Finite Group Databasebeta

← #128476

Subgroups — #128476

Order 512.

455 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 104 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#22260C211yesmaximal, characteristic
2C4×#353C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#8160C212yesmaximal, complemented
5#7205C212yesmaximal, complemented
6C2×C8×D4C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8C4×#127C2211yescharacteristic
9C2×#353C412yes—
10#1954C411yescharacteristic, complemented
11#1954C2211yescharacteristic
12#1869—22no—
13#1869—22no—
14#1831C411yescharacteristic, complemented
15C4×#99C2212yes—
16#1752—22no—
17#1744C411yescharacteristic, complemented
18#1459C2212yes—
19#1393C412yes—
20#1325C412yescomplemented
21#1318C412yescomplemented
22#1282—22no—
23C23×C8—21no—
24C23×C8D412yes—
25C2×#127C2×C411yescharacteristic
26C2×C4×D4—22no—
27C2×C4×D4C2×C411yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C8×D4—42no—
30C8×D4—42no—
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8C2×C411yescharacteristic
33C2×#99C2×C414yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8D412yes—
39C2×#97—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42#390—22no—
43#390C2×C411yescharacteristic
44#390C2×C411yescharacteristic
45#384C2×C411yescharacteristic
46C4×C22⋊C4—22no—
47C43C2×C411yescharacteristic
48#353—24no—
49#342—22no—
50#336—22no—
51#336—21no—
52#336—21no—
53#336—42no—
54#324C2×C414yes—
55#323—22no—
56C22×D4—22no—
57C22×D4C4211yescharacteristic
58C22×D4C22×C411yescharacteristic
59C23×C4—21no—
60C23×C4C2×D412yes—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—42no—
66C22×C8—42no—
67C22×C8—82no—
68C22×C8—82no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C412yes—
71C22×C8C22⋊C411yescharacteristic
72#127—22no—
73#127M1614yes—
74C4×D4—41no—
75C4×D4—41no—
76C4×D4—41no—
77C4×D4—41no—
78C4×D4—82no—
79C4×D4—82no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4C4211yescharacteristic
82C2×C4⋊C4C22×C411yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4#4112yes—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—42no—
91C2×C42C4211yescharacteristic
92C2×C42C4211yescharacteristic
93C2×C42C22×C411yescharacteristic
94C4⋊C8—41no—
95C4⋊C8—41no—
96#99—24no—
97#99—24no—
98C22⋊C8—24no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C22⋊C8—82no—
103C22⋊C8—82no—
104#97—84no—
105C4×C8—22no—
106C4×C8—41no—
107#95—42no—
108C24—21no—
109C24C4×D412yes—
110C2×D4—28no—
111C2×D4—21no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4#9712yescomplemented
119C2×D4C2×M1612yes—
120C2×D4#13112yes—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—44no—
133C22×C4C2×C4211yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4#11711yescharacteristic
136C22×C4C4×D412yes—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—41no—
140C2×C8—44no—
141C2×C8—42no—
142C2×C8—41no—
143C2×C8—44no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—82no—
150C2×C8—84no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8#9912yes—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—82no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—81no—
170C42—82no—
171C42—82no—
172C42#9712yescomplemented
173C23—22no—
174C23—22no—
175C23—22no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—41no—
182C23C4×C22⋊C411yescharacteristic
183C23#43514yes—
184D4—42no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—88no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4C2×#9711yescharacteristic
215C2×C4#40511yescharacteristic
216C2×C4C2×#9911yescharacteristic
217C2×C4#41011yescharacteristic
218C8—22no—
219C8—42no—
220C8—41no—
221C8—81no—
222C8—84no—
223C8—81no—
224C22—21no—
225C22—21no—
226C22—24no—
227C22—44no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—82no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22#175211yescharacteristic
237C22C4×#9911yescharacteristic
238C22#175711yescharacteristic
239C4—21no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—42no—
244C4—41no—
245C4—81no—
246C4—81no—
247C4—164no—
248C4#212711yescharacteristic
249C4#212811yescharacteristic
250C2—21no—
251C2—21no—
252C2—44no—
253C2—82no—
254C2#816011yescharacteristic
255C2#1262811yescharacteristic
256C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8160C2semidirect
5. #7205C2semidirect
6. C2×C8×D4C4semidirect
10. #1954C4semidirect
14. #1831C4semidirect
17. #1744C4semidirect
20. #1325C4semidirect
21. #1318C4semidirect
118. C2×D4#97semidirect
172. C42#97semidirect

Maximal subgroups

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

subgroupindex
#222602
C4×#3532
#114752
#81602
#72052