Finite Group Databasebeta

← #141163

Subgroups — #141163

Order 512.

450 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 102 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#12783C211yesmaximal, characteristic
3C2×#1350C214yesmaximal
4#10822C211yesmaximal, characteristic
5C2×#494C411yescharacteristic
6C2×#467—21no—
7C2×#467C2212yes—
8C22×C4×C8C2211yescharacteristic
9#2857—21no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12#1969—21no—
13#1969C412yes—
14#1956—21no—
15#1845C411yescharacteristic
16C2×#341—21no—
17C2×#341C2212yes—
18#1728—21no—
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—44no—
34C2×#107—21no—
35C2×#107C2×C412yes—
36C2×#103—22no—
37C2×#103—22no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#391—22no—
44#389—41no—
45#389—41no—
46#384—21no—
47#380—41no—
48#380—41no—
49C4⋊C16—42no—
50C22⋊C16—24no—
51#341—22no—
52#341—42no—
53#341D414yes—
54#336—41no—
55#336—41no—
56C22×Q8—22no—
57C23×C4SD1611yescharacteristic
58C23×C4SD1611yescharacteristic
59C23×C4C2×D411yescharacteristic
60C2×Q16—24no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×M16—24no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—42no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D412yes—
83C22×C8#4112yes—
84C4⋊Q8—42no—
85C4⋊Q8—42no—
86C22⋊Q8—44no—
87C22⋊Q8—44no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42D811yescharacteristic
100C2×C42D811yescharacteristic
101C2×C16—42no—
102C2×C16—42no—
103C2×C16—42no—
104#107—24no—
105#107—42no—
106#103—44no—
107#103—42no—
108#103—42no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—41no—
113#95—42no—
114#95—42no—
115C24C2×SD1611yescharacteristic
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4#10211yescharacteristic
135C22×C4#10211yescharacteristic
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4#12112yes—
140C22×C4C2×D811yescharacteristic
141C22×C4AGL(1,ℤ/8)12yes—
142Q16—84no—
143Q16—84no—
144M16—44no—
145M16—44no—
146M16—82no—
147M16—82no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8#9912yes—
164C2×C8C4×D412yes—
165C2×C8#12411yescharacteristic
166C2×C8#12711yescharacteristic
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C22⋊C4—84no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—41no—
180C42—41no—
181C16—82no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23C2×#10211yescharacteristic
189C23#46012yes—
190Q8—82no—
191Q8—82no—
192Q8—82no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#35912yes—
218C2×C4#36212yes—
219C2×C4#36212yes—
220C2×C4#39011yescharacteristic
221C2×C4C2×#9911yescharacteristic
222C2×C4#41011yescharacteristic
223C2×C4#41811yescharacteristic
224C2×C4#42111yescharacteristic
225C2×C4#42111yescharacteristic
226C2×C4#45912yes—
227C8—21no—
228C8—22no—
229C8—21no—
230C8—22no—
231C8—41no—
232C8—84no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22#151711yescharacteristic
241C22#151911yescharacteristic
242C22#198211yescharacteristic
243C22C2×#35911yescharacteristic
244C22#214311yescharacteristic
245C22C2×#36211yescharacteristic
246C22C2×#36211yescharacteristic
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—162no—
253C4—162no—
254C4#151011yescharacteristic
255C4#151411yescharacteristic
256C4#196311yescharacteristic
257C4#196411yescharacteristic
258C2—42no—
259C2#971511yescharacteristic
260C2#1186311yescharacteristic
261C2#1186511yescharacteristic
262C2#1189211yescharacteristic
263C2#1189611yescharacteristic
264C2#1325111yescharacteristic
265C2#1325611yescharacteristic
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
#127832
C2×#13502
#108222