Finite Group Databasebeta

← C22×#1387 #492875

Subgroups — C22×#1387 #492875

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M32C211yesmaximal, characteristic
2C23×#108C211yesmaximal, characteristic
3C23×#107C211yesmaximal, characteristic
4C2×#1387C2112yesmaximal, direct product factor
5C23×M16—21no—
6C24×C8C2211yescharacteristic
7C23×C4⋊C4—21no—
8C22×M32—23no—
9C22×M32—24no—
10C22×M32C418yescomplemented
11C22×M32C2216yes—
12C23×C16—21no—
13C22×#108—23no—
14C22×#108—24no—
15C22×#108C418yescomplemented
16C22×#108C2216yes—
17C22×#107—21no—
18C22×#107—23no—
19C22×#107—23no—
20C22×#107C416yes—
21C22×#107C412yes—
22C22×#107C2216yes—
23#1387C22116yesdirect product factor
24C24×C4—41no—
25C24×C4D411yescharacteristic
26C22×M16—28no—
27C22×M16—26no—
28C22×M16—43no—
29C22×M16—44no—
30C23×C8—41no—
31C23×C8C2×C414yes—
32C23×C8C2×C413yes—
33C23×C8C2×C411yescharacteristic
34C23×C8C2×C414yes—
35C23×C8D413yes—
36C23×C8D411yescharacteristic
37C23×C8D413yes—
38C23×C8D414yes—
39C23×C8Q814yes—
40C23×C8C2313yes—
41C22×C4⋊C4—26no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—26no—
44C22×C4⋊C4—43no—
45C22×C4⋊C4—41no—
46C22×C4⋊C4—43no—
47C2×M32—224no—
48C2×M32—212no—
49C2×M32—26no—
50C2×M32C2×C4124yescomplemented
51C2×M32C2314yes—
52C22×C16—28no—
53C22×C16—26no—
54C2×#108—224no—
55C2×#108—26no—
56C2×#108—212no—
57C2×#108C2×C4124yescomplemented
58C2×#108C2314yes—
59C2×#107—26no—
60C2×#107—212no—
61C2×#107—26no—
62C2×#107—26no—
63C2×#107—26no—
64C2×#107—26no—
65C2×#107C2×C4112yes—
66C2×#107C2×C4112yes—
67C2×#107C2314yes—
68C25SD1611yescharacteristic
69C23×C4—46no—
70C23×C4—42no—
71C23×C4—46no—
72C23×C4—88no—
73C23×C4C22⋊C414yes—
74C23×C4C22⋊C413yes—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C4⋊C414yes—
77C23×C4D813yes—
78C23×C4D811yescharacteristic
79C23×C4SD1613yes—
80C23×C4SD1614yes—
81C23×C4Q1614yes—
82C23×C4C2×D413yes—
83C2×M16—224no—
84C2×M16—24no—
85C2×M16—424no—
86C2×M16—46no—
87C2×M16—412no—
88C22×C8—48no—
89C22×C8—46no—
90C22×C8C42112yes—
91C22×C8C4214yes—
92C22×C8C22⋊C413yes—
93C22×C8C22⋊C416yes—
94C22×C8C22⋊C413yes—
95C22×C8C22⋊C4112yes—
96C22×C8C22⋊C414yes—
97C22×C8C22⋊C4112yes—
98C22×C8C4⋊C4112yes—
99C22×C8C4⋊C4112yes—
100C22×C8C4⋊C414yes—
101C22×C8C4⋊C414yes—
102C22×C8C4⋊C4112yes—
103C22×C8C22×C416yes—
104C22×C8C22×C413yes—
105C22×C8C22×C413yes—
106C22×C8C22×C416yes—
107C22×C8C2×D413yes—
108C22×C8C2×D413yes—
109C22×C8C2×D413yes—
110C22×C8C2×D416yes—
111C22×C8C2×Q816yes—
112C22×C8C2411yescharacteristic
113C2×C4⋊C4—212no—
114C2×C4⋊C4—212no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—412no—
117C2×C4⋊C4—46no—
118C2×C4⋊C4—46no—
119C2×C4⋊C4—46no—
120C2×C4⋊C4—46no—
121C2×C4⋊C4—46no—
122M32—224no—
123M32—24no—
124M32C22×C418yescomplemented
125C2×C16—224no—
126C2×C16—24no—
127#108—224no—
128#108—24no—
129#108C22×C418yescomplemented
130#107—212no—
131#107—212no—
132#107—24no—
133#107C22×C418yes—
134C24—24no—
135C24—24no—
136C24#10213yes—
137C24#10211yescharacteristic
138C24#10314yes—
139C24#10614yes—
140C24C2×SD1613yes—
141C22×C4—44no—
142C22×C4—412no—
143C22×C4—412no—
144C22×C4—824no—
145C22×C4—824no—
146C22×C4—88no—
147C22×C4#95112yes—
148C22×C4#9514yes—
149C22×C4#10216yes—
150C22×C4#10213yes—
151C22×C4#10213yes—
152C22×C4#10214yes—
153C22×C4#102112yes—
154C22×C4#103112yes—
155C22×C4#103112yes—
156C22×C4#10314yes—
157C22×C4#106112yes—
158C22×C4#107112yes—
159C22×C4#10714yes—
160C22×C4C2×C22⋊C416yes—
161C22×C4C2×C22⋊C413yes—
162C22×C4C2×C22⋊C413yes—
163C22×C4C2×C4⋊C416yes—
164C22×C4C2×D813yes—
165C22×C4C2×D813yes—
166C22×C4C2×SD1616yes—
167C22×C4C2×SD1613yes—
168C22×C4C2×Q1616yes—
169C22×C4C22×D411yescharacteristic
170M16—28no—
171M16—44no—
172M16—424no—
173C2×C8—44no—
174C2×C8—424no—
175C2×C8#95112yes—
176C2×C8#95124yes—
177C2×C8#95112yes—
178C2×C8C2×C4216yes—
179C2×C8C2×C4216yes—
180C2×C8C2×C22⋊C416yes—
181C2×C8C2×C22⋊C416yes—
182C2×C8C2×C22⋊C413yes—
183C2×C8C2×C22⋊C413yes—
184C2×C8C2×C22⋊C416yes—
185C2×C8C2×C4⋊C416yes—
186C2×C8C2×C4⋊C416yes—
187C2×C8C2×C4⋊C416yes—
188C2×C8C2×C4⋊C416yes—
189C2×C8C2×C4⋊C416yes—
190C2×C8C23×C411yescharacteristic
191C2×C8C23×C411yescharacteristic
192C2×C8C23×C411yescharacteristic
193C2×C8C22×D411yescharacteristic
194C2×C8C22×D411yescharacteristic
195C2×C8C22×Q811yescharacteristic
196C4⋊C4—28no—
197C4⋊C4—44no—
198C4⋊C4—412no—
199C4⋊C4—412no—
200C16—28no—
201C23—26no—
202C23—26no—
203C23—26no—
204C23—26no—
205C23—216no—
206C23—216no—
207C23#340112yes—
208C23#34014yes—
209C23#36114yes—
210C23#36214yes—
211C23C2×#10213yes—
212C23C2×#10213yes—
213C23C2×#10316yes—
214C23C2×#10616yes—
215C23C22×SD1611yescharacteristic
216C2×C4—48no—
217C2×C4—88no—
218C2×C4—824no—
219C2×C4—824no—
220C2×C4#340124yes—
221C2×C4#340112yes—
222C2×C4#340112yes—
223C2×C4C2×#9516yes—
224C2×C4C2×#9516yes—
225C2×C4C2×#10216yes—
226C2×C4C2×#10216yes—
227C2×C4C2×#10213yes—
228C2×C4C2×#10213yes—
229C2×C4C2×#10316yes—
230C2×C4C2×#10316yes—
231C2×C4C2×#10316yes—
232C2×C4C2×#10616yes—
233C2×C4C2×#10716yes—
234C2×C4C2×#10716yes—
235C2×C4C22×C22⋊C411yescharacteristic
236C2×C4C22×C22⋊C411yescharacteristic
237C2×C4C22×C4⋊C411yescharacteristic
238C2×C4C22×D811yescharacteristic
239C2×C4C22×SD1611yescharacteristic
240C2×C4C22×Q1611yescharacteristic
241C8—48no—
242C8C2×#9516yes—
243C8C2×#9516yes—
244C8C22×C4211yescharacteristic
245C8C22×C22⋊C411yescharacteristic
246C8C22×C4⋊C411yescharacteristic
247C8C22×C4⋊C411yescharacteristic
248C22—224no—
249C22—212no—
250C22—212no—
251C22—21no—
252C22—23no—
253C22—21no—
254C22—23no—
255C22#1387116yesdirect product factor
256C22C2×#34016yes—
257C22C2×#34016yes—
258C22C2×#36116yes—
259C22C2×#36216yes—
260C22C22×#10211yescharacteristic
261C22C22×#10311yescharacteristic
262C22C22×#10611yescharacteristic
263C4—88no—
264C4C2×#34016yes—
265C4C2×#34016yes—
266C4C22×#9511yescharacteristic
267C4C22×#10211yescharacteristic
268C4C22×#10311yescharacteristic
269C4C22×#10711yescharacteristic
270C2—21no—
271C2—21no—
272C2—26no—
273C2C2×#1387112yesdirect product factor
274C2C22×#34011yescharacteristic
275C2C22×#36111yescharacteristic
276C2C22×#36211yescharacteristic
subgroupG/H
center, socleC24#102
commutatorC8C22×C42
Frattini, cosocleC22×C8C24
prime radicalC25SD16
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×#1387C2direct (N × H)
10. C22×M32C4semidirect
15. C22×#108C4semidirect
23. #1387C22direct (N × H)
50. C2×M32C2×C4semidirect
57. C2×#108C2×C4semidirect
124. M32C22×C4semidirect
129. #108C22×C4semidirect
255. C22#1387direct (N × H)
273. C2C2×#1387direct (N × H)

Maximal subgroups

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

subgroupindex
C23×M322
C23×#1082
C23×#1072
C2×#13872