Finite Group Databasebeta

← #62876

Subgroups — #62876

Order 256.

3707 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3562C218yesmaximal, complemented
2#3558C218yesmaximal, complemented
3#3546C218yesmaximal, complemented
4#3543C218yesmaximal, complemented
5#3534C218yesmaximal, complemented
6#3532C218yesmaximal, complemented
7#3510C212yesmaximal, complemented
8#3508C212yesmaximal, complemented
9#3500C212yesmaximal, complemented
10#3498C212yesmaximal, complemented
11#3487C211yesmaximal, characteristic, complemented
12#3485C211yesmaximal, characteristic, complemented
13#3484C212yesmaximal, complemented
14#3455C212yesmaximal, complemented
15C2×#535C211yesmaximal, characteristic, complemented
16C2×2−1+4C2211yescharacteristic, complemented
17C2×H(5,𝔽2)C2212yescomplemented
18C2×H(5,𝔽2)C2211yescharacteristic, complemented
19C22×#41C2211yescharacteristic, complemented
20C23×D4C2211yescharacteristic, complemented
21#562C22116yescomplemented
22H(3,𝔽4)C2218yescomplemented
23#560C22116yescomplemented
24#560C2218yescomplemented
25#560C22116yescomplemented
26#559C2218yescomplemented
27#559C2218yescomplemented
28#556C2218yescomplemented
29#553C22116yescomplemented
30#551C22116yescomplemented
31#551C2218yescomplemented
32#550C2218yescomplemented
33#550C2218yescomplemented
34#550C2218yescomplemented
35#548C2218yescomplemented
36#548C2218yescomplemented
37#547C2218yescomplemented
38#546C2218yescomplemented
39#546C2218yescomplemented
40#546C22116yescomplemented
41#546C22116yescomplemented
42#546C22116yescomplemented
43#546C2218yescomplemented
44#546C22116yescomplemented
45#546C2218yescomplemented
46D42C2218yescomplemented
47D42C2218yescomplemented
48D42C22116yescomplemented
49D42C2218yescomplemented
50D42C2218yescomplemented
51#542C2214yescomplemented
52#542C2214yescomplemented
53#542C2218yescomplemented
54#541C2214yescomplemented
55#540C2214yescomplemented
56#540C2218yescomplemented
57#540C22116yescomplemented
58#539C2218yescomplemented
59#539C2218yescomplemented
60#538C2218yescomplemented
61#538C22116yescomplemented
62#538C22116yescomplemented
63#538C22116yescomplemented
64#538C2218yescomplemented
65#537C2218yescomplemented
66#536C2212yescomplemented
67#536C2214yescomplemented
68#535C2212yes—
69#535C2218yescomplemented
70#535C2218yescomplemented
71#535C2212yescomplemented
72#535C2214yescomplemented
73#535C2212yescomplemented
74#535C22116yescomplemented
75#535C2214yescomplemented
76#535C2214yescomplemented
77#535C2214yescomplemented
78#534C2218yescomplemented
79#534C2214yescomplemented
80#534C22116yescomplemented
81#534C2218yescomplemented
82#533C2212yescomplemented
83#532C2212yescomplemented
84#532C2218yescomplemented
85#532C2214yescomplemented
86#532C2214yescomplemented
87C2×#127C2212yescomplemented
88C2×#127C2214yescomplemented
89#529C2218yescomplemented
90#529C2218yescomplemented
91#529C2218yescomplemented
92C2×#124C2212yescomplemented
93C2×#124C2214yescomplemented
94#525C2212yescomplemented
95#525C2212yescomplemented
96#525C2214yescomplemented
97C2×#123C2214yescomplemented
98C2×#121C2214yescomplemented
99C2×#120C2214yescomplemented
100C2×#120C2218yescomplemented
101#519C2211yescharacteristic, complemented
102#518C2212yescomplemented
103#518C2211yescharacteristic, complemented
104C4×#41C2212yescomplemented
105C4×#41C2212yescomplemented
106C2×#117C2211yescharacteristic, complemented
107C25C2312yes—
1082−1+4—44no—
109H(5,𝔽2)—48no—
110H(5,𝔽2)—44no—
111C2×#41—44no—
112C2×#41C2312yes—
113C2×#41C2312yes—
114C2×#41C2314yes—
115C2×#41C2314yes—
116C2×#41C2314yes—
117C2×#41C2318yes—
118C2×#41C2314yescomplemented
119C2×#41C2314yescomplemented
120C2×#41C23116yescomplemented
121C22×Q8C2311yescharacteristic
122C22×Q8C2314yescomplemented
123C22×D4—44no—
124C22×D4C2312yes—
125C22×D4C2311yescharacteristic
126C22×D4C2312yes—
127C22×D4C2318yes—
128C22×D4C2312yes—
129C22×D4C2314yes—
130C22×D4C2314yes—
131C22×D4C2314yes—
132C22×D4C2318yes—
133C22×D4C2318yes—
134C22×D4C23116yes—
135C22×D4C2314yescomplemented
136C22×D4C23116yescomplemented
137C23×C4C2311yescharacteristic
138C23×C4C2312yes—
139C4⋊Q8C2312yescomplemented
140C4⋊Q8C2314yescomplemented
141C4⋊Q8C2314yescomplemented
142#127C2314yes—
143#127C2314yes—
144#127C2314yes—
145#127C2318yes—
146#127C2314yes—
147#127C2318yes—
148#127C2312yescomplemented
149#127C2314yescomplemented
150#127C23116yescomplemented
151#126C2318yes—
152#126C2318yes—
153#126C2318yescomplemented
154#126C23116yescomplemented
155#126C2318yescomplemented
156#126C23116yescomplemented
157#125C2314yescomplemented
158#125C2318yescomplemented
159#125C2318yescomplemented
160#124C2314yes—
161#124C2314yes—
162#124C2318yes—
163#124C2314yes—
164#124C2318yes—
165#124C2318yes—
166#124C2314yes—
167#124C2318yes—
168#124C2312yescomplemented
169#124C2314yescomplemented
170#124C23116yescomplemented
171#124C2312yescomplemented
172#124C2314yescomplemented
173#124C23116yescomplemented
174#124C23116yescomplemented
175#123C2314yes—
176#123C2318yes—
177#123C2314yes—
178#123C2318yes—
179#123C2318yes—
180#123C2318yes—
181#123C23116yes—
182#123C23116yescomplemented
183#123C2318yescomplemented
184#123C23116yescomplemented
185#123C2318yescomplemented
186#123C23116yescomplemented
187#123C23116yescomplemented
188#123C23116yescomplemented
189#123C23116yescomplemented
190C22⋊Q8C2314yes—
191C22⋊Q8C2314yes—
192C22⋊Q8C2318yes—
193C22⋊Q8C2318yescomplemented
194C22⋊Q8C2318yescomplemented
195C22⋊Q8C2318yescomplemented
196C22⋊Q8C2318yescomplemented
197C22⋊Q8C23116yescomplemented
198C22⋊Q8C23116yescomplemented
199#121C2318yes—
200#121C2318yes—
201#121C23116yes—
202#121C2314yes—
203#121C2314yes—
204#121C2318yes—
205#121C2318yes—
206#121C2318yes—
207#121C2318yes—
208#121C2318yes—
209#121C23116yes—
210#121C23116yes—
211#121C23116yes—
212#121C23116yes—
213#121C2318yescomplemented
214#121C23116yescomplemented
215#121C23116yescomplemented
216#121C2318yescomplemented
217#121C23116yescomplemented
218#121C23116yescomplemented
219#121C23116yescomplemented
220#121C23116yescomplemented
221#121C23116yescomplemented
222#121C23116yescomplemented
223#120C2314yes—
224#120C2314yes—
225#120C2318yes—
226#120C23116yes—
227#120C2318yes—
228#120C23116yes—
229#120C2318yes—
230#120C2318yes—
231#120C23116yes—
232#120C2318yes—
233#120C2318yes—
234#120C23116yes—
235#120C23116yescomplemented
236#120C23116yescomplemented
237#120C23116yescomplemented
238C4×Q8C2314yescomplemented
239C4×Q8C2314yescomplemented
240C4×Q8C2318yescomplemented
241C4×Q8C2318yescomplemented
242C4×D4C2318yes—
243C4×D4C2318yes—
244C4×D4C2318yes—
245C4×D4C2318yes—
246C4×D4C2318yes—
247C4×D4C2318yes—
248C4×D4C2314yescomplemented
249C4×D4C2314yescomplemented
250C4×D4C2318yescomplemented
251C4×D4C23116yescomplemented
252C4×D4C2318yescomplemented
253C4×D4C23116yescomplemented
254C4×D4C23116yescomplemented
255C4×D4C23116yescomplemented
256C4×D4C23116yescomplemented
257#117C2314yes—
258#117C2314yes—
259#117C2312yes—
260#117C2312yes—
261#117C2314yes—
262#117C2314yescomplemented
263#117C2314yescomplemented
264#117C23116yescomplemented
265C2×C4⋊C4C2312yes—
266C2×C4⋊C4C2318yescomplemented
267C2×C22⋊C4C2312yes—
268C2×C22⋊C4C2318yes—
269C2×C22⋊C4C2314yes—
270C2×C22⋊C4C2318yes—
271C2×C22⋊C4C23116yes—
272C2×C22⋊C4C2318yescomplemented
273C2×C22⋊C4C23116yescomplemented
274C2×C42C2312yes—
275C2×C42C2314yescomplemented
276C2×C42C2314yescomplemented
277C24—44no—
278C24—84no—
279C24C2414yes—
280C24C2411yescharacteristic
281C24C2412yes—
282C24C2418yes—
283C24C2418yes—
284C24C2414yes—
285#41—28no—
286#41—28no—
287#41—48no—
288#41—416no—
289#41—48no—
290#41—48no—
291#41—48no—
292#41—48no—
293#41—816no—
294C2×Q8—24no—
295C2×Q8—48no—
296C2×Q8C2412yes—
297C2×Q8C2412yes—
298C2×Q8C2414yes—
299C2×Q8C2418yes—
300C2×Q8C2414yes—
301C2×D4—28no—
302C2×D4—24no—
303C2×D4—416no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—416no—
311C2×D4—88no—
312C2×D4—816no—
313C2×D4—816no—
314C2×D4C2414yes—
315C2×D4C2414yes—
316C2×D4C2414yes—
317C2×D4C2414yes—
318C2×D4C2412yes—
319C2×D4C2412yes—
320C2×D4C2414yes—
321C2×D4C2418yes—
322C2×D4C2418yes—
323C2×D4C24116yes—
324C2×D4C24116yes—
325C2×D4C24116yes—
326C2×D4C24116yes—
327C2×D4C24116yes—
328C2×D4C2418yes—
329C2×D4C2418yes—
330C2×D4C2418yes—
331C2×D4C2418yes—
332C2×D4C2418yes—
333C2×D4C2414yes—
334C2×D4C2418yes—
335C22×C4—24no—
336C22×C4—44no—
337C22×C4C2414yes—
338C22×C4C2414yes—
339C22×C4C2412yes—
340C22×C4C2411yescharacteristic
341C22×C4C2414yes—
342C22×C4C2414yes—
343C22×C4C2418yes—
344C22×C4C2418yes—
345C22×C4C24116yes—
346C22×C4C2418yes—
347C22×C4C2418yes—
348C22×C4C2418yes—
349C22×C4C2418yes—
350C22×C4C2414yes—
351C22×C4C2411yescharacteristic
352C4⋊C4C2412yes—
353C4⋊C4C2414yes—
354C4⋊C4C2412yes—
355C4⋊C4C2418yes—
356C4⋊C4C24116yes—
357C4⋊C4C2414yes—
358C4⋊C4C2414yes—
359C4⋊C4C2418yes—
360C4⋊C4C2418yes—
361C4⋊C4C2418yes—
362C4⋊C4C2418yes—
363C4⋊C4C2418yes—
364C4⋊C4C2418yes—
365C4⋊C4C2418yes—
366C4⋊C4C2418yes—
367C22⋊C4C2414yes—
368C22⋊C4C2414yes—
369C22⋊C4C2418yes—
370C22⋊C4C2418yes—
371C22⋊C4C24116yes—
372C22⋊C4C24116yes—
373C22⋊C4C2412yes—
374C22⋊C4C2414yes—
375C22⋊C4C2412yes—
376C22⋊C4C24116yes—
377C22⋊C4C24116yes—
378C22⋊C4C24116yes—
379C22⋊C4C2418yes—
380C22⋊C4C2418yes—
381C22⋊C4C24116yes—
382C22⋊C4C2418yes—
383C22⋊C4C2418yes—
384C22⋊C4C2418yes—
385C22⋊C4C2418yes—
386C22⋊C4C24116yes—
387C22⋊C4C2418yes—
388C42C2412yes—
389C42C2412yes—
390C42C2414yes—
391C42C2414yes—
392C42C2414yes—
393C42C2418yes—
394C42C2418yes—
395C42C2418yes—
396C23—22no—
397C23—24no—
398C23—28no—
399C23—42no—
400C23—48no—
401C23—44no—
402C23—48no—
403C23—416no—
404C23—48no—
405C23—44no—
406C23—416no—
407C23—48no—
408C23—816no—
409C23C2512yes—
410C23C2514yes—
411C23C2518yes—
412C23C2514yes—
413C23C2514yes—
414C23C2511yescharacteristic
415Q8—28no—
416Q8—24no—
417Q8—24no—
418Q8—44no—
419Q8—48no—
420D4—28no—
421D4—28no—
422D4—28no—
423D4—28no—
424D4—28no—
425D4—24no—
426D4—24no—
427D4—48no—
428D4—48no—
429D4—48no—
430D4—48no—
431D4—48no—
432D4—48no—
433D4—416no—
434D4—48no—
435D4—48no—
436D4—416no—
437D4—416no—
438D4—416no—
439D4—416no—
440D4—44no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—416no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—48no—
455C2×C4H(5,𝔽2)14yes—
456C2×C4C2514yes—
457C2×C4C2518yes—
458C2×C4C2514yes—
459C2×C4C2518yes—
460C2×C4C2514yes—
461C2×C4C2514yes—
462C2×C4C2514yes—
463C2×C4C2512yes—
464C2×C4C2512yes—
465C22—22no—
466C22—22no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—28no—
471C22—28no—
472C22—28no—
473C22—216no—
474C22—28no—
475C22—24no—
476C22—44no—
477C22—84no—
478C22—88no—
479C22—816no—
480C22—168no—
481C22—164no—
482C22C2×H(5,𝔽2)12yes—
483C22C2611yescharacteristic
484C4—22no—
485C4—24no—
486C4—24no—
487C4—24no—
488C4—28no—
489C4—24no—
490C4—28no—
491C4—24no—
492C4C2×H(5,𝔽2)14yes—
493C2—22no—
494C2—42no—
495C2—48no—
496C2—44no—
497C2—44no—
498C2—44no—
499C2C22×H(5,𝔽2)11yescharacteristic
500C2H(7,𝔽2)12yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3562C2semidirect
2. #3558C2semidirect
3. #3546C2semidirect
4. #3543C2semidirect
5. #3534C2semidirect
6. #3532C2semidirect
7. #3510C2semidirect
8. #3508C2semidirect
9. #3500C2semidirect
10. #3498C2semidirect
11. #3487C2semidirect
12. #3485C2semidirect
13. #3484C2semidirect
14. #3455C2semidirect
15. C2×#535C2semidirect
16. C2×2−1+4C22semidirect
17. C2×H(5,𝔽2)C22semidirect
18. C2×H(5,𝔽2)C22semidirect
19. C22×#41C22semidirect
20. C23×D4C22semidirect
21. #562C22semidirect
22. H(3,𝔽4)C22semidirect
23. #560C22semidirect
24. #560C22semidirect
25. #560C22semidirect
26. #559C22semidirect
27. #559C22semidirect
28. #556C22semidirect
29. #553C22semidirect
30. #551C22semidirect
31. #551C22semidirect
32. #550C22semidirect
33. #550C22semidirect
34. #550C22semidirect
35. #548C22semidirect
36. #548C22semidirect
37. #547C22semidirect
38. #546C22semidirect
39. #546C22semidirect
40. #546C22semidirect
41. #546C22semidirect
42. #546C22semidirect
43. #546C22semidirect
44. #546C22semidirect
45. #546C22semidirect
46. D42C22semidirect
47. D42C22semidirect
48. D42C22semidirect
49. D42C22semidirect
50. D42C22semidirect
51. #542C22semidirect
52. #542C22semidirect
53. #542C22semidirect
54. #541C22semidirect
55. #540C22semidirect
56. #540C22semidirect
57. #540C22semidirect
58. #539C22semidirect
59. #539C22semidirect
60. #538C22semidirect
61. #538C22semidirect
62. #538C22semidirect
63. #538C22semidirect
64. #538C22semidirect
65. #537C22semidirect
66. #536C22semidirect
67. #536C22semidirect
69. #535C22semidirect
70. #535C22semidirect
71. #535C22semidirect
72. #535C22semidirect
73. #535C22semidirect
74. #535C22semidirect
75. #535C22semidirect
76. #535C22semidirect
77. #535C22semidirect
78. #534C22semidirect
79. #534C22semidirect
80. #534C22semidirect
81. #534C22semidirect
82. #533C22semidirect
83. #532C22semidirect
84. #532C22semidirect
85. #532C22semidirect
86. #532C22semidirect
87. C2×#127C22semidirect
88. C2×#127C22semidirect
89. #529C22semidirect
90. #529C22semidirect
91. #529C22semidirect
92. C2×#124C22semidirect
93. C2×#124C22semidirect
94. #525C22semidirect
95. #525C22semidirect
96. #525C22semidirect
97. C2×#123C22semidirect
98. C2×#121C22semidirect
99. C2×#120C22semidirect
100. C2×#120C22semidirect
101. #519C22semidirect
102. #518C22semidirect
103. #518C22semidirect
104. C4×#41C22semidirect
105. C4×#41C22semidirect
106. C2×#117C22semidirect
118. C2×#41C23semidirect
119. C2×#41C23semidirect
120. C2×#41C23semidirect
122. C22×Q8C23semidirect
135. C22×D4C23semidirect
136. C22×D4C23semidirect
139. C4⋊Q8C23semidirect
140. C4⋊Q8C23semidirect
141. C4⋊Q8C23semidirect
148. #127C23semidirect
149. #127C23semidirect
150. #127C23semidirect
153. #126C23semidirect
154. #126C23semidirect
155. #126C23semidirect
156. #126C23semidirect
157. #125C23semidirect
158. #125C23semidirect
159. #125C23semidirect
168. #124C23semidirect
169. #124C23semidirect
170. #124C23semidirect
171. #124C23semidirect
172. #124C23semidirect
173. #124C23semidirect
174. #124C23semidirect
182. #123C23semidirect
183. #123C23semidirect
184. #123C23semidirect
185. #123C23semidirect
186. #123C23semidirect
187. #123C23semidirect
188. #123C23semidirect
189. #123C23semidirect
193. C22⋊Q8C23semidirect
194. C22⋊Q8C23semidirect
195. C22⋊Q8C23semidirect
196. C22⋊Q8C23semidirect
197. C22⋊Q8C23semidirect
198. C22⋊Q8C23semidirect
213. #121C23semidirect
214. #121C23semidirect
215. #121C23semidirect
216. #121C23semidirect
217. #121C23semidirect
218. #121C23semidirect
219. #121C23semidirect
220. #121C23semidirect
221. #121C23semidirect
222. #121C23semidirect
235. #120C23semidirect
236. #120C23semidirect
237. #120C23semidirect
238. C4×Q8C23semidirect
239. C4×Q8C23semidirect
240. C4×Q8C23semidirect
241. C4×Q8C23semidirect
248. C4×D4C23semidirect
249. C4×D4C23semidirect
250. C4×D4C23semidirect
251. C4×D4C23semidirect
252. C4×D4C23semidirect
253. C4×D4C23semidirect
254. C4×D4C23semidirect
255. C4×D4C23semidirect
256. C4×D4C23semidirect
262. #117C23semidirect
263. #117C23semidirect
264. #117C23semidirect
266. C2×C4⋊C4C23semidirect
272. C2×C22⋊C4C23semidirect
273. C2×C22⋊C4C23semidirect
275. C2×C42C23semidirect
276. C2×C42C23semidirect

Maximal subgroups

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

subgroupindex
#35622
#35582
#35462
#35432
#35342
#35322
#35102
#35082
#35002
#34982
#34872
#34852
#34842
#34552
C2×#5352