Finite Group Databasebeta

← C2×#2745 #52181

Subgroups — C2×#2745 #52181

Order 256.

1067 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2745C2116yesmaximal, direct product factor
2C2×#401C211yesmaximal, characteristic, complemented
3C2×#400C211yesmaximal, characteristic, complemented
4C2×#400C211yesmaximal, characteristic, complemented
5C2×#397C211yesmaximal, characteristic
6C2×#395C211yesmaximal, characteristic, complemented
7C2×#393C211yesmaximal, characteristic
8C2×#391C211yesmaximal, characteristic, complemented
9C2×#388C211yesmaximal, characteristic
10C2×#388C211yesmaximal, characteristic
11C2×#387C211yesmaximal, characteristic, complemented
12C2×#387C211yesmaximal, characteristic, complemented
13C2×#387C211yesmaximal, characteristic, complemented
14C2×#385C211yesmaximal, characteristic
15C2×#381C211yesmaximal, characteristic
16C2×#380C211yesmaximal, characteristic
17C23×Q8C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C42C2211yescharacteristic
33C22×C42C2211yescharacteristic
34#401C2218yescomplemented
35#400C2218yescomplemented
36#400C2218yescomplemented
37#397C2218yes—
38#395C2218yescomplemented
39#393C2218yes—
40#391C2218yescomplemented
41#388C2218yes—
42#388C2218yes—
43#387C2218yescomplemented
44#387C2218yescomplemented
45#387C2218yescomplemented
46#385C2218yes—
47#381C2218yes—
48#380C2218yes—
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C2×#95C2211yescharacteristic
59C2×#95C2211yescharacteristic
60C2×#95C2211yescharacteristic
61C2×#95C2211yescharacteristic
62C2×#95C2211yescharacteristic
63C2×#95C2211yescharacteristic
64C2×#95C2211yescharacteristic
65C2×#95C2211yescharacteristic
66C2×#95C2211yescharacteristic
67C25C2311yescharacteristic
68C22×Q8—24no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8C2314yes—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—42no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2314yes—
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4C2314yes—
98C2×C4⋊C4C2314yes—
99C2×C4⋊C4C2314yes—
100C2×C4⋊C4C2314yes—
101C2×C4⋊C4C2314yes—
102C2×C4⋊C4C2314yes—
103C2×C4⋊C4C2314yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4C2314yes—
111C2×C22⋊C4C2314yes—
112C2×C22⋊C4C2314yes—
113C2×C22⋊C4C2314yes—
114C2×C22⋊C4C2314yes—
115C2×C22⋊C4C2314yes—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42C2314yes—
119C2×C42C2314yes—
120#95C2314yes—
121#95C2314yes—
122#95C2314yes—
123#95C2314yes—
124#95C2314yes—
125#95C2314yes—
126#95C2314yes—
127#95C2314yes—
128#95C2314yes—
129#95C2314yes—
130#95C2314yes—
131#95C2314yes—
132#95C2314yes—
133#95C2314yes—
134#95C2314yes—
135#95C2314yes—
136#95C2314yes—
137#95C2314yes—
138C24—22no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24—22no—
143C24—22no—
144C24#4114yes—
145C24C2412yes—
146C24C2411yescharacteristic
147C2×Q8—28no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—416no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—28no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—28no—
193C22×C4—44no—
194C22×C4—416no—
195C22×C4#4114yes—
196C22×C4#4114yes—
197C22×C4C2412yes—
198C22×C4C2412yes—
199C22×C4C2412yes—
200C22×C4C2412yes—
201C22×C4C2412yes—
202C22×C4C2412yes—
203C22×C4C2412yes—
204C22×C4C2412yes—
205C22×C4C2412yes—
206C22×C4C2412yes—
207C22×C4C2412yes—
208C22×C4C2412yes—
209C22×C4C2412yes—
210C22×C4C2412yes—
211C4⋊C4—28no—
212C4⋊C4—28no—
213C4⋊C4—28no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C22⋊C4—28no—
219C22⋊C4—28no—
220C22⋊C4—28no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C42—44no—
225C42—44no—
226C23—28no—
227C23—28no—
228C23—28no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—44no—
239C23C2×#4114yes—
240C23C2×#4112yes—
241C23C2×#4112yes—
242C23C2×#4112yes—
243C23H(5,𝔽2)12yes—
244C23H(5,𝔽2)12yes—
245C232−1+412yes—
246C232−1+412yes—
247C23C2511yescharacteristic
248Q8—416no—
249C2×C4—28no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—416no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4C2×#4114yes—
305C2×C4C2×#4114yes—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—416no—
313C22—42no—
314C22#53814yes—
315C22#53914yes—
316C22#54214yes—
317C22#55114yes—
318C22#55214yes—
319C22#55614yes—
320C22#56314yes—
321C22C22×#4111yescharacteristic
322C22C22×#4111yescharacteristic
323C22C22×#4111yescharacteristic
324C22C2×H(5,𝔽2)11yescharacteristic
325C22C2×H(5,𝔽2)11yescharacteristic
326C22C2×2−1+411yescharacteristic
327C22C2×2−1+411yescharacteristic
328C4—24no—
329C4—24no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—42no—
342C2—44no—
343C2#274518yesdirect product factor
344C2C2×#53811yescharacteristic
345C2C2×#53911yescharacteristic
346C2C2×#54211yescharacteristic
347C2C2×#55111yescharacteristic
348C2C2×#55211yescharacteristic
349C2C2×#55611yescharacteristic
350C2C2×#56311yescharacteristic
subgroupG/H
center, socleC24C24
commutator, Frattini, cosocleC23C25
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2745C2direct (N × H)
2. C2×#401C2semidirect
3. C2×#400C2semidirect
4. C2×#400C2semidirect
6. C2×#395C2semidirect
8. C2×#391C2semidirect
11. C2×#387C2semidirect
12. C2×#387C2semidirect
13. C2×#387C2semidirect
34. #401C22semidirect
35. #400C22semidirect
36. #400C22semidirect
38. #395C22semidirect
40. #391C22semidirect
43. #387C22semidirect
44. #387C22semidirect
45. #387C22semidirect
343. C2#2745direct (N × H)

Maximal subgroups

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

subgroupindex
#27452
C2×#4012
C2×#4002
C2×#3972
C2×#3952
C2×#3932
C2×#3912
C2×#3882
C2×#3872
C2×#3852
C2×#3812
C2×#3802