Finite Group Databasebeta

← #62930

Subgroups — #62930

Order 256.

3659 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3558C218yesmaximal, complemented
2#3549C214yesmaximal, complemented
3#3547C214yesmaximal, complemented
4#3542C218yesmaximal, complemented
5#3540C218yesmaximal, complemented
6#3537C214yesmaximal, complemented
7#3519C214yesmaximal, complemented
8#3511C212yesmaximal, complemented
9#3506C214yesmaximal, complemented
10#3496C211yesmaximal, characteristic, complemented
11#3494C214yesmaximal, complemented
12#3493C212yesmaximal, complemented
13#3488C214yesmaximal, complemented
14C2×#555C211yesmaximal, characteristic, complemented
15D4×#41C212yesmaximal, complemented
16#3460C211yesmaximal, characteristic, complemented
17#3449C212yesmaximal, complemented
18C2×H(5,𝔽2)C2212yescomplemented
19C22×#41C2214yescomplemented
20#562C22116yescomplemented
21H(3,𝔽4)C22116yescomplemented
22#560C22116yescomplemented
23#559C22116yescomplemented
24#556C2218yescomplemented
25#556C2214yescomplemented
26#555C2214yescomplemented
27#555C2218yescomplemented
28#555C2214yescomplemented
29#555C2218yescomplemented
30#555C2218yescomplemented
31#555C2214yescomplemented
32#554C2214yescomplemented
33#553C2218yescomplemented
34#553C2218yescomplemented
35#553C2218yescomplemented
36#553C2218yescomplemented
37#552C2218yescomplemented
38#551C22116yescomplemented
39#550C2212yescomplemented
40#550C2218yescomplemented
41D4×Q8C2212yescomplemented
42#548C2214yescomplemented
43#548C2218yescomplemented
44#548C2218yescomplemented
45#547C2214yescomplemented
46#547C2212yescomplemented
47#547C2214yescomplemented
48#547C2214yescomplemented
49#546C2214yescomplemented
50#546C2218yescomplemented
51#546C22116yescomplemented
52#546C2218yescomplemented
53#546C2218yescomplemented
54#546C22116yescomplemented
55D42C2214yescomplemented
56D42C2212yescomplemented
57D42C22116yescomplemented
58D42C2218yescomplemented
59D42C2214yescomplemented
60D42C2218yescomplemented
61#544C2211yescharacteristic, complemented
62#544C2214yescomplemented
63#542C2214yescomplemented
64#542C2218yescomplemented
65#542C2218yescomplemented
66#542C22116yescomplemented
67#540C2214yescomplemented
68#540C2218yescomplemented
69#540C2218yescomplemented
70#538C2214yescomplemented
71#538C2218yescomplemented
72#538C22116yescomplemented
73#538C22116yescomplemented
74#538C2218yescomplemented
75#537C2211yescharacteristic, complemented
76#537C2214yescomplemented
77#537C2212yescomplemented
78#537C2218yescomplemented
79#536C2218yescomplemented
80#535C2218yescomplemented
81#535C2212yescomplemented
82#535C2218yescomplemented
83#535C2214yescomplemented
84#535C2218yescomplemented
85#535C22116yescomplemented
86#534C2218yescomplemented
87#534C2214yescomplemented
88#534C2218yescomplemented
89#533C2214yescomplemented
90#532C2218yescomplemented
91#532C2214yescomplemented
92#532C2212yescomplemented
93#532C2214yescomplemented
94#532C2214yescomplemented
95C2×C4⋊Q8C2211yescharacteristic, complemented
96#529C2214yescomplemented
97#529C2218yescomplemented
98C2×#124C2214yescomplemented
99#525C2218yescomplemented
100#525C2218yescomplemented
101#525C2214yescomplemented
102#525C2214yescomplemented
103#525C2218yescomplemented
104#525C2218yescomplemented
105C2×#121C2212yescomplemented
106C2×#121C2212yescomplemented
107C2×#121C2218yescomplemented
108C2×#121C2214yescomplemented
109#520C2214yescomplemented
110#520C2214yescomplemented
111#518C2214yescomplemented
112C4×#41C2212yescomplemented
113C2×C4×D4C2211yescharacteristic, complemented
114C2×C4×D4C2211yescharacteristic, complemented
115C2×C4×D4C2214yescomplemented
116C2×#117C2212yescomplemented
117C2×#117C2212yescomplemented
118H(5,𝔽2)—48no—
119C2×#41—416no—
120C2×#41C2312yes—
121C2×#41C2314yes—
122C2×#41C2314yes—
123C2×#41C2314yes—
124C2×#41C2314yes—
125C2×#41C2318yes—
126C2×#41C2318yes—
127C2×#41C2318yes—
128C2×#41C2318yes—
129C22×Q8C2314yes—
130C22×Q8C2312yes—
131C22×D4C2311yescharacteristic
132C22×D4C2311yescharacteristic
133C22×D4C2314yes—
134C22×D4C2314yes—
135C22×D4C2318yes—
136C22×D4C2318yes—
137C22×D4C2318yes—
138C22×D4C2314yes—
139C22×D4C2314yes—
140C22×D4C23116yes—
141C23×C4C2312yes—
142C23×C4C2312yes—
143C23×C4C2318yes—
144C4⋊Q8C2314yes—
145C4⋊Q8C2314yes—
146C4⋊Q8C2312yes—
147C4⋊Q8C2318yes—
148C4⋊Q8C2314yes—
149C4⋊Q8C2314yes—
150#127C2312yes—
151#127C2318yes—
152#127C2314yes—
153#127C2314yes—
154#127C23116yes—
155#126C2318yes—
156#126C23116yes—
157#126C23116yes—
158#126C23116yes—
159#125C2314yes—
160#125C2318yes—
161#125C2318yes—
162#124C23116yes—
163#124C23116yes—
164#124C2314yes—
165#124C2318yes—
166#124C23116yes—
167#124C2318yes—
168#124C2318yes—
169#124C2314yes—
170#124C2318yes—
171#124C23116yes—
172#124C23116yes—
173#124C23116yes—
174#123C2318yes—
175#123C23116yes—
176#123C23116yes—
177#123C2318yes—
178#123C23116yes—
179#123C2318yes—
180#123C2318yes—
181#123C2318yes—
182#123C23116yes—
183#123C23116yes—
184#123C23116yes—
185C22⋊Q8C2314yes—
186C22⋊Q8C2314yes—
187C22⋊Q8C2318yes—
188C22⋊Q8C2314yes—
189C22⋊Q8C2318yes—
190C22⋊Q8C2318yes—
191C22⋊Q8C2314yes—
192C22⋊Q8C2314yes—
193C22⋊Q8C2318yes—
194C22⋊Q8C2318yes—
195C22⋊Q8C2318yes—
196C22⋊Q8C23116yes—
197C22⋊Q8C23116yes—
198#121C2314yes—
199#121C2318yes—
200#121C2314yes—
201#121C2314yes—
202#121C2318yes—
203#121C2318yes—
204#121C2318yes—
205#121C2318yes—
206#121C2318yes—
207#121C2318yes—
208#121C23116yes—
209#121C23116yes—
210#121C2318yes—
211#121C2314yes—
212#121C2314yes—
213#121C2314yes—
214#121C2314yes—
215#121C2314yes—
216#121C2318yes—
217#121C2318yes—
218#121C2318yes—
219#121C2318yes—
220#121C2314yes—
221#121C23116yes—
222#121C2318yes—
223#121C2314yes—
224#121C2314yes—
225#121C2318yes—
226#121C2318yes—
227#121C23116yes—
228#121C23116yes—
229#121C23116yes—
230#121C23116yes—
231#120C23116yes—
232#120C2318yes—
233#120C23116yes—
234#120C2318yes—
235#120C2318yes—
236#120C23116yes—
237#120C2318yes—
238#120C23116yes—
239#120C23116yes—
240#120C23116yes—
241C4×Q8C2312yes—
242C4×Q8C2312yes—
243C4×Q8C2318yes—
244C4×Q8C2318yes—
245C4×Q8C2318yes—
246C4×D4C2312yes—
247C4×D4C2314yes—
248C4×D4C2314yes—
249C4×D4C2318yes—
250C4×D4C2314yes—
251C4×D4C2314yes—
252C4×D4C2312yes—
253C4×D4C2318yes—
254C4×D4C2314yes—
255C4×D4C2314yes—
256C4×D4C2314yes—
257C4×D4C2312yes—
258C4×D4C2314yes—
259C4×D4C2312yes—
260C4×D4C2318yes—
261C4×D4C2318yes—
262C4×D4C2318yes—
263C4×D4C2318yes—
264C4×D4C23116yes—
265C4×D4C23116yes—
266C4×D4C2318yes—
267C4×D4C2318yes—
268C4×D4C2314yes—
269C4×D4C23116yes—
270C4×D4C2318yes—
271C4×D4C23116yes—
272#117C2314yes—
273#117C2318yes—
274#117C2318yes—
275#117C2314yes—
276#117C2318yes—
277#117C2312yes—
278#117C2314yes—
279#117C2314yes—
280#117C2318yes—
281#117C2318yes—
282#117C23116yes—
283C2×C4⋊C4C2311yescharacteristic
284C2×C4⋊C4C2311yescharacteristic
285C2×C4⋊C4C2312yes—
286C2×C4⋊C4C2312yes—
287C2×C4⋊C4C2314yes—
288C2×C4⋊C4C2318yes—
289C2×C4⋊C4C2314yes—
290C2×C22⋊C4C2314yes—
291C2×C22⋊C4C2314yes—
292C2×C22⋊C4C2312yes—
293C2×C22⋊C4C2312yes—
294C2×C22⋊C4C2318yes—
295C2×C22⋊C4C23116yes—
296C2×C22⋊C4C2318yes—
297C2×C42C2314yes—
298C2×C42C2311yescharacteristic
299C2×C42C2314yes—
300C24C2412yes—
301C24C2412yes—
302C24C2418yes—
303C24C2418yes—
304#41—28no—
305#41—216no—
306#41—416no—
307#41—48no—
308#41—416no—
309#41—48no—
310#41—48no—
311#41—416no—
312#41—88no—
313C2×Q8—48no—
314C2×Q8—82no—
315C2×Q8C2412yes—
316C2×Q8C2414yes—
317C2×Q8C2414yes—
318C2×Q8C2418yes—
319C2×Q8C2418yes—
320C2×D4—24no—
321C2×D4—42no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—416no—
326C2×D4—416no—
327C2×D4—416no—
328C2×D4—84no—
329C2×D4—816no—
330C2×D4C2412yes—
331C2×D4C2414yes—
332C2×D4C2414yes—
333C2×D4C2418yes—
334C2×D4C2414yes—
335C2×D4C2414yes—
336C2×D4C2412yes—
337C2×D4C2418yes—
338C2×D4C2412yes—
339C2×D4C2418yes—
340C2×D4C2414yes—
341C2×D4C2418yes—
342C2×D4C2414yes—
343C2×D4C24116yes—
344C2×D4C2418yes—
345C2×D4C24116yes—
346C2×D4C2418yes—
347C2×D4C2418yes—
348C2×D4C2418yes—
349C2×D4C2418yes—
350C2×D4C24116yes—
351C22×C4—28no—
352C22×C4—44no—
353C22×C4—416no—
354C22×C4C2412yes—
355C22×C4C2411yescharacteristic
356C22×C4C2414yes—
357C22×C4C2414yes—
358C22×C4C2414yes—
359C22×C4C2418yes—
360C22×C4C2414yes—
361C22×C4C2414yes—
362C22×C4C2412yes—
363C22×C4C2411yescharacteristic
364C22×C4C2414yes—
365C22×C4C2414yes—
366C22×C4C2414yes—
367C22×C4C2418yes—
368C22×C4C24116yes—
369C22×C4C2418yes—
370C22×C4C2418yes—
371C22×C4C2418yes—
372C22×C4C2411yescharacteristic
373C4⋊C4C2411yescharacteristic
374C4⋊C4C2412yes—
375C4⋊C4C2414yes—
376C4⋊C4C2418yes—
377C4⋊C4C2414yes—
378C4⋊C4C2411yescharacteristic
379C4⋊C4C2414yes—
380C4⋊C4C2412yes—
381C4⋊C4C2414yes—
382C4⋊C4C2418yes—
383C4⋊C4C24116yes—
384C4⋊C4C2418yes—
385C4⋊C4C2412yes—
386C4⋊C4C2418yes—
387C4⋊C4C2418yes—
388C4⋊C4C2414yes—
389C4⋊C4C2414yes—
390C4⋊C4C2414yes—
391C4⋊C4C2414yes—
392C4⋊C4C24116yes—
393C4⋊C4C2414yes—
394C22⋊C4C2418yes—
395C22⋊C4C2418yes—
396C22⋊C4C2414yes—
397C22⋊C4C2418yes—
398C22⋊C4C2418yes—
399C22⋊C4C2418yes—
400C22⋊C4C24116yes—
401C22⋊C4C24116yes—
402C22⋊C4C2418yes—
403C22⋊C4C24116yes—
404C22⋊C4C2418yes—
405C22⋊C4C2414yes—
406C22⋊C4C2418yes—
407C22⋊C4C2418yes—
408C22⋊C4C2414yes—
409C22⋊C4C24116yes—
410C22⋊C4C24116yes—
411C22⋊C4C2414yes—
412C22⋊C4C2414yes—
413C22⋊C4C2418yes—
414C22⋊C4C2418yes—
415C22⋊C4C2418yes—
416C42C2412yes—
417C42C2418yes—
418C42C2418yes—
419C42C2412yes—
420C42C2414yes—
421C42C2414yes—
422C42C2414yes—
423C42C24116yes—
424C23—24no—
425C23—24no—
426C23—48no—
427C23—42no—
428C23—42no—
429C23—48no—
430C23—42no—
431C23—42no—
432C23—48no—
433C23—48no—
434C23—48no—
435C23—48no—
436C23C2514yes—
437C23C2518yes—
438C23C2514yes—
439C23C2514yes—
440C23C2511yescharacteristic
441Q8—24no—
442Q8—28no—
443Q8—48no—
444Q8—48no—
445Q8—44no—
446D4—216no—
447D4—28no—
448D4—28no—
449D4—28no—
450D4—24no—
451D4—24no—
452D4—28no—
453D4—416no—
454D4—42no—
455D4—48no—
456D4—48no—
457D4—48no—
458D4—48no—
459D4—48no—
460D4—48no—
461D4—48no—
462D4—416no—
463D4—416no—
464D4—44no—
465D4—48no—
466D4—44no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—216no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—416no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4H(5,𝔽2)14yes—
486C2×C4C2514yes—
487C2×C4C2512yes—
488C2×C4C2514yes—
489C2×C4C2518yes—
490C2×C4C2512yes—
491C2×C4C2512yes—
492C2×C4C2511yescharacteristic
493C2×C4C2514yes—
494C2×C4C2514yes—
495C2×C4C2518yes—
496C2×C4C2512yes—
497C2×C4C2511yescharacteristic
498C22—21no—
499C22—21no—
500C22—24no—
501C22—24no—
502C22—28no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—28no—
508C22—28no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—84no—
513C22—84no—
514C22—816no—
515C22—168no—
516C22C2×H(5,𝔽2)12yes—
517C22C2611yescharacteristic
518C4—22no—
519C4—22no—
520C4—28no—
521C4—24no—
522C4—21no—
523C4—24no—
524C4—24no—
525C4—24no—
526C4—28no—
527C4—22no—
528C4C2×H(5,𝔽2)14yes—
529C4#58512yes—
530C2—22no—
531C2—44no—
532C2—44no—
533C2—48no—
534C2—44no—
535C2C22×H(5,𝔽2)11yescharacteristic
536C2C2×#58511yescharacteristic
537C2H(7,𝔽2)11yescharacteristic
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. #3549C2semidirect
3. #3547C2semidirect
4. #3542C2semidirect
5. #3540C2semidirect
6. #3537C2semidirect
7. #3519C2semidirect
8. #3511C2semidirect
9. #3506C2semidirect
10. #3496C2semidirect
11. #3494C2semidirect
12. #3493C2semidirect
13. #3488C2semidirect
14. C2×#555C2semidirect
15. D4×#41C2semidirect
16. #3460C2semidirect
17. #3449C2semidirect
18. C2×H(5,𝔽2)C22semidirect
19. C22×#41C22semidirect
20. #562C22semidirect
21. H(3,𝔽4)C22semidirect
22. #560C22semidirect
23. #559C22semidirect
24. #556C22semidirect
25. #556C22semidirect
26. #555C22semidirect
27. #555C22semidirect
28. #555C22semidirect
29. #555C22semidirect
30. #555C22semidirect
31. #555C22semidirect
32. #554C22semidirect
33. #553C22semidirect
34. #553C22semidirect
35. #553C22semidirect
36. #553C22semidirect
37. #552C22semidirect
38. #551C22semidirect
39. #550C22semidirect
40. #550C22semidirect
41. D4×Q8C22semidirect
42. #548C22semidirect
43. #548C22semidirect
44. #548C22semidirect
45. #547C22semidirect
46. #547C22semidirect
47. #547C22semidirect
48. #547C22semidirect
49. #546C22semidirect
50. #546C22semidirect
51. #546C22semidirect
52. #546C22semidirect
53. #546C22semidirect
54. #546C22semidirect
55. D42C22semidirect
56. D42C22semidirect
57. D42C22semidirect
58. D42C22semidirect
59. D42C22semidirect
60. D42C22semidirect
61. #544C22semidirect
62. #544C22semidirect
63. #542C22semidirect
64. #542C22semidirect
65. #542C22semidirect
66. #542C22semidirect
67. #540C22semidirect
68. #540C22semidirect
69. #540C22semidirect
70. #538C22semidirect
71. #538C22semidirect
72. #538C22semidirect
73. #538C22semidirect
74. #538C22semidirect
75. #537C22semidirect
76. #537C22semidirect
77. #537C22semidirect
78. #537C22semidirect
79. #536C22semidirect
80. #535C22semidirect
81. #535C22semidirect
82. #535C22semidirect
83. #535C22semidirect
84. #535C22semidirect
85. #535C22semidirect
86. #534C22semidirect
87. #534C22semidirect
88. #534C22semidirect
89. #533C22semidirect
90. #532C22semidirect
91. #532C22semidirect
92. #532C22semidirect
93. #532C22semidirect
94. #532C22semidirect
95. C2×C4⋊Q8C22semidirect
96. #529C22semidirect
97. #529C22semidirect
98. C2×#124C22semidirect
99. #525C22semidirect
100. #525C22semidirect
101. #525C22semidirect
102. #525C22semidirect
103. #525C22semidirect
104. #525C22semidirect
105. C2×#121C22semidirect
106. C2×#121C22semidirect
107. C2×#121C22semidirect
108. C2×#121C22semidirect
109. #520C22semidirect
110. #520C22semidirect
111. #518C22semidirect
112. C4×#41C22semidirect
113. C2×C4×D4C22semidirect
114. C2×C4×D4C22semidirect
115. C2×C4×D4C22semidirect
116. C2×#117C22semidirect
117. C2×#117C22semidirect

Maximal subgroups

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

subgroupindex
#35582
#35492
#35472
#35422
#35402
#35372
#35192
#35112
#35062
#34962
#34942
#34932
#34882
C2×#5552
D4×#412
#34602
#34492