Finite Group Databasebeta

← C22×#1288 #181208

Subgroups — C22×#1288 #181208

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C22×C4×M16C211yesmaximal, characteristic
3C22×#383C211yesmaximal, characteristic
4C2×#1288C2112yesmaximal, direct product factor
5C23×M16—21no—
6C23×C42C2211yescharacteristic
7C2×#432—23no—
8C2×#432—23no—
9C2×#432C416yescomplemented
10C2×#432C2216yes—
11C22×C4⋊C8—21no—
12C22×C22⋊C8—21no—
13C2×C4×M16—23no—
14C2×C4×M16—23no—
15C2×C4×M16C416yescomplemented
16C2×C4×M16C2216yes—
17C22×#97—21no—
18C22×#97C412yescomplemented
19C22×C4×C8C412yescomplemented
20C2×#383—23no—
21C2×#383—23no—
22C2×#383C416yes—
23C2×#383C2216yes—
24C22×#95—21no—
25C22×#95—21no—
26C22×#95C412yes—
27#1288C22116yesdirect product factor
28C24×C4—22no—
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C24×C4D411yescharacteristic
32C22×M16—216no—
33C22×M16—26no—
34C22×M16—26no—
35C23×C8—22no—
36C23×C8—22no—
37C22×C42C2×C413yes—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C413yes—
41C22×C42C2×C413yes—
42C22×C42D413yes—
43C22×C42D413yes—
44C22×C42D411yescharacteristic
45C22×C42D413yes—
46C22×C42Q813yes—
47C22×C42Q811yescharacteristic
48C22×C42C2313yes—
49#432—212no—
50#432—26no—
51#432—26no—
52#432C2×C4112yescomplemented
53#432C2314yes—
54C2×C4⋊C8—26no—
55C2×C4⋊C8—26no—
56C2×C22⋊C8—26no—
57C2×C22⋊C8—26no—
58C4×M16—212no—
59C4×M16—26no—
60C4×M16—26no—
61C4×M16C2×C4112yescomplemented
62C4×M16C2314yes—
63C2×#97—26no—
64C2×#97—26no—
65C2×#97—26no—
66C2×#97C2×C4112yescomplemented
67C2×C4×C8—26no—
68C2×C4×C8C2×C4112yescomplemented
69#383—212no—
70#383—26no—
71#383—26no—
72#383C2×C4112yes—
73#383C2314yes—
74C2×#95—26no—
75C2×#95—26no—
76C2×#95—26no—
77C2×#95—26no—
78C2×#95—26no—
79C2×#95C2×C4112yes—
80C25C22⋊C411yescharacteristic
81C23×C4—28no—
82C23×C4—212no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4—48no—
86C23×C4—48no—
87C23×C4—48no—
88C23×C4C4211yescharacteristic
89C23×C4C4213yes—
90C23×C4C22⋊C413yes—
91C23×C4C22⋊C413yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C413yes—
95C23×C4C22⋊C413yes—
96C23×C4C22⋊C413yes—
97C23×C4C4⋊C413yes—
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C413yes—
102C23×C4C4⋊C413yes—
103C23×C4C22×C413yes—
104C23×C4C2×D413yes—
105C23×C4C2×D413yes—
106C2×M16—248no—
107C2×M16—248no—
108C2×M16—212no—
109C2×M16—24no—
110C22×C8—216no—
111C22×C8—212no—
112C22×C8—212no—
113C22×C8—48no—
114C2×C42C4213yes—
115C2×C42C4213yes—
116C2×C42C4216yes—
117C2×C42C22⋊C416yes—
118C2×C42C22⋊C413yes—
119C2×C42C22⋊C413yes—
120C2×C42C22⋊C413yes—
121C2×C42C22⋊C416yes—
122C2×C42C22⋊C413yes—
123C2×C42C22⋊C416yes—
124C2×C42C4⋊C413yes—
125C2×C42C4⋊C413yes—
126C2×C42C4⋊C413yes—
127C2×C42C4⋊C413yes—
128C2×C42C4⋊C413yes—
129C2×C42C4⋊C413yes—
130C2×C42C4⋊C416yes—
131C2×C42C4⋊C416yes—
132C2×C42C4⋊C416yes—
133C2×C42C22×C413yes—
134C2×C42C22×C413yes—
135C2×C42C22×C413yes—
136C2×C42C22×C413yes—
137C2×C42C22×C413yes—
138C2×C42C2×D413yes—
139C2×C42C2×D413yes—
140C2×C42C2×D413yes—
141C2×C42C2×D413yes—
142C2×C42C2×Q813yes—
143C2×C42C2×Q813yes—
144C2×C42C2411yescharacteristic
145C4⋊C8—212no—
146C4⋊C8—24no—
147C22⋊C8—212no—
148C22⋊C8—24no—
149#97—212no—
150#97—212no—
151#97—24no—
152#97C22×C418yescomplemented
153C4×C8—212no—
154C4×C8C22×C418yescomplemented
155#95—212no—
156#95—212no—
157#95—212no—
158#95—24no—
159#95—24no—
160#95C22×C418yes—
161C24—24no—
162C24—24no—
163C24#9513yes—
164C24#9511yescharacteristic
165C24#10418yes—
166C24C2×C22⋊C413yes—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—212no—
171C22×C4—212no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—212no—
176C22×C4—212no—
177C22×C4—28no—
178C22×C4—46no—
179C22×C4—46no—
180C22×C4—46no—
181C22×C4—46no—
182C22×C4—46no—
183C22×C4—46no—
184C22×C4—46no—
185C22×C4—46no—
186C22×C4—424no—
187C22×C4—424no—
188C22×C4—424no—
189C22×C4—832no—
190C22×C4#9513yes—
191C22×C4#9513yes—
192C22×C4#9516yes—
193C22×C4#9513yes—
194C22×C4#9513yes—
195C22×C4#9513yes—
196C22×C4#9513yes—
197C22×C4#9516yes—
198C22×C4#9516yes—
199C22×C4C2×C4213yes—
200C22×C4C2×C4213yes—
201C22×C4C2×C22⋊C413yes—
202C22×C4C2×C22⋊C413yes—
203C22×C4C2×C22⋊C413yes—
204C22×C4C2×C22⋊C413yes—
205C22×C4C2×C22⋊C413yes—
206C22×C4C2×C22⋊C413yes—
207C22×C4C2×C22⋊C413yes—
208C22×C4C2×C4⋊C413yes—
209C22×C4C2×C4⋊C413yes—
210C22×C4C2×C4⋊C413yes—
211C22×C4C2×C4⋊C413yes—
212C22×C4C2×C4⋊C413yes—
213C22×C4C2×C4⋊C413yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D411yescharacteristic
217M16—216no—
218M16—248no—
219C2×C8—248no—
220C2×C8—28no—
221C2×C8—28no—
222C2×C8—424no—
223C42#9516yes—
224C42#9516yes—
225C42#9516yes—
226C42#9516yes—
227C42C2×C4213yes—
228C42C2×C4213yes—
229C42C2×C22⋊C413yes—
230C42C2×C22⋊C413yes—
231C42C2×C22⋊C413yes—
232C42C2×C22⋊C413yes—
233C42C2×C4⋊C413yes—
234C42C2×C4⋊C413yes—
235C42C2×C4⋊C413yes—
236C42C2×C4⋊C413yes—
237C42C2×C4⋊C413yes—
238C42C2×C4⋊C413yes—
239C42C23×C411yescharacteristic
240C42C23×C411yescharacteristic
241C42C22×D411yescharacteristic
242C42C22×Q811yescharacteristic
243C23—232no—
244C23—26no—
245C23—26no—
246C23—26no—
247C23—26no—
248C23H(3,ℤ/4)14yes—
249C23#339112yes—
250C23#33914yes—
251C23#34414yes—
252C23C2×#9513yes—
253C23C2×#9513yes—
254C23C2×#104112yes—
255C23C22×C22⋊C411yescharacteristic
256C2×C4—22no—
257C2×C4—26no—
258C2×C4—26no—
259C2×C4—26no—
260C2×C4—26no—
261C2×C4—26no—
262C2×C4—26no—
263C2×C4—26no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—26no—
270C2×C4—26no—
271C2×C4—26no—
272C2×C4—26no—
273C2×C4—46no—
274C2×C4—46no—
275C2×C4—43no—
276C2×C4—43no—
277C2×C4—43no—
278C2×C4—43no—
279C2×C4—43no—
280C2×C4—43no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—848no—
287C2×C4C2×#9513yes—
288C2×C4C2×#9513yes—
289C2×C4C2×#9513yes—
290C2×C4C2×#9513yes—
291C2×C4C2×#9513yes—
292C2×C4C2×#9513yes—
293C2×C4C22×C4211yescharacteristic
294C2×C4C22×C22⋊C411yescharacteristic
295C2×C4C22×C22⋊C411yescharacteristic
296C2×C4C22×C4⋊C411yescharacteristic
297C2×C4C22×C4⋊C411yescharacteristic
298C2×C4C22×C4⋊C411yescharacteristic
299C8—216no—
300C8—48no—
301C22—224no—
302C22—224no—
303C22—21no—
304C22—23no—
305C22—21no—
306C22—23no—
307C22#1288116yesdirect product factor
308C22C2×H(3,ℤ/4)16yes—
309C22C2×#33916yes—
310C22C2×#33916yes—
311C22C2×#34416yes—
312C22C22×#9511yescharacteristic
313C22C22×#10412yes—
314C4—21no—
315C4—21no—
316C4—23no—
317C4—23no—
318C4—21no—
319C4—21no—
320C4—23no—
321C4—23no—
322C4—21no—
323C4—23no—
324C4—23no—
325C4—21no—
326C4—88no—
327C2—22no—
328C2—26no—
329C2C2×#1288112yesdirect product factor
330C2C22×H(3,ℤ/4)11yescharacteristic
331C2C22×#33911yescharacteristic
332C2C22×#34411yescharacteristic
subgroupG/H
center, socleC24#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1288C2direct (N × H)
9. C2×#432C4semidirect
15. C2×C4×M16C4semidirect
18. C22×#97C4semidirect
19. C22×C4×C8C4semidirect
27. #1288C22direct (N × H)
52. #432C2×C4semidirect
61. C4×M16C2×C4semidirect
66. C2×#97C2×C4semidirect
68. C2×C4×C8C2×C4semidirect
152. #97C22×C4semidirect
154. C4×C8C22×C4semidirect
307. C22#1288direct (N × H)
329. C2C2×#1288direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4322
C22×C4×M162
C22×#3832
C2×#12882