Finite Group Databasebeta

← #63021

Subgroups — #63021

Order 256.

3636 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3558C218yesmaximal, complemented
2#3554C214yesmaximal, complemented
3#3550C211yesmaximal, characteristic, complemented
4#3543C212yesmaximal, complemented
5#3540C214yesmaximal, complemented
6#3533C218yesmaximal, complemented
7#3522C214yesmaximal, complemented
8#3514C214yesmaximal, complemented
9#3513C218yesmaximal, complemented
10#3508C212yesmaximal, complemented
11#3506C214yesmaximal, complemented
12#3498C211yesmaximal, characteristic, complemented
13#3485C214yesmaximal, complemented
14#3482C212yesmaximal, complemented
15#3481C211yesmaximal, characteristic, complemented
16C2×#559C212yesmaximal, complemented
17D4×#41C214yesmaximal, complemented
18C2×H(5,𝔽2)C2212yescomplemented
19C22×#41C2214yescomplemented
20#563C2214yescomplemented
21#562C22116yescomplemented
22H(3,𝔽4)C2218yescomplemented
23#560C2218yescomplemented
24#559C2211yescharacteristic, complemented
25#559C2218yescomplemented
26#559C2214yescomplemented
27#559C2218yescomplemented
28#559C2212yescomplemented
29#559C2218yescomplemented
30#559C2218yescomplemented
31#559C2214yescomplemented
32Q82C2212yescomplemented
33#557C2212yescomplemented
34#556C2212yescomplemented
35#556C2214yescomplemented
36#556C2214yescomplemented
37#555C2218yescomplemented
38#554C2212yescomplemented
39#554C2218yescomplemented
40#553C2214yescomplemented
41#553C22116yescomplemented
42#552C2214yescomplemented
43#552C2218yescomplemented
44#551C2214yescomplemented
45#551C2218yescomplemented
46#551C22116yescomplemented
47#550C2214yescomplemented
48#550C2214yescomplemented
49#550C2214yescomplemented
50D4×Q8C2214yescomplemented
51D4×Q8C2212yescomplemented
52#548C2218yescomplemented
53#548C2214yescomplemented
54#548C2218yescomplemented
55#547C2212yescomplemented
56#547C2218yescomplemented
57#547C2214yescomplemented
58#547C2218yescomplemented
59#547C22116yescomplemented
60#546C2218yescomplemented
61#546C22116yescomplemented
62#546C2214yescomplemented
63#546C2218yescomplemented
64#546C22116yescomplemented
65#546C2218yescomplemented
66#546C2218yescomplemented
67D42C2214yescomplemented
68D42C2218yescomplemented
69D42C2211yescharacteristic, complemented
70D42C2218yescomplemented
71#542C2218yescomplemented
72#542C2218yescomplemented
73#542C22116yescomplemented
74#542C2218yescomplemented
75#540C2218yescomplemented
76#540C2218yescomplemented
77#539C22116yescomplemented
78#538C22116yescomplemented
79#538C22116yescomplemented
80#538C22116yescomplemented
81#536C2214yescomplemented
82#536C2214yescomplemented
83#535C2214yescomplemented
84#535C2212yescomplemented
85#535C2218yescomplemented
86#535C2214yescomplemented
87#534C2218yescomplemented
88#534C2218yescomplemented
89#533C2214yescomplemented
90#533C2212yescomplemented
91#532C2211yescharacteristic
92#532C2214yescomplemented
93#532C2218yescomplemented
94#532C2212yescomplemented
95#532C2214yescomplemented
96C2×#127C2212yes—
97C2×#127C2211yescharacteristic, complemented
98#529C2214yescomplemented
99#529C2218yescomplemented
100#529C2214yescomplemented
101#529C2218yescomplemented
102C2×#124C2218yescomplemented
103#525C2218yescomplemented
104#525C22116yescomplemented
105#525C2214yescomplemented
106#525C2214yescomplemented
107C2×#123C2218yescomplemented
108C2×#123C2218yescomplemented
109#520C2218yescomplemented
110#518C2214yescomplemented
111#518C2214yescomplemented
112C4×#41C2214yescomplemented
113C4×#41C2212yescomplemented
114C4×#41C2211yescharacteristic, complemented
115C2×C4×Q8C2214yescomplemented
116C2×C4×D4C2218yescomplemented
117C2×C4×D4C2214yescomplemented
118H(5,𝔽2)—48no—
119C2×#41—416no—
120C2×#41C2314yes—
121C2×#41C2311yescharacteristic
122C2×#41C2312yes—
123C2×#41C2318yes—
124C2×#41C2312yes—
125C2×#41C23116yes—
126C2×#41C2318yes—
127C2×#41C2314yes—
128C2×#41C2314yes—
129C22×Q8C2314yes—
130C22×Q8C2314yes—
131C22×D4C2314yes—
132C22×D4C2318yes—
133C22×D4C2314yes—
134C22×D4C2314yes—
135C22×D4C2314yes—
136C22×D4C2318yes—
137C22×D4C2312yes—
138C22×D4C2318yes—
139C23×C4C2318yes—
140C23×C4C2314yes—
141C4⋊Q8C2311yescharacteristic
142C4⋊Q8C2312yes—
143C4⋊Q8C2312yes—
144C4⋊Q8C2318yes—
145C4⋊Q8C2314yes—
146C4⋊Q8C2314yes—
147C4⋊Q8C2318yes—
148#127C2311yescharacteristic
149#127C2312yes—
150#127C2314yes—
151#127C2318yes—
152#127C2314yes—
153#127C2312yes—
154#127C2318yes—
155#126C23116yes—
156#126C2318yes—
157#126C2318yes—
158#126C2318yes—
159#126C2318yes—
160#126C23116yes—
161#125C2314yes—
162#125C2318yes—
163#125C2314yes—
164#125C2318yes—
165#125C2314yes—
166#124C2314yes—
167#124C2318yes—
168#124C2318yes—
169#124C2318yes—
170#124C23116yes—
171#124C23116yes—
172#124C2314yes—
173#124C2318yes—
174#124C2312yes—
175#124C2314yes—
176#124C2318yes—
177#124C23116yes—
178#124C2318yes—
179#124C23116yes—
180#123C2314yes—
181#123C2314yes—
182#123C2318yes—
183#123C2318yes—
184#123C23116yes—
185#123C2318yes—
186#123C2318yes—
187#123C2318yes—
188#123C2318yes—
189#123C23116yes—
190#123C23116yes—
191#123C2318yes—
192#123C23116yes—
193#123C2318yes—
194#123C2318yes—
195#123C23116yes—
196#123C23116yes—
197#123C23116yes—
198C22⋊Q8C2314yes—
199C22⋊Q8C2314yes—
200C22⋊Q8C2318yes—
201C22⋊Q8C23116yes—
202C22⋊Q8C23116yes—
203C22⋊Q8C2318yes—
204C22⋊Q8C2314yes—
205C22⋊Q8C2318yes—
206C22⋊Q8C2314yes—
207C22⋊Q8C2314yes—
208C22⋊Q8C2318yes—
209C22⋊Q8C2314yes—
210C22⋊Q8C23116yes—
211C22⋊Q8C23116yes—
212#121C2318yes—
213#121C2318yes—
214#121C2318yes—
215#121C23116yes—
216#121C23116yes—
217#121C2314yes—
218#121C2318yes—
219#121C2318yes—
220#121C2314yes—
221#121C2314yes—
222#121C23116yes—
223#121C2318yes—
224#121C2318yes—
225#121C2318yes—
226#121C23116yes—
227#121C23116yes—
228#121C23116yes—
229#121C23116yes—
230#120C2318yes—
231#120C2314yes—
232#120C2314yes—
233#120C23116yes—
234#120C23116yes—
235#120C23116yes—
236#120C23116yes—
237#120C2318yes—
238C4×Q8C2312yes—
239C4×Q8C2314yes—
240C4×Q8C2314yes—
241C4×Q8C2318yes—
242C4×Q8C2314yes—
243C4×Q8C2318yes—
244C4×Q8C2312yes—
245C4×Q8C2314yes—
246C4×Q8C2312yes—
247C4×Q8C2314yes—
248C4×Q8C2318yes—
249C4×D4C2314yes—
250C4×D4C2312yes—
251C4×D4C2318yes—
252C4×D4C2318yes—
253C4×D4C23116yes—
254C4×D4C2318yes—
255C4×D4C2314yes—
256C4×D4C2318yes—
257C4×D4C2314yes—
258C4×D4C23116yes—
259C4×D4C2318yes—
260C4×D4C2312yes—
261C4×D4C2314yes—
262C4×D4C2314yes—
263C4×D4C2314yes—
264C4×D4C2318yes—
265C4×D4C2312yes—
266C4×D4C2314yes—
267C4×D4C2318yes—
268C4×D4C2318yes—
269C4×D4C23116yes—
270C4×D4C2318yes—
271C4×D4C2318yes—
272C4×D4C2318yes—
273C4×D4C23116yes—
274C4×D4C23116yes—
275C4×D4C2318yes—
276#117C2312yes—
277#117C2318yes—
278#117C2314yes—
279#117C2312yes—
280#117C2311yescharacteristic
281#117C2314yes—
282#117C2314yes—
283#117C2314yes—
284#117C2314yes—
285#117C2318yes—
286#117C23116yes—
287C2×C4⋊C4C2314yes—
288C2×C4⋊C4C2314yes—
289C2×C4⋊C4C2318yes—
290C2×C4⋊C4C23116yes—
291C2×C22⋊C4C23116yes—
292C2×C22⋊C4C2318yes—
293C2×C22⋊C4C2318yes—
294C2×C22⋊C4C2318yes—
295C2×C22⋊C4C23116yes—
296C2×C42C2318yes—
297C2×C42C2314yes—
298C2×C42C2312yes—
299C2×C42C2311yescharacteristic
300C24C2418yes—
301C24C2414yes—
302C24C2414yes—
303#41—216no—
304#41—24no—
305#41—44no—
306#41—432no—
307#41—48no—
308#41—416no—
309#41—44no—
310#41—416no—
311#41—48no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8C2412yes—
315C2×Q8C2414yes—
316C2×Q8C2414yes—
317C2×Q8C2411yescharacteristic
318C2×Q8C2418yes—
319C2×Q8C2418yes—
320C2×Q8C2412yes—
321C2×D4—216no—
322C2×D4—416no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—416no—
328C2×D4—416no—
329C2×D4C2414yes—
330C2×D4C2412yes—
331C2×D4C2418yes—
332C2×D4C2418yes—
333C2×D4C2418yes—
334C2×D4C2412yes—
335C2×D4C2414yes—
336C2×D4C2414yes—
337C2×D4C24116yes—
338C2×D4C2418yes—
339C2×D4C24116yes—
340C2×D4C2418yes—
341C2×D4C2414yes—
342C2×D4C24116yes—
343C2×D4C2412yes—
344C2×D4C2418yes—
345C2×D4C2414yes—
346C2×D4C2411yescharacteristic
347C22×C4—48no—
348C22×C4—416no—
349C22×C4C2414yes—
350C22×C4C2418yes—
351C22×C4C2412yes—
352C22×C4C2414yes—
353C22×C4C2412yes—
354C22×C4C2414yes—
355C22×C4C2411yescharacteristic
356C22×C4C2414yes—
357C22×C4C2418yes—
358C22×C4C2414yes—
359C22×C4C24116yes—
360C22×C4C2418yes—
361C22×C4C24116yes—
362C22×C4C2418yes—
363C22×C4C2414yes—
364C22×C4C2418yes—
365C4⋊C4C2412yes—
366C4⋊C4C2418yes—
367C4⋊C4C2418yes—
368C4⋊C4C2414yes—
369C4⋊C4C2418yes—
370C4⋊C4C2414yes—
371C4⋊C4C2414yes—
372C4⋊C4C2418yes—
373C4⋊C4C24116yes—
374C4⋊C4C2418yes—
375C4⋊C4C2418yes—
376C4⋊C4C2414yes—
377C4⋊C4C2412yes—
378C4⋊C4C2418yes—
379C4⋊C4C2412yes—
380C4⋊C4C2414yes—
381C4⋊C4C2414yes—
382C4⋊C4C2418yes—
383C4⋊C4C2414yes—
384C4⋊C4C2418yes—
385C4⋊C4C2418yes—
386C4⋊C4C2414yes—
387C22⋊C4C2414yes—
388C22⋊C4C2418yes—
389C22⋊C4C2418yes—
390C22⋊C4C2414yes—
391C22⋊C4C24116yes—
392C22⋊C4C2418yes—
393C22⋊C4C2414yes—
394C22⋊C4C2418yes—
395C22⋊C4C2414yes—
396C22⋊C4C2418yes—
397C22⋊C4C24116yes—
398C22⋊C4C24116yes—
399C22⋊C4C24116yes—
400C22⋊C4C2418yes—
401C22⋊C4C24116yes—
402C22⋊C4C2414yes—
403C22⋊C4C2412yes—
404C22⋊C4C2414yes—
405C22⋊C4C2418yes—
406C22⋊C4C2418yes—
407C22⋊C4C24116yes—
408C22⋊C4C2418yes—
409C42C2414yes—
410C42C2412yes—
411C42C2411yescharacteristic
412C42C2411yescharacteristic
413C42C2414yes—
414C42C2418yes—
415C42C2414yes—
416C42C2418yes—
417C42C2414yes—
418C42C2412yes—
419C42C2418yes—
420C42C2418yes—
421C23—216no—
422C23—44no—
423C23—416no—
424C23—48no—
425C23—44no—
426C23—48no—
427C23C2511yescharacteristic
428C23C2514yes—
429C23C2514yes—
430C23C2518yes—
431C23C2512yes—
432Q8—28no—
433Q8—22no—
434Q8—24no—
435Q8—28no—
436Q8—48no—
437Q8—48no—
438Q8—42no—
439D4—216no—
440D4—24no—
441D4—216no—
442D4—28no—
443D4—28no—
444D4—24no—
445D4—216no—
446D4—42no—
447D4—44no—
448D4—44no—
449D4—48no—
450D4—48no—
451D4—48no—
452D4—416no—
453D4—416no—
454D4—416no—
455D4—44no—
456D4H(5,𝔽2)14yes—
457C2×C4—216no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—416no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—416no—
472C2×C4—44no—
473C2×C4C2514yes—
474C2×C4C2514yes—
475C2×C4C2518yes—
476C2×C4C2512yes—
477C2×C4C2514yes—
478C2×C4C2511yescharacteristic
479C2×C4C2514yes—
480C2×C4C2514yes—
481C2×C4C2518yes—
482C2×C4C2512yes—
483C2×C4C2511yescharacteristic
484C2×C4C2512yes—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—21no—
489C22—28no—
490C22—24no—
491C22—22no—
492C22—24no—
493C22—216no—
494C22—28no—
495C22—816no—
496C22—88no—
497C22—164no—
498C22C2×H(5,𝔽2)14yes—
499C22C2611yescharacteristic
500C4—24no—
501C4—21no—
502C4—24no—
503C4—22no—
504C4—28no—
505C4—24no—
506C4—22no—
507C4—24no—
508C4—28no—
509C4—24no—
510C4C2×H(5,𝔽2)12yes—
511C4#58514yes—
512C2—24no—
513C2—44no—
514C2—44no—
515C2—48no—
516C2—41no—
517C2C22×H(5,𝔽2)11yescharacteristic
518C2C2×#58512yes—
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. #3558C2semidirect
2. #3554C2semidirect
3. #3550C2semidirect
4. #3543C2semidirect
5. #3540C2semidirect
6. #3533C2semidirect
7. #3522C2semidirect
8. #3514C2semidirect
9. #3513C2semidirect
10. #3508C2semidirect
11. #3506C2semidirect
12. #3498C2semidirect
13. #3485C2semidirect
14. #3482C2semidirect
15. #3481C2semidirect
16. C2×#559C2semidirect
17. D4×#41C2semidirect
18. C2×H(5,𝔽2)C22semidirect
19. C22×#41C22semidirect
20. #563C22semidirect
21. #562C22semidirect
22. H(3,𝔽4)C22semidirect
23. #560C22semidirect
24. #559C22semidirect
25. #559C22semidirect
26. #559C22semidirect
27. #559C22semidirect
28. #559C22semidirect
29. #559C22semidirect
30. #559C22semidirect
31. #559C22semidirect
32. Q82C22semidirect
33. #557C22semidirect
34. #556C22semidirect
35. #556C22semidirect
36. #556C22semidirect
37. #555C22semidirect
38. #554C22semidirect
39. #554C22semidirect
40. #553C22semidirect
41. #553C22semidirect
42. #552C22semidirect
43. #552C22semidirect
44. #551C22semidirect
45. #551C22semidirect
46. #551C22semidirect
47. #550C22semidirect
48. #550C22semidirect
49. #550C22semidirect
50. D4×Q8C22semidirect
51. D4×Q8C22semidirect
52. #548C22semidirect
53. #548C22semidirect
54. #548C22semidirect
55. #547C22semidirect
56. #547C22semidirect
57. #547C22semidirect
58. #547C22semidirect
59. #547C22semidirect
60. #546C22semidirect
61. #546C22semidirect
62. #546C22semidirect
63. #546C22semidirect
64. #546C22semidirect
65. #546C22semidirect
66. #546C22semidirect
67. D42C22semidirect
68. D42C22semidirect
69. D42C22semidirect
70. D42C22semidirect
71. #542C22semidirect
72. #542C22semidirect
73. #542C22semidirect
74. #542C22semidirect
75. #540C22semidirect
76. #540C22semidirect
77. #539C22semidirect
78. #538C22semidirect
79. #538C22semidirect
80. #538C22semidirect
81. #536C22semidirect
82. #536C22semidirect
83. #535C22semidirect
84. #535C22semidirect
85. #535C22semidirect
86. #535C22semidirect
87. #534C22semidirect
88. #534C22semidirect
89. #533C22semidirect
90. #533C22semidirect
92. #532C22semidirect
93. #532C22semidirect
94. #532C22semidirect
95. #532C22semidirect
97. C2×#127C22semidirect
98. #529C22semidirect
99. #529C22semidirect
100. #529C22semidirect
101. #529C22semidirect
102. C2×#124C22semidirect
103. #525C22semidirect
104. #525C22semidirect
105. #525C22semidirect
106. #525C22semidirect
107. C2×#123C22semidirect
108. C2×#123C22semidirect
109. #520C22semidirect
110. #518C22semidirect
111. #518C22semidirect
112. C4×#41C22semidirect
113. C4×#41C22semidirect
114. C4×#41C22semidirect
115. C2×C4×Q8C22semidirect
116. C2×C4×D4C22semidirect
117. C2×C4×D4C22semidirect

Maximal subgroups

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

subgroupindex
#35582
#35542
#35502
#35432
#35402
#35332
#35222
#35142
#35132
#35082
#35062
#34982
#34852
#34822
#34812
C2×#5592
D4×#412