Finite Group Databasebeta

← C2×#3539 #62727

Subgroups — C2×#3539 #62727

Order 256.

3677 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3539C2132yesmaximal, direct product factor
2C2×#560C214yesmaximal, complemented
3C2×#553C212yesmaximal, complemented
4C2×#553C214yesmaximal, complemented
5C2×#552C212yesmaximal, complemented
6C2×#550C212yesmaximal, complemented
7C2×#547C212yesmaximal, complemented
8C2×#541C211yesmaximal, characteristic, complemented
9C2×#540C211yesmaximal, characteristic, complemented
10C2×#540C214yesmaximal, complemented
11C2×#539C214yesmaximal, complemented
12C2×#537C212yesmaximal, complemented
13C2×#525C211yesmaximal, characteristic, complemented
14C2×#525C212yesmaximal, complemented
15C22×#41C2211yescharacteristic
16C22×#41C2212yes—
17C22×#41C2212yescomplemented
18C24×C4C2211yescharacteristic
19#560C22164yescomplemented
20#553C22132yescomplemented
21#553C22164yescomplemented
22#552C22132yescomplemented
23#550C22132yescomplemented
24#547C22132yescomplemented
25#541C22116yescomplemented
26#540C22116yescomplemented
27#540C22164yescomplemented
28#539C22164yescomplemented
29#537C22132yescomplemented
30C2×C4⋊Q8C2211yescharacteristic, complemented
31C2×#127C2214yes—
32C2×#127C2211yescharacteristic, complemented
33C2×#126C2214yes—
34C2×#126C2214yes—
35C2×#126C2214yescomplemented
36C2×#125C2214yescomplemented
37C2×#125C2214yescomplemented
38C2×#125C2211yescharacteristic, complemented
39C2×#125C2211yescharacteristic, complemented
40#525C22116yescomplemented
41#525C22132yescomplemented
42C2×#123C2214yes—
43C2×#123C2212yes—
44C2×#123C2214yes—
45C2×#123C2214yes—
46C2×#123C2214yes—
47C2×#123C2214yescomplemented
48C2×#123C2214yescomplemented
49C2×C22⋊Q8C2212yes—
50C2×C22⋊Q8C2212yes—
51C2×C22⋊Q8C2212yes—
52C2×C22⋊Q8C2214yescomplemented
53C2×C22⋊Q8C2212yescomplemented
54C2×#121C2212yes—
55C2×#121C2212yes—
56C2×#121C2212yes—
57C2×#121C2214yes—
58C2×#121C2214yes—
59C2×#121C2214yes—
60C2×#121C2214yes—
61C2×#121C2214yes—
62C2×#121C2214yes—
63C2×#121C2214yescomplemented
64C2×#121C2212yescomplemented
65C2×#121C2214yescomplemented
66C2×#120C2212yes—
67C2×#120C2214yes—
68C2×C4×Q8C2212yescomplemented
69C2×C4×D4C2214yes—
70C2×C4×D4C2212yes—
71C2×C4×D4C2212yes—
72C2×C4×D4C2214yes—
73C2×C4×D4C2214yescomplemented
74C2×C4×D4C2212yescomplemented
75C2×C4×D4C2214yescomplemented
76C2×#117C2211yescharacteristic
77C2×#117C2212yes—
78C2×#117C2214yescomplemented
79C22×C4⋊C4C2212yescomplemented
80C22×C4⋊C4C2214yescomplemented
81C25C2311yescharacteristic
82C2×#41—44no—
83C2×#41—48no—
84C2×#41—48no—
85C2×#41C2318yes—
86C2×#41C23116yes—
87C2×#41C23116yescomplemented
88C22×Q8C2312yes—
89C22×Q8C2311yescharacteristic
90C22×D4C2312yes—
91C22×D4C2312yes—
92C22×D4C2314yes—
93C22×D4C2314yes—
94C22×D4C2312yes—
95C22×D4C2314yes—
96C22×D4C2314yes—
97C22×D4C2312yes—
98C22×D4C2314yes—
99C22×D4C2311yescharacteristic
100C22×D4C2314yes—
101C23×C4—44no—
102C23×C4C2312yes—
103C23×C4C2312yes—
104C23×C4C2311yescharacteristic
105C23×C4C2314yes—
106C23×C4C2314yes—
107C23×C4C2314yes—
108C23×C4C2314yes—
109C23×C4C2311yescharacteristic
110C23×C4C2312yes—
111C23×C4C2318yes—
112C4⋊Q8C2318yescomplemented
113#127C23132yes—
114#127C2318yescomplemented
115#126C23132yes—
116#126C23132yes—
117#126C23132yescomplemented
118#125C23132yescomplemented
119#125C23132yescomplemented
120#125C2318yescomplemented
121#125C2318yescomplemented
122#123C23132yes—
123#123C23116yes—
124#123C23132yes—
125#123C23132yes—
126#123C23132yes—
127#123C23132yescomplemented
128#123C23132yescomplemented
129C22⋊Q8C23116yes—
130C22⋊Q8C23116yes—
131C22⋊Q8C23116yes—
132C22⋊Q8C23132yescomplemented
133C22⋊Q8C23116yescomplemented
134#121C23116yes—
135#121C23116yes—
136#121C23116yes—
137#121C23132yes—
138#121C23132yes—
139#121C23132yes—
140#121C23132yes—
141#121C23132yes—
142#121C23132yes—
143#121C23132yescomplemented
144#121C23116yescomplemented
145#121C23132yescomplemented
146#120C23116yes—
147#120C23132yes—
148C4×Q8C23116yescomplemented
149C4×D4C23132yes—
150C4×D4C23116yes—
151C4×D4C23116yes—
152C4×D4C23132yes—
153C4×D4C23132yescomplemented
154C4×D4C23116yescomplemented
155C4×D4C23132yescomplemented
156#117C2318yes—
157#117C23116yes—
158#117C23132yescomplemented
159C2×C4⋊C4C2312yes—
160C2×C4⋊C4C2312yes—
161C2×C4⋊C4C2312yes—
162C2×C4⋊C4C2314yes—
163C2×C4⋊C4C2314yes—
164C2×C4⋊C4C2314yes—
165C2×C4⋊C4C2314yes—
166C2×C4⋊C4C2312yes—
167C2×C4⋊C4C2314yes—
168C2×C4⋊C4C2312yes—
169C2×C4⋊C4C2314yes—
170C2×C4⋊C4C2314yes—
171C2×C4⋊C4C2314yes—
172C2×C4⋊C4C23116yescomplemented
173C2×C4⋊C4C23132yescomplemented
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2314yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2314yes—
178C2×C22⋊C4C2314yes—
179C2×C22⋊C4C2312yes—
180C2×C22⋊C4C2312yes—
181C2×C22⋊C4C2314yes—
182C2×C22⋊C4C2314yes—
183C2×C22⋊C4C2312yes—
184C2×C22⋊C4C2314yes—
185C2×C22⋊C4C2314yes—
186C2×C22⋊C4C2314yes—
187C2×C42C2314yes—
188C2×C42C2311yescharacteristic
189C2×C42C2311yescharacteristic
190C2×C42C2314yes—
191C24—42no—
192C24—44no—
193C24C2411yescharacteristic
194C24C2412yes—
195C24C2412yes—
196C24C2414yes—
197C24C2414yes—
198#41—416no—
199#41—432no—
200#41—432no—
201C2×Q8—28no—
202C2×Q8—42no—
203C2×Q8C2418yes—
204C2×Q8C2414yes—
205C2×D4—28no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—48no—
215C2×D4—48no—
216C2×D4C2418yes—
217C2×D4C2418yes—
218C2×D4C24116yes—
219C2×D4C24116yes—
220C2×D4C2418yes—
221C2×D4C24116yes—
222C2×D4C24116yes—
223C2×D4C2418yes—
224C2×D4C24116yes—
225C2×D4C2414yes—
226C2×D4C24116yes—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—416no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4C2418yes—
238C22×C4C2418yes—
239C22×C4C2414yes—
240C22×C4C2412yes—
241C22×C4C2414yes—
242C22×C4C2412yes—
243C22×C4C2414yes—
244C22×C4C2412yes—
245C22×C4C2414yes—
246C22×C4C2412yes—
247C22×C4C2411yescharacteristic
248C22×C4C24116yes—
249C22×C4C24116yes—
250C22×C4C24116yes—
251C22×C4C24116yes—
252C22×C4C2414yes—
253C22×C4C2418yes—
254C22×C4C2411yescharacteristic
255C4⋊C4C2418yes—
256C4⋊C4C2418yes—
257C4⋊C4C2418yes—
258C4⋊C4C24116yes—
259C4⋊C4C24116yes—
260C4⋊C4C24116yes—
261C4⋊C4C24116yes—
262C4⋊C4C2418yes—
263C4⋊C4C24116yes—
264C4⋊C4C2418yes—
265C4⋊C4C24116yes—
266C4⋊C4C24116yes—
267C4⋊C4C24116yes—
268C22⋊C4C2418yes—
269C22⋊C4C24116yes—
270C22⋊C4C2418yes—
271C22⋊C4C24116yes—
272C22⋊C4C24116yes—
273C22⋊C4C2418yes—
274C22⋊C4C2418yes—
275C22⋊C4C24116yes—
276C22⋊C4C24116yes—
277C22⋊C4C2418yes—
278C22⋊C4C24116yes—
279C22⋊C4C24116yes—
280C22⋊C4C24116yes—
281C42C24116yes—
282C42C2414yes—
283C42C2414yes—
284C42C24116yes—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—28no—
289C23—24no—
290C23—28no—
291C23—24no—
292C23—28no—
293C23—24no—
294C23—24no—
295C23—24no—
296C23—44no—
297C23—48no—
298C23—416no—
299C23C2512yes—
300C23C2514yes—
301C23C2514yes—
302C23C2518yes—
303C23C2511yescharacteristic
304Q8—216no—
305Q8—44no—
306D4—216no—
307D4—48no—
308D4—416no—
309D4—416no—
310D4—416no—
311D4—48no—
312D4—416no—
313D4—48no—
314D4—44no—
315D4—416no—
316D4—416no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—24no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—216no—
326C2×C4—28no—
327C2×C4—216no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—416no—
331C2×C4—416no—
332C2×C4—416no—
333C2×C4—416no—
334C2×C4H(5,𝔽2)14yes—
335C2×C4C2514yes—
336C2×C4C2518yes—
337C2×C4C2514yes—
338C2×C4C2518yes—
339C2×C4C2514yes—
340C2×C4C2518yes—
341C2×C4C2514yes—
342C2×C4C2512yes—
343C2×C4C2512yes—
344C22—24no—
345C22—22no—
346C22—24no—
347C22—28no—
348C22—24no—
349C22—28no—
350C22—24no—
351C22—28no—
352C22—24no—
353C22—24no—
354C22—24no—
355C22—416no—
356C22—48no—
357C22—416no—
358C22—48no—
359C22—44no—
360C22C2×H(5,𝔽2)12yes—
361C22#58514yes—
362C22C2611yescharacteristic
363C4—24no—
364C4—22no—
365C4—24no—
366C4—28no—
367C4—24no—
368C4—28no—
369C4—28no—
370C4—24no—
371C4C2×H(5,𝔽2)14yes—
372C2—44no—
373C2—48no—
374C2—44no—
375C2—42no—
376C2#353914yesdirect product factor
377C2C22×H(5,𝔽2)11yescharacteristic
378C2C2×#58512yes—
subgroupG/H
center, socleC23C25
commutator, Frattini, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3539C2direct (N × H)
2. C2×#560C2semidirect
3. C2×#553C2semidirect
4. C2×#553C2semidirect
5. C2×#552C2semidirect
6. C2×#550C2semidirect
7. C2×#547C2semidirect
8. C2×#541C2semidirect
9. C2×#540C2semidirect
10. C2×#540C2semidirect
11. C2×#539C2semidirect
12. C2×#537C2semidirect
13. C2×#525C2semidirect
14. C2×#525C2semidirect
17. C22×#41C22semidirect
19. #560C22semidirect
20. #553C22semidirect
21. #553C22semidirect
22. #552C22semidirect
23. #550C22semidirect
24. #547C22semidirect
25. #541C22semidirect
26. #540C22semidirect
27. #540C22semidirect
28. #539C22semidirect
29. #537C22semidirect
30. C2×C4⋊Q8C22semidirect
32. C2×#127C22semidirect
35. C2×#126C22semidirect
36. C2×#125C22semidirect
37. C2×#125C22semidirect
38. C2×#125C22semidirect
39. C2×#125C22semidirect
40. #525C22semidirect
41. #525C22semidirect
47. C2×#123C22semidirect
48. C2×#123C22semidirect
52. C2×C22⋊Q8C22semidirect
53. C2×C22⋊Q8C22semidirect
63. C2×#121C22semidirect
64. C2×#121C22semidirect
65. C2×#121C22semidirect
68. C2×C4×Q8C22semidirect
73. C2×C4×D4C22semidirect
74. C2×C4×D4C22semidirect
75. C2×C4×D4C22semidirect
78. C2×#117C22semidirect
79. C22×C4⋊C4C22semidirect
80. C22×C4⋊C4C22semidirect
87. C2×#41C23semidirect
112. C4⋊Q8C23semidirect
114. #127C23semidirect
117. #126C23semidirect
118. #125C23semidirect
119. #125C23semidirect
120. #125C23semidirect
121. #125C23semidirect
127. #123C23semidirect
128. #123C23semidirect
132. C22⋊Q8C23semidirect
133. C22⋊Q8C23semidirect
143. #121C23semidirect
144. #121C23semidirect
145. #121C23semidirect
148. C4×Q8C23semidirect
153. C4×D4C23semidirect
154. C4×D4C23semidirect
155. C4×D4C23semidirect
158. #117C23semidirect
172. C2×C4⋊C4C23semidirect
173. C2×C4⋊C4C23semidirect
376. C2#3539direct (N × H)

Maximal subgroups

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

subgroupindex
#35392
C2×#5602
C2×#5532
C2×#5522
C2×#5502
C2×#5472
C2×#5412
C2×#5402
C2×#5392
C2×#5372
C2×#5252