Finite Group Databasebeta

← #140907

Subgroups — #140907

Order 512.

480 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 114 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#33172C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3C2×#1350C214yesmaximal
4#10844C211yesmaximal, characteristic
5C2×#494C411yescharacteristic
6C2×#467—21no—
7C2×#467C2212yes—
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1969—21no—
14#1969C412yes—
15#1956—21no—
16#1845C411yescharacteristic
17C2×#341—21no—
18C2×#341C2212yes—
19#1374—21no—
20#1350—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16—22no—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26C2×C4⋊Q8—21no—
27C2×C22⋊Q8—22no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#494—42no—
31#467—24no—
32#467—24no—
33#467—42no—
34#467—42no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37#423—24no—
38#423—42no—
39C2×C4⋊C8—42no—
40C2×#103—22no—
41C2×#103—22no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—22no—
48#389—42no—
49#384—21no—
50#380—42no—
51C22⋊C16—22no—
52C22⋊C16D414yes—
53#346—42no—
54#341—24no—
55#341—42no—
56C22×Q8—22no—
57C23×C4SD1612yes—
58C23×C4C2×D411yescharacteristic
59C2×Q16—24no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×M16—24no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8D812yes—
79C22×C8SD1612yes—
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D412yes—
83C22×C8#4112yes—
84C4⋊Q8—42no—
85C4⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—42no—
89C22⋊Q8—42no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42D812yes—
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103#107—24no—
104#107—42no—
105C4⋊C8—48no—
106#103—44no—
107#103—44no—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—41no—
111C4×C8—41no—
112#95—44no—
113C24C2×SD1611yescharacteristic
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4#10212yes—
130C22×C4#10212yes—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4#12411yescharacteristic
133C22×C4#12711yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4C2×SD1611yescharacteristic
137Q16—84no—
138Q16—84no—
139M16—44no—
140M16—44no—
141M16—82no—
142M16—82no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—44no—
156C2×C8—82no—
157C2×C8—82no—
158C2×C8#10212yes—
159C2×C8#10212yes—
160C2×C8C4×D412yes—
161C2×C8#12112yes—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C42—24no—
171C42—21no—
172C42—21no—
173C42#9912yes—
174C42#10412yes—
175C16—44no—
176C23—22no—
177C23—22no—
178C23—21no—
179C23—21no—
180C23C2×#10211yescharacteristic
181C23#48611yescharacteristic
182C23#49211yescharacteristic
183Q8—82no—
184Q8—82no—
185Q8—82no—
186Q8—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—48no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4#32714yes—
208C2×C4#36214yes—
209C2×C4#39011yescharacteristic
210C2×C4C2×#9911yescharacteristic
211C2×C4#41011yescharacteristic
212C2×C4C2×#10211yescharacteristic
213C2×C4C2×#10411yescharacteristic
214C2×C4#42111yescharacteristic
215C2×C4#48611yescharacteristic
216C2×C4#49311yescharacteristic
217C8—22no—
218C8—22no—
219C8—22no—
220C8—22no—
221C8—84no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—42no—
230C22C2×#32711yescharacteristic
231C22#151711yescharacteristic
232C22#195711yescharacteristic
233C22#214312yes—
234C22C2×#36212yes—
235C4—24no—
236C4—24no—
237C4—21no—
238C4—21no—
239C4—162no—
240C4—162no—
241C4#150211yescharacteristic
242C4#151911yescharacteristic
243C4#196411yescharacteristic
244C4#198011yescharacteristic
245C2—41no—
246C2—41no—
247C2#968011yescharacteristic
248C2#1171212yes—
249C2#1186512yes—
250C2#1320911yescharacteristic
251C2#1321211yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#331722
#125992
C2×#13502
#108442