Finite Group Databasebeta

← C2×#12920 #494235

Subgroups — C2×#12920 #494235

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic
2C22×#367C212yesmaximal, complemented
3C22×C22⋊C16C211yesmaximal, characteristic
4C2×#1808C211yesmaximal, characteristic
5#12920C218yesmaximal, direct product factor
6C2×#1380C212yesmaximal, complemented
7C24×C8C2211yescharacteristic
8C23×C4⋊C4—21no—
9C23×C16C2212yes—
10C22×#107—28no—
11C22×#107—22no—
12C22×#107—21no—
13C22×#107C2211yescharacteristic
14C22×#107C2214yes—
15C22×#107C2211yescharacteristic
16C22×#107C2212yes—
17C2×#380—21no—
18C2×#367—28no—
19C2×#367C2218yescomplemented
20C2×C22⋊C16C418yescomplemented
21C2×C22⋊C16C2214yes—
22#1808—22no—
23#1808C2214yes—
24C2×#340—21no—
25#1380C2218yescomplemented
26C24×C4—41no—
27C24×C4D411yescharacteristic
28C23×C8—24no—
29C23×C8—24no—
30C23×C8C2×C414yes—
31C23×C8D412yes—
32C23×C8D411yescharacteristic
33C23×C8Q814yes—
34C23×C8C2312yes—
35C23×C8C2311yescharacteristic
36C22×C4⋊C4—28no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—44no—
42C22×C4⋊C4—42no—
43C22×C4⋊C4—41no—
44C22×C22⋊C4—41no—
45C22×C16—24no—
46C22×C16—24no—
47C22×C16C2×C418yescomplemented
48C22×C16C2314yes—
49C2×#107—28no—
50C2×#107—216no—
51C2×#107—24no—
52C2×#107—22no—
53C2×#107—24no—
54C2×#107—216no—
55C2×#107—24no—
56C2×#107—24no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107D418yes—
60C2×#107C2312yes—
61C2×#107C2312yes—
62C2×#107C2314yes—
63#380—24no—
64#380—42no—
65C2×#95—41no—
66#367—216no—
67C22⋊C16C2×C4116yescomplemented
68#340—24no—
69C25D811yescharacteristic
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—48no—
75C23×C4—44no—
76C23×C4—42no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—84no—
80C23×C4—84no—
81C23×C4C4⋊C414yes—
82C23×C4D812yes—
83C23×C4D811yescharacteristic
84C23×C4Q1614yes—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—216no—
90C22×C8—24no—
91C22×C8—216no—
92C22×C8—24no—
93C22×C8—24no—
94C22×C8—24no—
95C22×C8C4⋊C414yes—
96C22×C8C4⋊C418yes—
97C22×C8C22×C414yes—
98C22×C8C22×C412yes—
99C22×C8C2×D412yes—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D414yes—
102C22×C8C2×Q814yes—
103C22×C8C2×Q812yes—
104C22×C8#4114yes—
105C22×C8C2411yescharacteristic
106C2×C4⋊C4—216no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—48no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—416no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—84no—
125C2×C16—216no—
126C2×C16—24no—
127C2×C16—24no—
128C2×C16C22×C418yescomplemented
129#107—216no—
130#107—28no—
131#107—24no—
132#107—24no—
133#107—24no—
134#107—42no—
135#107—42no—
136#107C2×D418yes—
137#95—44no—
138C24—22no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24#10714yes—
143C24SD3214yes—
144C24SD3214yes—
145C24C2×D812yes—
146C24C2×D811yescharacteristic
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—28no—
154C22×C4—28no—
155C22×C4—24no—
156C22×C4—28no—
157C22×C4—24no—
158C22×C4—28no—
159C22×C4—416no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—88no—
165C22×C4—832no—
166C22×C4—88no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4#10714yes—
176C22×C4#10718yes—
177C22×C4C2×C4⋊C414yes—
178C22×C4C2×C4⋊C412yes—
179C22×C4#12012yes—
180C22×C4#12312yes—
181C22×C4C2×D812yes—
182C22×C4C2×D811yescharacteristic
183C22×C4C2×Q1614yes—
184C22×C4C2×Q1612yes—
185C22×C4AGL(1,ℤ/8)12yes—
186C22×C4#13712yes—
187C22×C4C22×D411yescharacteristic
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—28no—
191C2×C8—28no—
192C2×C8—28no—
193C2×C8—216no—
194C2×C8C2×C4⋊C414yes—
195C2×C8C2×C4⋊C412yes—
196C2×C8C4×D418yes—
197C2×C8#12012yes—
198C2×C8C22⋊Q818yes—
199C2×C8#12312yes—
200C2×C8C23×C411yescharacteristic
201C2×C8C22×D412yes—
202C2×C8C22×Q811yescharacteristic
203C2×C8C2×#4112yes—
204C4⋊C4—28no—
205C4⋊C4—416no—
206C4⋊C4—44no—
207C4⋊C4—48no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C22⋊C4—88no—
215C16—28no—
216C23—22no—
217C23—28no—
218C23—22no—
219C23—22no—
220C23—28no—
221C23—28no—
222C23—28no—
223C23—22no—
224C23—28no—
225C23—28no—
226C23#367116yescomplemented
227C23C2×#10714yes—
228C23C2×#10712yes—
229C23#44712yes—
230C23#48012yes—
231C23C2×SD3214yes—
232C23C2×SD3214yes—
233C23C2×SD3212yes—
234C23C2×SD3212yes—
235C23#50912yes—
236C23#51012yes—
237C23C22×D811yescharacteristic
238C2×C4—22no—
239C2×C4—28no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—216no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—48no—
247C2×C4—832no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4—816no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—164no—
258C2×C4—164no—
259C2×C4#38514yes—
260C2×C4C2×#10714yes—
261C2×C4C2×#10712yes—
262C2×C4#42814yes—
263C2×C4#44712yes—
264C2×C4#45114yes—
265C2×C4#48012yes—
266C2×C4#48414yes—
267C2×C4C22×C4⋊C411yescharacteristic
268C2×C4C2×#12011yescharacteristic
269C2×C4C2×#12311yescharacteristic
270C2×C4C22×Q1611yescharacteristic
271C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
272C2×C4C2×#13711yescharacteristic
273C8—24no—
274C8#38514yes—
275C8C2×C4×D412yes—
276C8C2×C22⋊Q812yes—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—232no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22#185214yes—
285C22C2×#36714yes—
286C22C2×#36718yescomplemented
287C22#215914yes—
288C22#218514yes—
289C22#218714yes—
290C22#223314yes—
291C22#223514yes—
292C22C22×#10711yescharacteristic
293C22C2×#44711yescharacteristic
294C22C2×#48011yescharacteristic
295C22C22×SD3211yescharacteristic
296C22C22×SD3211yescharacteristic
297C22C2×#50911yescharacteristic
298C22C2×#51011yescharacteristic
299C4—24no—
300C4—88no—
301C4—162no—
302C4—162no—
303C4#185214yes—
304C4C2×#38511yescharacteristic
305C4C2×#42811yescharacteristic
306C4C2×#45111yescharacteristic
307C4C2×#48411yescharacteristic
308C2—28no—
309C2#1292018yesdirect product factor
310C2C2×#185211yescharacteristic
311C2C22×#36711yescharacteristic
312C2C2×#215911yescharacteristic
313C2C2×#218711yescharacteristic
314C2C2×#218511yescharacteristic
315C2C2×#223511yescharacteristic
316C2C2×#223311yescharacteristic
subgroupG/H
center, socleC24C2×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C2×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
2. C22×#367C2semidirect
5. #12920C2direct (N × H)
6. C2×#1380C2semidirect
19. C2×#367C22semidirect
20. C2×C22⋊C16C4semidirect
25. #1380C22semidirect
47. C22×C16C2×C4semidirect
67. C22⋊C16C2×C4semidirect
128. C2×C16C22×C4semidirect
226. C23#367semidirect
286. C22C2×#367semidirect
309. C2#12920direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1072
C22×#3672
C22×C22⋊C162
C2×#18082
#129202
C2×#13802