Finite Group Databasebeta

← #129681

Subgroups — #129681

Order 512.

837 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 837 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C2×#2972C211yesmaximal, characteristic
2#11629C214yesmaximal, complemented
3#11459C212yesmaximal, complemented
4C22×C4×D4C411yescharacteristic
5#2972—28no—
6C2×C8×D4—22no—
7C2×C8×D4C412yes—
8C2×C8×D4C2212yes—
9C2×#431C411yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C22⋊C8—22no—
12#2162C2214yescomplemented
13D4⋊C16—28no—
14#1322—22no—
15#1314—22no—
16C23×D4C811yescharacteristic
17C24×C4—21no—
18C23×C8—22no—
19C23×C8—22no—
20C2×C4×D4—24no—
21C2×C4×D4—24no—
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×C4×D4C812yes—
25C2×C4×D4C2×C412yes—
26C22×C4⋊C4C811yescharacteristic
27C22×C22⋊C4—21no—
28C22×C42C2×C411yescharacteristic
29C8×D4—216no—
30C8×D4—24no—
31C8×D4—24no—
32C8×D4—24no—
33C8×D4D418yes—
34#431—24no—
35#431—24no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8D412yes—
38C2×C22⋊C8—216no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8D412yes—
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46C4⋊C16—24no—
47C22⋊C16—24no—
48C4×C16—24no—
49C25—21no—
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—21no—
53C22×D4—22no—
54C22×D4C1614yescomplemented
55C22×D4C2×C812yes—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—84no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C2×C811yescharacteristic
64C23×C4M1611yescharacteristic
65C22×C8—22no—
66C22×C8—28no—
67C22×C8—28no—
68C22×C8—84no—
69C22×C8—84no—
70C22×C8C22⋊C412yes—
71C22×C8C2×D412yes—
72C4×D4—24no—
73C4×D4—28no—
74C4×D4—24no—
75C4×D4—24no—
76C4×D4—24no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4C22⋊C414yes—
80C4×D4M1614yes—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4C2×C812yes—
84C2×C22⋊C4—28no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C2×C811yescharacteristic
91C2×C42M1611yescharacteristic
92C2×C42C22×C411yescharacteristic
93C2×C16—44no—
94C4⋊C8—24no—
95C4⋊C8—22no—
96C4⋊C8C22⋊C414yes—
97C22⋊C8—24no—
98C22⋊C8—28no—
99C22⋊C8—24no—
100C22⋊C8—28no—
101C4×C8—24no—
102C4×C8—22no—
103C4×C8C2×D414yes—
104C24—24no—
105C24—24no—
106C24—22no—
107C24—24no—
108C24—82no—
109C24—82no—
110C24C22⋊C811yescharacteristic
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—216no—
114C2×D4—216no—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4C22⋊C814yes—
120C2×D4C2×C1614yes—
121C2×D4M3214yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—28no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—88no—
136C22×C4—88no—
137C22×C4—82no—
138C22×C4—84no—
139C22×C4—81no—
140C22×C4C22⋊C811yescharacteristic
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C2×C1612yes—
144C22×C4M3212yes—
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C22×C811yescharacteristic
147C22×C4C2×M1611yescharacteristic
148C2×C8—22no—
149C2×C8—24no—
150C2×C8—42no—
151C2×C8—44no—
152C2×C8—42no—
153C2×C8—88no—
154C2×C8—88no—
155C2×C8#12014yes—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4C22⋊C814yes—
159C22⋊C4—28no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C42—21no—
163C42—22no—
164C42—21no—
165C42—22no—
166C42C2×C22⋊C412yes—
167C42C2×M1612yes—
168C16—84no—
169C23—216no—
170C23—24no—
171C23—21no—
172C23—24no—
173C23—24no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23—44no—
178C23—44no—
179C23—84no—
180C23—88no—
181C23—88no—
182C23—84no—
183C23C22⋊C1612yes—
184C23C2×C22⋊C811yescharacteristic
185D4—216no—
186D4—24no—
187D4—24no—
188D4—24no—
189D4C22⋊C1618yes—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—41no—
199C2×C4—48no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—82no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4C22⋊C1612yes—
211C2×C4#37911yescharacteristic
212C2×C4C2×C22⋊C812yes—
213C2×C4#40712yes—
214C2×C4#40711yescharacteristic
215C2×C4C22×C1611yescharacteristic
216C2×C4C2×M3212yes—
217C2×C4C2×M3211yescharacteristic
218C8—24no—
219C8—44no—
220C8—42no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—28no—
225C22—42no—
226C22—48no—
227C22—48no—
228C22—42no—
229C22—41no—
230C22—44no—
231C22—41no—
232C22—44no—
233C22—816no—
234C22#177911yescharacteristic
235C22C2×C22⋊C1611yescharacteristic
236C22#211211yescharacteristic
237C4—21no—
238C4—22no—
239C4—22no—
240C4—44no—
241C4—41no—
242C4—41no—
243C4#179512yes—
244C4C2×C22⋊C1612yes—
245C4#211212yes—
246C2—21no—
247C2—21no—
248C2—48no—
249C2—42no—
250C2#832211yescharacteristic
251C2#1267911yescharacteristic
252C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC23×D4C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11629C2semidirect
3. #11459C2semidirect
12. #2162C22semidirect
54. C22×D4C16semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#116292
#114592