Finite Group Databasebeta

← #6374064

Subgroups — #6374064

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34644C211yesmaximal, characteristic
2#21717C214yesmaximal, complemented
3#21679C214yesmaximal, complemented
4#21605C218yesmaximal
5#21352C212yesmaximal
6#21341C214yesmaximal, complemented
7#20811C211yesmaximal, characteristic
8#20518C214yesmaximal, complemented
9#20476C212yesmaximal
10C4×#405C211yesmaximal, characteristic
11C2×#446C2218yes—
12C2×#432C2212yes—
13C2×#405C2214yes—
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2667C414yes—
17#2666C418yes—
18#2665C414yes—
19#2402C2212yes—
20#2399C2212yes—
21#2396C2214yes—
22#2295C2214yes—
23#2283C2212yes—
24C4×#117C2211yescharacteristic
25#1984C2218yes—
26#1940C2214yes—
27#1940C2214yes—
28#1939C2218yes—
29#1916C2218yes—
30#1916C2218yes—
31#1851C2218yes—
32#1843C2212yes—
33#1832C2212yes—
34#1798C2214yes—
35#1771C2218yes—
36#1771C2218yes—
37#1770C2218yes—
38#1770C2214yes—
39C8×C4⋊C4C2214yes—
40#1768C2218yes—
41#1767C2214yes—
42#1752C2214yes—
43C8×C22⋊C4C2214yes—
44#1746C2214yes—
45#1726C2214yes—
46#1726C2212yes—
47#1726C2212yes—
48C4×#97C2212yes—
49C42×C8C2212yes—
50C22×M16C2312yes—
51C23×C8C2312yes—
52C2×C4×Q8C2314yes—
53C2×#117C2312yes—
54C22×C42C2311yescharacteristic
55#446—232no—
56#432—28no—
57C2×C4⋊C8C2318yes—
58C2×C4⋊C8C2314yes—
59C2×C4⋊C8C2318yes—
60C2×C4⋊C8C2318yes—
61C2×C22⋊C8C2314yes—
62#405—216no—
63C4×M16—24no—
64C2×#97C2312yes—
65C2×#97C2314yes—
66C2×#97C2314yes—
67C2×#97C2318yes—
68C2×#97C2314yes—
69C2×C4×C8Q818yescomplemented
70C2×C4×C8C2312yes—
71C2×C4×C8C2314yes—
72C2×C4×C8C2314yes—
73C2×C4×C8C2318yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77#400C2×C414yes—
78#400C2×C418yes—
79#399C2×C414yes—
80#398C2×C418yes—
81#398C2×C418yes—
82#398C2×C418yes—
83#396C2×C418yes—
84#395C2×C414yes—
85#395C2×C418yes—
86#393C2×C414yes—
87#391C2×C418yes—
88#389C2×C418yes—
89#387C2×C418yes—
90#387C2×C418yes—
91#383C2×C414yes—
92#382C2×C414yes—
93#381C2×C412yes—
94#381C2×C412yes—
95#380C2×C414yes—
96C4×C4⋊C4C2×C414yes—
97C4×C4⋊C4C2314yes—
98C4×C4⋊C4C2318yes—
99C4×C4⋊C4C2314yes—
100C4×C22⋊C4C2×C414yes—
101C4×C22⋊C4C2314yes—
102#376C2311yescharacteristic
103#376C2311yescharacteristic
104#376C2314yes—
105C43C2312yes—
106#336C2314yes—
107#336C2314yes—
108#336C2318yes—
109#336C2318yes—
110#336C2318yes—
111#336C2318yes—
112C22×Q8C22×C414yes—
113C23×C4C22×C412yes—
114C23×C4C2411yescharacteristic
115C2×M16—24no—
116C2×M16—28no—
117C22×C8—28no—
118C22×C8C2×Q818yes—
119C22×C8C2418yes—
120C22×C8C2412yes—
121C22×C8C2414yes—
122C22×C8C2412yes—
123C4×Q8—216no—
124#117—28no—
125C2×C4⋊C4C22×C414yes—
126C2×C4⋊C4C22×C418yes—
127C2×C4⋊C4C22×C414yes—
128C2×C4⋊C4C22×C418yes—
129C2×C4⋊C4C22×C418yes—
130C2×C4⋊C4C22×C418yes—
131C2×C4⋊C4C22×C418yes—
132C2×C22⋊C4C22×C414yes—
133C2×C22⋊C4C22×C414yes—
134C2×C22⋊C4C22×C414yes—
135C2×C42C22×C418yes—
136C2×C42C22×C412yes—
137C2×C42C2×Q814yes—
138C2×C42#4114yes—
139C2×C42C2412yes—
140C2×C42C2418yes—
141C2×C42C2411yescharacteristic
142C2×C42C2411yescharacteristic
143C2×C42C2412yes—
144C4⋊C8—216no—
145C4⋊C8—216no—
146C4⋊C8—28no—
147C4⋊C8—216no—
148C22⋊C8—28no—
149#97—28no—
150#97—216no—
151#97—24no—
152#97—28no—
153#97#41116yes—
154C4×C8—28no—
155C4×C8—216no—
156C4×C8—24no—
157C4×C8—28no—
158C4×C8C2×Q814yes—
159C4×C8C2×Q818yes—
160C4×C8C2×Q814yes—
161#95C22×C414yes—
162#95C22×C414yes—
163#95C22×C418yes—
164#95C22×C414yes—
165#95C22×C414yes—
166#95C22×C418yes—
167#95C22×C414yes—
168#95C22×C414yes—
169#95C22×C414yes—
170#95C22×C414yes—
171#95C22×C418yes—
172#95C22×C418yes—
173C24C23×C411yescharacteristic
174C2×Q8—28no—
175C2×Q8—48no—
176C22×C4—24no—
177C22×C4—44no—
178C22×C4C4×Q818yes—
179C22×C4C23×C418yes—
180C22×C4C23×C414yes—
181C22×C4C23×C418yes—
182C22×C4C23×C412yes—
183C22×C4C23×C411yescharacteristic
184C22×C4C23×C412yes—
185C22×C4C23×C414yes—
186C22×C4C22×Q812yes—
187C22×C4C2×#4112yes—
188C22×C4C2511yescharacteristic
189M16—216no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—216no—
194C2×C8—28no—
195C2×C8—28no—
196C2×C8—28no—
197C2×C8C22×Q814yes—
198C2×C8C22×Q814yes—
199C2×C8C2×#4118yes—
200C2×C8C2×#4118yes—
201C4⋊C4—28no—
202C4⋊C4—28no—
203C4⋊C4—216no—
204C4⋊C4—48no—
205C4⋊C4—48no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C22⋊C4—28no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C42—24no—
212C42—28no—
213C42—28no—
214C42—48no—
215C42C4×Q818yes—
216C42C22×Q812yes—
217C42C22×Q812yes—
218C42C2×#4112yes—
219C42C2×#4112yes—
220C42C2×#4118yes—
221C23—22no—
222C23—24no—
223C23C2×C4×Q812yes—
224C23C24×C411yescharacteristic
225Q8—88no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—28no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—28no—
235C2×C4—28no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—28no—
240C2×C4—48no—
241C2×C4C2×C4×Q812yes—
242C2×C4C2×C4×Q814yes—
243C2×C4C2×C4×Q814yes—
244C2×C4C4×#4118yes—
245C2×C4C23×Q811yescharacteristic
246C2×C4C22×#4114yes—
247C2×C4C22×#4111yescharacteristic
248C8—24no—
249C8—28no—
250C8—48no—
251C8#53318yes—
252C22—24no—
253C22—22no—
254C22—44no—
255C22C22×C4×Q811yescharacteristic
256C22C2×C4×#4112yes—
257C22#357314yes—
258C4—24no—
259C4—24no—
260C4—28no—
261C4—22no—
262C4—44no—
263C4—48no—
264C4—48no—
265C4C2×#53312yes—
266C4#344412yes—
267C2—42no—
268C2C4×#53311yescharacteristic
269C2#6019112yes—
270C2#6021214yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21717C2semidirect
3. #21679C2semidirect
6. #21341C2semidirect
8. #20518C2semidirect
69. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#346442
#217172
#216792
#216052
#213522
#213412
#208112
#205182
#204762
C4×#4052