Finite Group Databasebeta

← #141968

Subgroups — #141968

Order 512.

330 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 84 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#21026C211yesmaximal, characteristic
2#12789C211yesmaximal, characteristic
3#12606C211yesmaximal, characteristic
4#7185C212yesmaximal
5#7184C212yesmaximal
6C2×#424C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2285C411yescharacteristic
9C2×#364—21no—
10#2112C2212yes—
11#2112C2212yes—
12C2×#347—21no—
13#1917—21no—
14#1917—21no—
15#1916—21no—
16#1916—21no—
17#1729C411yescharacteristic
18#1729C2211yescharacteristic
19#1728C411yescharacteristic
20C2×#336C2211yescharacteristic
21#1376—21no—
22#1375—21no—
23C22×M16C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×#117C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C2×M32—22no—
31C2×M32—22no—
32#424—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35#405—42no—
36C2×#97D411yescharacteristic
37C2×#97D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#384—21no—
41#382—21no—
42#380—21no—
43#380—21no—
44C4×C4⋊C4—21no—
45C4×C4⋊C4—21no—
46#364—42no—
47C22⋊C16—22no—
48C22⋊C16—24no—
49C22⋊C16Q814yes—
50#347—42no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59C23×C4—21no—
60C23×C4C22×C411yescharacteristic
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—22no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×Q812yes—
79C22×C8#4112yes—
80#117—41no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4M1611yescharacteristic
92C2×C22⋊C4M1611yescharacteristic
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42M1611yescharacteristic
100C2×C42M1611yescharacteristic
101M32—42no—
102M32—42no—
103M32—42no—
104M32—42no—
105C2×C16—24no—
106C2×C16—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109#97—22no—
110#97—41no—
111C4×C8—22no—
112C4×C8—41no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117C24C2×M1611yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—81no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C4×Q812yes—
133C22×C4C2×M1611yescharacteristic
134C22×C4#13112yes—
135M16—44no—
136M16—82no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—82no—
155C2×C8#9912yes—
156C2×C8C22⋊Q812yes—
157C2×C8#12411yescharacteristic
158C2×C8C4⋊Q811yescharacteristic
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C22⋊C4—42no—
166C22⋊C4—41no—
167C22⋊C4—41no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—41no—
173C42—41no—
174C42—41no—
175C42—41no—
176C42—41no—
177C42—41no—
178C16—44no—
179C16—82no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23#40711yescharacteristic
185C23#44612yes—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—84no—
211C2×C4#39111yescharacteristic
212C2×C4#40811yescharacteristic
213C2×C4C2×#9911yescharacteristic
214C2×C4#41011yescharacteristic
215C2×C4#41311yescharacteristic
216C2×C4#41311yescharacteristic
217C2×C4#44612yes—
218C8—22no—
219C8—22no—
220C8—21no—
221C8—21no—
222C8—41no—
223C8—81no—
224C8—81no—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—42no—
231C22—42no—
232C22#146411yescharacteristic
233C22#146511yescharacteristic
234C22#198511yescharacteristic
235C4—22no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4—81no—
240C4—81no—
241C4—81no—
242C4#146811yescharacteristic
243C4#147311yescharacteristic
244C4#199611yescharacteristic
245C4#199811yescharacteristic
246C2—21no—
247C2—21no—
248C2—42no—
249C2#988111yescharacteristic
250C2#1326911yescharacteristic
251C2#1327011yescharacteristic
subgroupG/H
centerC2×C4#391
commutatorC23×C4C22×C4
Frattini, cosocleC23×C8C23
socleC22#1985
prime radicalC24C2×M16
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×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
#210262
#127892
#126062
#71852
#71842