Finite Group Databasebeta

← C2×#3557 #62738

Subgroups — C2×#3557 #62738

Order 256.

3277 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
1#3557C2132yesmaximal, direct product factor
2C2×#563C214yesmaximal, complemented
3C2×#559C212yesmaximal, complemented
4C2×#557C212yesmaximal, complemented
5C2×#556C212yesmaximal, complemented
6C2×#554C212yesmaximal, complemented
7C2×#552C214yesmaximal, complemented
8C2×#542C212yesmaximal, complemented
9C2×#542C214yesmaximal, complemented
10C2×#541C211yesmaximal, characteristic, complemented
11C2×#540C211yesmaximal, characteristic
12C2×#539C214yesmaximal, complemented
13C2×#533C211yesmaximal, characteristic, complemented
14C2×#529C212yesmaximal, complemented
15#563C22164yescomplemented
16#559C22132yescomplemented
17#557C22132yescomplemented
18#556C22132yescomplemented
19#554C22132yescomplemented
20#552C22164yescomplemented
21#542C22132yescomplemented
22#542C22164yescomplemented
23#541C22116yescomplemented
24#540C22116yes—
25#539C22164yescomplemented
26#533C22116yescomplemented
27C2×C4⋊Q8C2212yes—
28C2×C4⋊Q8C2211yescharacteristic
29C2×C4⋊Q8C2212yes—
30C2×C4⋊Q8C2214yes—
31C2×#127C2211yescharacteristic
32#529C22132yescomplemented
33C2×#126C2214yes—
34C2×#126C2214yes—
35C2×#126C2214yes—
36C2×#126C2214yes—
37C2×#126C2214yes—
38C2×#125C2212yes—
39C2×#125C2212yes—
40C2×#125C2214yes—
41C2×#125C2214yes—
42C2×#125C2211yescharacteristic
43C2×#125C2214yes—
44C2×#125C2211yescharacteristic
45C2×#124C2212yes—
46C2×#124C2214yes—
47C2×#124C2212yes—
48C2×#124C2214yes—
49C2×#123C2214yes—
50C2×#123C2214yes—
51C2×#123C2214yes—
52C2×#123C2214yes—
53C2×#123C2214yes—
54C2×C22⋊Q8C2212yes—
55C2×C22⋊Q8C2212yes—
56C2×C22⋊Q8C2212yes—
57C2×C22⋊Q8C2214yes—
58C2×C22⋊Q8C2214yes—
59C2×C22⋊Q8C2214yes—
60C2×C22⋊Q8C2214yes—
61C2×C22⋊Q8C2214yes—
62C2×#121C2212yes—
63C2×#121C2214yes—
64C2×C4×Q8C2212yes—
65C2×C4×Q8C2212yes—
66C2×C4×Q8C2212yes—
67C2×C4×Q8C2212yes—
68C2×C4×Q8C2214yes—
69C2×C4×Q8C2212yes—
70C2×C4×D4C2214yes—
71C2×C4×D4C2212yes—
72C2×C4×D4C2214yes—
73C2×C4×D4C2212yes—
74C2×C4×D4C2212yes—
75C2×#117C2214yes—
76C2×#117C2211yescharacteristic
77C2×#117C2211yescharacteristic
78C2×#117C2214yes—
79C22×C4⋊C4C2214yes—
80C22×C42C2211yescharacteristic
81C22×Q8C2312yes—
82C22×Q8C2312yes—
83C22×Q8C2314yes—
84C22×Q8C2312yes—
85C22×D4C2312yes—
86C22×D4C2314yes—
87C22×D4C2314yes—
88C23×C4C2311yescharacteristic
89C23×C4C2314yes—
90C23×C4C2314yes—
91C23×C4C2314yes—
92C23×C4C2312yes—
93C4⋊Q8C23116yes—
94C4⋊Q8C2318yes—
95C4⋊Q8C23116yes—
96C4⋊Q8C23132yes—
97#127C2318yes—
98#126C23132yes—
99#126C23132yes—
100#126C23132yes—
101#126C23132yes—
102#126C23132yes—
103#125C23116yes—
104#125C23116yes—
105#125C23132yes—
106#125C23132yes—
107#125C2318yes—
108#125C23132yes—
109#125C2318yes—
110#124C23116yes—
111#124C23132yes—
112#124C23116yes—
113#124C23132yes—
114#123C23132yes—
115#123C23132yes—
116#123C23132yes—
117#123C23132yes—
118#123C23132yes—
119C22⋊Q8C23116yes—
120C22⋊Q8C23116yes—
121C22⋊Q8C23116yes—
122C22⋊Q8C23132yes—
123C22⋊Q8C23132yes—
124C22⋊Q8C23132yes—
125C22⋊Q8C23132yes—
126C22⋊Q8C23132yes—
127#121C23116yes—
128#121C23132yes—
129C4×Q8C23116yes—
130C4×Q8C23116yes—
131C4×Q8C23116yes—
132C4×Q8C23116yes—
133C4×Q8C23132yes—
134C4×Q8C23116yes—
135C4×D4C23132yes—
136C4×D4C23116yes—
137C4×D4C23132yes—
138C4×D4C23116yes—
139C4×D4C23116yes—
140#117C23132yes—
141#117C2318yes—
142#117C2318yes—
143#117C23132yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2314yes—
146C2×C4⋊C4C2314yes—
147C2×C4⋊C4C2314yes—
148C2×C4⋊C4C23132yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2314yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2314yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2312yes—
157C2×C4⋊C4C2314yes—
158C2×C4⋊C4C2312yes—
159C2×C4⋊C4C2314yes—
160C2×C4⋊C4C2312yes—
161C2×C4⋊C4C2314yes—
162C2×C4⋊C4C2312yes—
163C2×C4⋊C4C2314yes—
164C2×C4⋊C4C2314yes—
165C2×C22⋊C4C2314yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2314yes—
168C2×C22⋊C4C2311yescharacteristic
169C2×C22⋊C4C2311yescharacteristic
170C2×C22⋊C4C2314yes—
171C2×C22⋊C4C2314yes—
172C2×C22⋊C4C2314yes—
173C2×C22⋊C4C2314yes—
174C2×C22⋊C4C2314yes—
175C2×C22⋊C4C2314yes—
176C2×C22⋊C4C2314yes—
177C2×C42C2311yescharacteristic
178C2×C42C2314yes—
179C2×C42C2311yescharacteristic
180C2×C42C2312yes—
181C2×C42C2311yescharacteristic
182C2×C42C2314yes—
183C2×C42C2312yes—
184C2×C42C2314yes—
185C2×C42C2311yescharacteristic
186C2×C42C2318yes—
187C24C2411yescharacteristic
188C24C2414yes—
189C2×Q8—28no—
190C2×Q8—44no—
191C2×Q8—48no—
192C2×Q8—44no—
193C2×Q8C2418yes—
194C2×Q8C2418yes—
195C2×Q8C24116yes—
196C2×Q8C2418yes—
197C2×D4—28no—
198C2×D4—48no—
199C2×D4—48no—
200C2×D4C2418yes—
201C2×D4C24116yes—
202C2×D4C24116yes—
203C22×C4—24no—
204C22×C4—48no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4C2414yes—
209C22×C4C2412yes—
210C22×C4C2414yes—
211C22×C4C2412yes—
212C22×C4C2412yes—
213C22×C4C2411yescharacteristic
214C22×C4C2412yes—
215C22×C4C2414yes—
216C22×C4C2412yes—
217C22×C4C2412yes—
218C22×C4C2414yes—
219C22×C4C2418yes—
220C22×C4C24116yes—
221C22×C4C24116yes—
222C22×C4C24116yes—
223C22×C4C2411yescharacteristic
224C4⋊C4C2418yes—
225C4⋊C4C24116yes—
226C4⋊C4C24116yes—
227C4⋊C4C24116yes—
228C4⋊C4C2418yes—
229C4⋊C4C24116yes—
230C4⋊C4C2418yes—
231C4⋊C4C24116yes—
232C4⋊C4C2418yes—
233C4⋊C4C2418yes—
234C4⋊C4C2418yes—
235C4⋊C4C2418yes—
236C4⋊C4C24116yes—
237C4⋊C4C2418yes—
238C4⋊C4C24116yes—
239C4⋊C4C2418yes—
240C4⋊C4C24116yes—
241C4⋊C4C2418yes—
242C4⋊C4C24116yes—
243C4⋊C4C24116yes—
244C22⋊C4C24116yes—
245C22⋊C4C2418yes—
246C22⋊C4C24116yes—
247C22⋊C4C2414yes—
248C22⋊C4C2414yes—
249C22⋊C4C24116yes—
250C22⋊C4C24116yes—
251C22⋊C4C24116yes—
252C22⋊C4C24116yes—
253C22⋊C4C24116yes—
254C22⋊C4C24116yes—
255C22⋊C4C24116yes—
256C42C2414yes—
257C42C2418yes—
258C42C2414yes—
259C42C24116yes—
260C42C2414yes—
261C42C2418yes—
262C42C2414yes—
263C42C24116yes—
264C42C24116yes—
265C23—22no—
266C23—28no—
267C23—28no—
268C23—24no—
269C23—28no—
270C23C2512yes—
271C23C2518yes—
272C23C2511yescharacteristic
273Q8—216no—
274Q8—48no—
275Q8—416no—
276Q8—48no—
277D4—216no—
278D4—416no—
279D4—416no—
280C2×C4—22no—
281C2×C4—28no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—416no—
292C2×C4—48no—
293C2×C4—416no—
294C2×C4—416no—
295C2×C42−1+414yes—
296C2×C4C2514yes—
297C2×C4C2518yes—
298C2×C4C2514yes—
299C2×C4C2514yes—
300C2×C4C2512yes—
301C2×C4C2514yes—
302C2×C4C2518yes—
303C2×C4C2514yes—
304C2×C4C2514yes—
305C2×C4C2518yes—
306C2×C4C2512yes—
307C22—22no—
308C22—28no—
309C22—28no—
310C22—24no—
311C22—28no—
312C22—44no—
313C22—416no—
314C22C2×2−1+412yes—
315C22#58514yes—
316C22C2611yescharacteristic
317C4—22no—
318C4—28no—
319C4—24no—
320C4—28no—
321C4—24no—
322C4—24no—
323C4—28no—
324C4—24no—
325C4—24no—
326C4—24no—
327C4C2×2−1+414yes—
328C2—42no—
329C2—48no—
330C2#355714yesdirect product factor
331C2C22×2−1+411yescharacteristic
332C2C2×#58512yes—
subgroupG/H
center, socleC23C25
commutator, Frattini, cosocleC22C26
prime radicalC2×#540C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3557C2direct (N × H)
2. C2×#563C2semidirect
3. C2×#559C2semidirect
4. C2×#557C2semidirect
5. C2×#556C2semidirect
6. C2×#554C2semidirect
7. C2×#552C2semidirect
8. C2×#542C2semidirect
9. C2×#542C2semidirect
10. C2×#541C2semidirect
12. C2×#539C2semidirect
13. C2×#533C2semidirect
14. C2×#529C2semidirect
15. #563C22semidirect
16. #559C22semidirect
17. #557C22semidirect
18. #556C22semidirect
19. #554C22semidirect
20. #552C22semidirect
21. #542C22semidirect
22. #542C22semidirect
23. #541C22semidirect
25. #539C22semidirect
26. #533C22semidirect
32. #529C22semidirect
330. C2#3557direct (N × H)

Maximal subgroups

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

subgroupindex
#35572
C2×#5632
C2×#5592
C2×#5572
C2×#5562
C2×#5542
C2×#5522
C2×#5422
C2×#5412
C2×#5402
C2×#5392
C2×#5332
C2×#5292