Finite Group Databasebeta

← C22×#1386 #492872

Subgroups — C22×#1386 #492872

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 271 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M32C211yesmaximal, characteristic
2C22×#427C211yesmaximal, characteristic
3C23×#106C211yesmaximal, characteristic
4C2×#1386C2112yesmaximal, direct product factor
5C24×C8C2211yescharacteristic
6C22×#117—21no—
7C23×C4⋊C4—21no—
8C22×M32—24no—
9C22×M32—23no—
10C22×M32C418yescomplemented
11C22×M32C2216yes—
12C23×C16—21no—
13C2×#427—23no—
14C2×#427—23no—
15C2×#427C416yescomplemented
16C2×#427C2216yes—
17C22×#107C412yescomplemented
18C22×#106—24no—
19C22×#106—23no—
20C22×#106—21no—
21C22×#106C418yescomplemented
22C22×#106C2216yes—
23#1386C22116yesdirect product factor
24C24×C4—41no—
25C24×C4D411yescharacteristic
26C23×C8C2×C414yes—
27C23×C8C2×C414yes—
28C23×C8C2×C411yescharacteristic
29C23×C8C2×C413yes—
30C23×C8D413yes—
31C23×C8D411yescharacteristic
32C23×C8D414yes—
33C23×C8D413yes—
34C23×C8Q814yes—
35C23×C8C2313yes—
36C2×#117—26no—
37C2×#117—26no—
38C2×#117—43no—
39C2×#117—43no—
40C22×C4⋊C4—28no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—26no—
43C22×C4⋊C4—43no—
44C22×C4⋊C4—44no—
45C22×C22⋊C4—41no—
46C22×C42—41no—
47C2×M32—224no—
48C2×M32—212no—
49C2×M32—26no—
50C2×M32C2×C4124yescomplemented
51C2×M32C2314yes—
52C22×C16—28no—
53C22×C16—26no—
54#427—212no—
55#427—26no—
56#427—26no—
57#427C2×C4112yescomplemented
58#427C2314yes—
59C2×#107—26no—
60C2×#107C2×C4112yescomplemented
61C2×#106—224no—
62C2×#106—26no—
63C2×#106—26no—
64C2×#106—212no—
65C2×#106—26no—
66C2×#106C2×C4124yescomplemented
67C2×#106C2314yes—
68C25D811yescharacteristic
69C23×C4—48no—
70C23×C4—42no—
71C23×C4—46no—
72C23×C4—88no—
73C23×C4C22⋊C414yes—
74C23×C4C22⋊C413yes—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C4⋊C414yes—
77C23×C4D813yes—
78C23×C4SD1614yes—
79C23×C4SD1613yes—
80C23×C4SD1611yescharacteristic
81C23×C4Q1614yes—
82C23×C4C2×D413yes—
83C22×C8C4214yes—
84C22×C8C42112yes—
85C22×C8C22⋊C4112yes—
86C22×C8C22⋊C413yes—
87C22×C8C22⋊C413yes—
88C22×C8C22⋊C416yes—
89C22×C8C22⋊C414yes—
90C22×C8C22⋊C4112yes—
91C22×C8C4⋊C4112yes—
92C22×C8C4⋊C414yes—
93C22×C8C4⋊C4112yes—
94C22×C8C4⋊C414yes—
95C22×C8C4⋊C4112yes—
96C22×C8C22×C416yes—
97C22×C8C22×C416yes—
98C22×C8C22×C413yes—
99C22×C8C22×C413yes—
100C22×C8C2×D413yes—
101C22×C8C2×D413yes—
102C22×C8C2×D416yes—
103C22×C8C2×D413yes—
104C22×C8C2×Q816yes—
105C22×C8C2411yescharacteristic
106#117—212no—
107#117—24no—
108#117—412no—
109#117—46no—
110#117—46no—
111C2×C4⋊C4—224no—
112C2×C4⋊C4—212no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—424no—
115C2×C4⋊C4—46no—
116C2×C4⋊C4—46no—
117C2×C4⋊C4—412no—
118C2×C22⋊C4—46no—
119C2×C22⋊C4—46no—
120C2×C42—46no—
121C2×C42—46no—
122M32—224no—
123M32—24no—
124M32C22×C418yescomplemented
125C2×C16—224no—
126C2×C16—24no—
127#107—212no—
128#107C22×C418yescomplemented
129#106—224no—
130#106—212no—
131#106—24no—
132#106—24no—
133#106C22×C418yescomplemented
134C24—28no—
135C24#10213yes—
136C24#10214yes—
137C24#10211yescharacteristic
138C24#10714yes—
139C24C2×D813yes—
140C22×C4—44no—
141C22×C4—412no—
142C22×C4—424no—
143C22×C4—824no—
144C22×C4—832no—
145C22×C4—88no—
146C22×C4#95112yes—
147C22×C4#9514yes—
148C22×C4#10213yes—
149C22×C4#102112yes—
150C22×C4#10213yes—
151C22×C4#10216yes—
152C22×C4#103112yes—
153C22×C4#103112yes—
154C22×C4#10314yes—
155C22×C4#10314yes—
156C22×C4#106112yes—
157C22×C4#10614yes—
158C22×C4#107112yes—
159C22×C4C2×C22⋊C416yes—
160C22×C4C2×C22⋊C413yes—
161C22×C4C2×C22⋊C413yes—
162C22×C4C2×C4⋊C416yes—
163C22×C4C2×D813yes—
164C22×C4C2×SD1613yes—
165C22×C4C2×SD1616yes—
166C22×C4C2×SD1613yes—
167C22×C4C2×Q1616yes—
168C22×C4C22×D411yescharacteristic
169C2×C8#95112yes—
170C2×C8#95112yes—
171C2×C8#95124yes—
172C2×C8C2×C4216yes—
173C2×C8C2×C4216yes—
174C2×C8C2×C22⋊C416yes—
175C2×C8C2×C22⋊C416yes—
176C2×C8C2×C22⋊C416yes—
177C2×C8C2×C22⋊C413yes—
178C2×C8C2×C22⋊C413yes—
179C2×C8C2×C4⋊C416yes—
180C2×C8C2×C4⋊C416yes—
181C2×C8C2×C4⋊C416yes—
182C2×C8C2×C4⋊C416yes—
183C2×C8C2×C4⋊C416yes—
184C2×C8C23×C411yescharacteristic
185C2×C8C23×C411yescharacteristic
186C2×C8C23×C411yescharacteristic
187C2×C8C22×D411yescharacteristic
188C2×C8C22×D411yescharacteristic
189C2×C8C22×Q811yescharacteristic
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—44no—
193C4⋊C4—424no—
194C4⋊C4—412no—
195C22⋊C4—44no—
196C22⋊C4—412no—
197C42—44no—
198C42—412no—
199C16—28no—
200C23—212no—
201C23—212no—
202C23—216no—
203C23—216no—
204C23#340112yes—
205C23#34014yes—
206C23#36018yes—
207C23C2×#10213yes—
208C23C2×#10216yes—
209C23C2×#10213yes—
210C23C2×#10716yes—
211C23C22×D811yescharacteristic
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—848no—
215C2×C4—88no—
216C2×C4—824no—
217C2×C4#340124yes—
218C2×C4#340112yes—
219C2×C4#340112yes—
220C2×C4C2×#9516yes—
221C2×C4C2×#9516yes—
222C2×C4C2×#10213yes—
223C2×C4C2×#10216yes—
224C2×C4C2×#10213yes—
225C2×C4C2×#10316yes—
226C2×C4C2×#10316yes—
227C2×C4C2×#10316yes—
228C2×C4C2×#10316yes—
229C2×C4C2×#10616yes—
230C2×C4C2×#10616yes—
231C2×C4C2×#10716yes—
232C2×C4C22×C22⋊C411yescharacteristic
233C2×C4C22×C22⋊C411yescharacteristic
234C2×C4C22×C4⋊C411yescharacteristic
235C2×C4C22×SD1611yescharacteristic
236C2×C4C22×SD1611yescharacteristic
237C2×C4C22×Q1611yescharacteristic
238C8C2×#9516yes—
239C8C2×#9516yes—
240C8C22×C4211yescharacteristic
241C8C22×C22⋊C411yescharacteristic
242C8C22×C4⋊C411yescharacteristic
243C8C22×C4⋊C411yescharacteristic
244C22—212no—
245C22—224no—
246C22—212no—
247C22—22no—
248C22—26no—
249C22#1386116yesdirect product factor
250C22C2×#34016yes—
251C22C2×#34016yes—
252C22C2×#360112yes—
253C22C22×#10211yescharacteristic
254C22C22×#10211yescharacteristic
255C22C22×#10711yescharacteristic
256C4—88no—
257C4—88no—
258C4C2×#34016yes—
259C4C2×#34016yes—
260C4C22×#9511yescharacteristic
261C4C22×#10311yescharacteristic
262C4C22×#10311yescharacteristic
263C4C22×#10611yescharacteristic
264C2—21no—
265C2—21no—
266C2—26no—
267C2C2×#1386112yesdirect product factor
268C2C22×#34011yescharacteristic
269C2C22×#36012yes—
subgroupG/H
center, socleC24#102
commutatorC8C22×C42
Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1386C2direct (N × H)
10. C22×M32C4semidirect
15. C2×#427C4semidirect
17. C22×#107C4semidirect
21. C22×#106C4semidirect
23. #1386C22direct (N × H)
50. C2×M32C2×C4semidirect
57. #427C2×C4semidirect
60. C2×#107C2×C4semidirect
66. C2×#106C2×C4semidirect
124. M32C22×C4semidirect
128. #107C22×C4semidirect
133. #106C22×C4semidirect
249. C22#1386direct (N × H)
267. C2C2×#1386direct (N × H)

Maximal subgroups

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

subgroupindex
C23×M322
C22×#4272
C23×#1062
C2×#13862