7186C212yesmaximal
5C2×C4×M16C411yescharacteristic
6C22×C4×C8C2211yescharacteristic
7C4×#117C411yescharacteristic
8C2×C22⋊C16C2214yes—
9#1746—22no—
10#1744—22no—
11#1729C411yescharacteristic
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1729C2211yescharacteristic
15#1374—22no—
16#1374—22no—
17C22×M16C2×C411yescharacteristic
18C22×M16C2×C411yescharacteristic
19C23×C8C2×C411yescharacteristic
20C23×C8C2311yescharacteristic
21C2×#117C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C22×C42C2×C411yescharacteristic
24C22×C16—24no—
25C4×M16—22no—
26C4×M16—22no—
27C2×#97—21no—
28C2×C4×C8—22no—
29C2×C4×C8—22no—
30C2×C4×C8—21no—
31C2×C4×C8D412yes—
32C2×C4×C8D412yes—
33C4×C4⋊C4C812yes—
34C4×C4⋊C4C812yes—
35C4×C22⋊C4—22no—
36#376—21no—
37C43—21no—
38C22⋊C16—24no—
39C22⋊C16—24no—
40#336—22no—
41#336—22no—
42#336—22no—
43#336—22no—
44C23×C4M1612yes—
45C23×C4C22×C411yescharacteristic
46C2×M16—21no—
47C2×M16—21no—
48C2×M16—41no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16M1612yes—
53C2×M16M1612yes—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8C22⋊C412yes—
63C22×C8C2×D412yes—
64C22×C8#4112yes—
65C22×C8#4112yes—
66#117—41no—
67#117—41no—
68#117—41no—
69#117—41no—
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C4⋊C4C2×C812yes—
72C2×C4⋊C4C2×C811yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C2×C812yes—
86C2×C16—44no—
87C2×C16—44no—
88C2×C16—44no—
89#97—24no—
90C4×C8—24no—
91C4×C8—24no—
92C4×C8—24no—
93#95—22no—
94#95—22no—
95C24C2×M1611yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—42no—
107C22×C4C22⋊C812yes—
108C22×C4C22⋊C812yes—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4#11711yescharacteristic
111C22×C4#11711yescharacteristic
112C22×C4C22×C811yescharacteristic
113C22×C4C2×M1611yescharacteristic
114C22×C4C2×M1611yescharacteristic
115M16—42no—
116M16—42no—
117M16—42no—
118M16—42no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8#12314yes—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—22no—
145C42—22no—
146C42—44no—
147C42—44no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42—41no—
152C16—84no—
153C23—22no—
154C23—22no—
155C23—21no—
156C23—21no—
157C23C2×C22⋊C811yescharacteristic
158C23#43211yescharacteristic
159C23#43211yescharacteristic
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4#35014yes—
185C2×C4#38111yescharacteristic
186C2×C4#40711yescharacteristic
187C2×C4#41011yescharacteristic
188C2×C4#41011yescharacteristic
189C2×C4#41311yescharacteristic
190C2×C4#41311yescharacteristic
191C2×C4#43111yescharacteristic
192C2×C4#43111yescharacteristic
193C8—24no—
194C8—42no—
195C8—42no—
196C8—42no—
197C22—22no—
198C22—22no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22#145811yescharacteristic
205C22#145811yescharacteristic
206C22#181711yescharacteristic
207C22C2×#35012yes—
208C22#211712yes—
209C4—24no—
210C4—21no—
211C4—21no—
212C4—42no—
213C4—81no—
214C4—81no—
215C4—82no—
216C4#146411yescharacteristic
217C4#146411yescharacteristic
218C4#182011yescharacteristic
219C4#182211yescharacteristic
220C2—41no—
221C2—41no—
222C2#869911yescharacteristic
223C2#1146211yescharacteristic
224C2#1146311yescharacteristic
225C2#1146412yes—
226C2#1284812yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#203662
#128442
#71862