Finite Group Databasebeta

← C2×#12557 #492856

Subgroups — C2×#12557 #492856

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M32C211yesmaximal, characteristic
2C22×#427C211yesmaximal, characteristic
3C23×#107C211yesmaximal, characteristic
4#12557C218yesmaximal, direct product factor
5C2×#1380C214yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C22×#117—21no—
8C23×C4⋊C4—21no—
9C22×M32—24no—
10C22×M32—24no—
11C22×M32C418yescomplemented
12C22×M32C2214yes—
13C23×C16C2212yes—
14C2×#427—24no—
15C2×#427—22no—
16C2×#427C418yescomplemented
17C2×#427C2214yes—
18C22×#107—24no—
19C22×#107—21no—
20C22×#107—22no—
21C22×#107C418yescomplemented
22C22×#107C2212yes—
23C22×#107C2212yes—
24C22×#107C2214yes—
25C22×#106—21no—
26#1380C22116yescomplemented
27C24×C4—41no—
28C24×C4D411yescharacteristic
29C23×C8C2×C414yes—
30C23×C8C2×C414yes—
31C23×C8C2×C414yes—
32C23×C8D411yescharacteristic
33C23×C8D412yes—
34C23×C8D414yes—
35C23×C8D414yes—
36C23×C8Q814yes—
37C23×C8C2311yescharacteristic
38C23×C8C2312yes—
39C2×#117—28no—
40C2×#117—24no—
41C2×#117—42no—
42C2×#117—44no—
43C22×C4⋊C4—28no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—42no—
48C22×C4⋊C4—41no—
49C22×C4⋊C4—44no—
50C22×C22⋊C4—41no—
51C22×C42—41no—
52C2×M32—232no—
53C2×M32—28no—
54C2×M32—28no—
55C2×M32C2×C4116yescomplemented
56C22×C16—24no—
57C22×C16—24no—
58C22×C16C2×C418yescomplemented
59C22×C16C2314yes—
60#427—216no—
61#427—28no—
62#427C2×C4116yescomplemented
63C2×#107—216no—
64C2×#107—28no—
65C2×#107—22no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—28no—
69C2×#107—22no—
70C2×#107—24no—
71C2×#107C2×C418yes—
72C2×#107C2×C4116yescomplemented
73C2×#107C2×C418yescomplemented
74C2×#107C2314yes—
75C2×#107C2314yes—
76C2×#106—28no—
77C2×#106—24no—
78C25D811yescharacteristic
79C23×C4—48no—
80C23×C4—44no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—88no—
84C23×C4C22⋊C414yes—
85C23×C4C22⋊C414yes—
86C23×C4C4⋊C414yes—
87C23×C4D812yes—
88C23×C4D811yescharacteristic
89C23×C4SD1614yes—
90C23×C4SD1614yes—
91C23×C4Q1614yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C22×C8C42116yes—
95C22×C8C22⋊C414yes—
96C22×C8C22⋊C414yes—
97C22×C8C22⋊C418yes—
98C22×C8C22⋊C418yes—
99C22×C8C22⋊C4116yes—
100C22×C8C4⋊C414yes—
101C22×C8C4⋊C418yes—
102C22×C8C4⋊C4116yes—
103C22×C8C4⋊C4116yes—
104C22×C8C22×C412yes—
105C22×C8C22×C414yes—
106C22×C8C22×C412yes—
107C22×C8C22×C414yes—
108C22×C8C22×C412yes—
109C22×C8C22×C414yes—
110C22×C8C2×D412yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C22×C8C2×D414yes—
115C22×C8C2×D414yes—
116C22×C8C2×Q812yes—
117C22×C8C2×Q814yes—
118C22×C8C2411yescharacteristic
119#117—216no—
120#117—416no—
121#117—48no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—216no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—416no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—44no—
135C2×C22⋊C4—48no—
136C2×C22⋊C4—44no—
137C2×C42—48no—
138C2×C42—44no—
139M32—232no—
140C2×C16—216no—
141C2×C16—24no—
142C2×C16—24no—
143C2×C16C22×C418yescomplemented
144#107—28no—
145#107—216no—
146#107—28no—
147#107—24no—
148#107—24no—
149#107C22×C418yes—
150#107C22×C418yescomplemented
151#106—216no—
152C24—24no—
153C24—24no—
154C24#10214yes—
155C24#10214yes—
156C24#10714yes—
157C24C2×D812yes—
158C24C2×D811yescharacteristic
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—416no—
163C22×C4—48no—
164C22×C4—816no—
165C22×C4—832no—
166C22×C4—84no—
167C22×C4—88no—
168C22×C4—84no—
169C22×C4#95116yes—
170C22×C4#10214yes—
171C22×C4#10218yes—
172C22×C4#10214yes—
173C22×C4#10218yes—
174C22×C4#103116yes—
175C22×C4#103116yes—
176C22×C4#106116yes—
177C22×C4#10718yes—
178C22×C4#10714yes—
179C22×C4C2×C22⋊C414yes—
180C22×C4C2×C22⋊C412yes—
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C22⋊C414yes—
183C22×C4C2×C4⋊C414yes—
184C22×C4C2×C4⋊C412yes—
185C22×C4C2×D812yes—
186C22×C4C2×D811yescharacteristic
187C22×C4C2×SD1614yes—
188C22×C4C2×SD1612yes—
189C22×C4C2×SD1612yes—
190C22×C4C2×SD1614yes—
191C22×C4C2×Q1614yes—
192C22×C4C2×Q1612yes—
193C22×C4C22×D411yescharacteristic
194C2×C8#95116yes—
195C2×C8#95132yes—
196C2×C8C2×C4214yes—
197C2×C8C2×C4218yes—
198C2×C8C2×C22⋊C414yes—
199C2×C8C2×C22⋊C418yes—
200C2×C8C2×C22⋊C414yes—
201C2×C8C2×C22⋊C412yes—
202C2×C8C2×C22⋊C414yes—
203C2×C8C2×C22⋊C412yes—
204C2×C8C2×C4⋊C414yes—
205C2×C8C2×C4⋊C414yes—
206C2×C8C2×C4⋊C418yes—
207C2×C8C2×C4⋊C418yes—
208C2×C8C2×C4⋊C412yes—
209C2×C8C2×C4⋊C414yes—
210C2×C8C23×C411yescharacteristic
211C2×C8C23×C411yescharacteristic
212C2×C8C23×C411yescharacteristic
213C2×C8C22×D411yescharacteristic
214C2×C8C22×D411yescharacteristic
215C2×C8C22×Q811yescharacteristic
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—48no—
221C4⋊C4—416no—
222C4⋊C4—48no—
223C22⋊C4—416no—
224C42—416no—
225C16—28no—
226C23—24no—
227C23—24no—
228C23—216no—
229C23—28no—
230C23—28no—
231C23—216no—
232C23#340116yes—
233C23C2×#10214yes—
234C23C2×#10214yes—
235C23C2×#10212yes—
236C23C2×#10212yes—
237C23C2×#10714yes—
238C23C2×#10712yes—
239C23#50914yes—
240C23#51014yes—
241C23C22×D811yescharacteristic
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—832no—
245C2×C4—84no—
246C2×C4—88no—
247C2×C4—84no—
248C2×C4—816no—
249C2×C4—816no—
250C2×C4#340132yes—
251C2×C4#340116yes—
252C2×C4C2×#9518yes—
253C2×C4C2×#9514yes—
254C2×C4C2×#10214yes—
255C2×C4C2×#10214yes—
256C2×C4C2×#10212yes—
257C2×C4C2×#10212yes—
258C2×C4C2×#10318yes—
259C2×C4C2×#10318yes—
260C2×C4C2×#10314yes—
261C2×C4C2×#10314yes—
262C2×C4C2×#10618yes—
263C2×C4C2×#10614yes—
264C2×C4C2×#10714yes—
265C2×C4C2×#10712yes—
266C2×C4C22×C22⋊C411yescharacteristic
267C2×C4C22×C22⋊C411yescharacteristic
268C2×C4C22×C4⋊C411yescharacteristic
269C2×C4C22×SD1611yescharacteristic
270C2×C4C22×SD1611yescharacteristic
271C2×C4C22×Q1611yescharacteristic
272C8C2×#9514yes—
273C8C2×#9518yes—
274C8C22×C4211yescharacteristic
275C8C22×C22⋊C411yescharacteristic
276C8C22×C4⋊C411yescharacteristic
277C8C22×C4⋊C411yescharacteristic
278C22—232no—
279C22—24no—
280C22—24no—
281C22—28no—
282C22—28no—
283C22C2×#34018yes—
284C22C2×#34014yes—
285C22#213918yes—
286C22#214014yes—
287C22#214114yes—
288C22#215918yes—
289C22C22×#10211yescharacteristic
290C22C22×#10211yescharacteristic
291C22C22×#10711yescharacteristic
292C22C2×#50912yes—
293C22C2×#51012yes—
294C4—88no—
295C4—88no—
296C4C2×#34018yes—
297C4C2×#34014yes—
298C4C22×#9511yescharacteristic
299C4C22×#10311yescharacteristic
300C4C22×#10311yescharacteristic
301C4C22×#10611yescharacteristic
302C2—28no—
303C2#1255718yesdirect product factor
304C2C22×#34011yescharacteristic
305C2C2×#214011yescharacteristic
306C2C2×#214111yescharacteristic
307C2C2×#213912yes—
308C2C2×#215912yes—
subgroupG/H
center, socleC24C2×D8
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. #12557C2direct (N × H)
5. C2×#1380C2semidirect
11. C22×M32C4semidirect
16. C2×#427C4semidirect
21. C22×#107C4semidirect
26. #1380C22semidirect
55. C2×M32C2×C4semidirect
58. C22×C16C2×C4semidirect
62. #427C2×C4semidirect
72. C2×#107C2×C4semidirect
73. C2×#107C2×C4semidirect
143. C2×C16C22×C4semidirect
150. #107C22×C4semidirect
303. C2#12557direct (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×#1072
#125572
C2×#13802