Finite Group Databasebeta

← #63024

Subgroups — #63024

Order 256.

3632 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3551C214yesmaximal, complemented
2#3549C211yesmaximal, characteristic, complemented
3#3547C214yesmaximal, complemented
4#3545C218yesmaximal, complemented
5#3544C212yesmaximal, complemented
6#3542C218yesmaximal, complemented
7#3519C214yesmaximal, complemented
8#3514C214yesmaximal, complemented
9#3511C212yesmaximal, complemented
10#3500C214yesmaximal, complemented
11#3499C211yesmaximal, characteristic, complemented
12#3496C214yesmaximal, complemented
13#3494C218yesmaximal, complemented
14#3483C211yesmaximal, characteristic, complemented
15#3481C212yesmaximal, complemented
16#3475C214yesmaximal, complemented
17C2×#555C212yesmaximal, complemented
18C2×2−1+4C2212yescomplemented
19C2×H(5,𝔽2)C2214yescomplemented
20#563C2218yescomplemented
21#562C22116yescomplemented
22H(3,𝔽4)C2214yescomplemented
23#560C2218yescomplemented
24#559C2212yescomplemented
25Q82C2211yescharacteristic, complemented
26#556C2214yescomplemented
27#555C2211yescharacteristic, complemented
28#555C2218yescomplemented
29#555C2214yescomplemented
30#555C2218yescomplemented
31#555C2212yescomplemented
32#555C2218yescomplemented
33#555C2214yescomplemented
34#555C2214yescomplemented
35#555C2214yescomplemented
36#554C2212yescomplemented
37#554C2212yescomplemented
38#554C2214yescomplemented
39#553C2218yescomplemented
40#553C2214yescomplemented
41#553C2214yescomplemented
42#553C2218yescomplemented
43#553C2218yescomplemented
44#552C2218yescomplemented
45#552C2218yescomplemented
46#551C22116yescomplemented
47#551C2218yescomplemented
48#551C2218yescomplemented
49#551C22116yescomplemented
50#550C2212yescomplemented
51#550C2214yescomplemented
52D4×Q8C2214yescomplemented
53#548C2218yescomplemented
54#548C2218yescomplemented
55#547C2214yescomplemented
56#547C2218yescomplemented
57#547C2218yescomplemented
58#547C2214yescomplemented
59#546C2218yescomplemented
60#546C2214yescomplemented
61#546C2214yescomplemented
62#546C22116yescomplemented
63#546C22116yescomplemented
64#546C2218yescomplemented
65#546C22116yescomplemented
66#546C2218yescomplemented
67D42C2212yes—
68D42C2214yescomplemented
69D42C2218yescomplemented
70D42C2212yescomplemented
71#544C2218yescomplemented
72#542C22116yescomplemented
73#542C2218yescomplemented
74#542C2218yescomplemented
75#540C22116yescomplemented
76#540C2218yescomplemented
77#538C22116yescomplemented
78#538C22116yescomplemented
79#537C2218yescomplemented
80#537C2214yescomplemented
81#536C2214yescomplemented
82#536C2212yescomplemented
83#536C2218yescomplemented
84#536C2214yescomplemented
85#535C2218yescomplemented
86#535C2218yescomplemented
87#535C2214yescomplemented
88#534C22116yescomplemented
89#533C2214yescomplemented
90#532C2212yescomplemented
91#532C2214yescomplemented
92#532C2211yescharacteristic, complemented
93#532C2214yescomplemented
94#532C2218yescomplemented
95#532C2214yescomplemented
96#532C2212yescomplemented
97C2×C4⋊Q8C2212yescomplemented
98C2×C4⋊Q8C2211yescharacteristic, complemented
99#529C22116yescomplemented
100#529C2218yescomplemented
101#529C2214yescomplemented
102#529C2214yescomplemented
103C2×#124C2218yescomplemented
104C2×#124C2214yescomplemented
105#525C2218yescomplemented
106#525C2218yescomplemented
107C2×#121C2218yescomplemented
108C2×#121C2218yescomplemented
109#520C2214yescomplemented
110#518C2218yescomplemented
111C4×#41C2214yescomplemented
112C4×#41C2214yescomplemented
113C4×#41C2211yescharacteristic, complemented
114C4×#41C2212yescomplemented
115C2×C4×D4C2214yescomplemented
116C2×C4×D4C2214yescomplemented
117C2×#117C2218yescomplemented
1182−1+4—48no—
119H(5,𝔽2)—416no—
120C2×#41C2311yescharacteristic
121C2×#41C2314yes—
122C2×#41C2314yes—
123C2×#41C2312yes—
124C2×#41C2314yes—
125C2×#41C2312yes—
126C2×#41C2314yes—
127C2×#41C2318yes—
128C2×#41C2314yes—
129C2×#41C2318yes—
130C2×#41C2318yes—
131C22×Q8C2314yes—
132C22×Q8C2314yes—
133C22×Q8C2312yes—
134C22×D4C2314yes—
135C22×D4C2314yes—
136C22×D4C2318yes—
137C22×D4C2318yes—
138C22×D4C23116yes—
139C22×D4C2314yes—
140C23×C4C2318yes—
141C4⋊Q8C2312yes—
142C4⋊Q8C2311yescharacteristic
143C4⋊Q8C2314yes—
144C4⋊Q8C2312yes—
145C4⋊Q8C2314yes—
146C4⋊Q8C2318yes—
147C4⋊Q8C2312yes—
148C4⋊Q8C2318yes—
149C4⋊Q8C2314yes—
150C4⋊Q8C2314yes—
151#127C2312yes—
152#127C2311yescharacteristic
153#127C2318yes—
154#127C2314yes—
155#127C2314yes—
156#126C2318yes—
157#126C23116yes—
158#126C23116yes—
159#126C23116yes—
160#126C2318yes—
161#125C2314yes—
162#125C23116yes—
163#125C2314yes—
164#125C2318yes—
165#124C2314yes—
166#124C2318yes—
167#124C2314yes—
168#124C2318yes—
169#124C2318yes—
170#124C2318yes—
171#124C2318yes—
172#124C23116yes—
173#124C23116yes—
174#124C23116yes—
175#124C2314yes—
176#124C2314yes—
177#124C2314yes—
178#124C2312yes—
179#124C2318yes—
180#124C2314yes—
181#123C2318yes—
182#123C23116yes—
183#123C2318yes—
184#123C23116yes—
185#123C23116yes—
186#123C2318yes—
187#123C2318yes—
188#123C23116yes—
189#123C23116yes—
190#123C23116yes—
191#123C2318yes—
192#123C23116yes—
193C22⋊Q8C2314yes—
194C22⋊Q8C2318yes—
195C22⋊Q8C2318yes—
196C22⋊Q8C2318yes—
197C22⋊Q8C2314yes—
198C22⋊Q8C2318yes—
199C22⋊Q8C2318yes—
200C22⋊Q8C2318yes—
201C22⋊Q8C23116yes—
202C22⋊Q8C23116yes—
203C22⋊Q8C23116yes—
204C22⋊Q8C23116yes—
205#121C2314yes—
206#121C2318yes—
207#121C2314yes—
208#121C2314yes—
209#121C2318yes—
210#121C23116yes—
211#121C23116yes—
212#121C23116yes—
213#121C2318yes—
214#121C2318yes—
215#121C23116yes—
216#121C2318yes—
217#121C2318yes—
218#121C2318yes—
219#121C2318yes—
220#121C2314yes—
221#121C2318yes—
222#121C2318yes—
223#121C2314yes—
224#121C2314yes—
225#121C2314yes—
226#121C2318yes—
227#121C2318yes—
228#121C23116yes—
229#121C23116yes—
230#121C23116yes—
231#120C2318yes—
232#120C23116yes—
233#120C2318yes—
234#120C23116yes—
235#120C23116yes—
236#120C2318yes—
237#120C2318yes—
238C4×Q8C2312yes—
239C4×Q8C2312yes—
240C4×Q8C2312yes—
241C4×Q8C2318yes—
242C4×Q8C2314yes—
243C4×Q8C2318yes—
244C4×D4C2312yes—
245C4×D4C2312yes—
246C4×D4C2314yes—
247C4×D4C2314yes—
248C4×D4C2318yes—
249C4×D4C2318yes—
250C4×D4C2318yes—
251C4×D4C2314yes—
252C4×D4C2314yes—
253C4×D4C2314yes—
254C4×D4C2318yes—
255C4×D4C2314yes—
256C4×D4C2314yes—
257C4×D4C2314yes—
258C4×D4C2314yes—
259C4×D4C23116yes—
260C4×D4C2312yes—
261C4×D4C2314yes—
262C4×D4C2318yes—
263C4×D4C2314yes—
264C4×D4C2318yes—
265C4×D4C23116yes—
266C4×D4C2318yes—
267C4×D4C2318yes—
268C4×D4C23116yes—
269C4×D4C23116yes—
270C4×D4C2318yes—
271#117C2314yes—
272#117C2318yes—
273#117C2318yes—
274#117C23116yes—
275#117C2318yes—
276#117C23116yes—
277#117C2312yes—
278#117C2318yes—
279#117C2318yes—
280#117C2314yes—
281#117C2312yes—
282#117C2311yescharacteristic
283#117C2314yes—
284#117C2318yes—
285C2×C4⋊C4C2314yes—
286C2×C4⋊C4C2318yes—
287C2×C4⋊C4C2314yes—
288C2×C4⋊C4C2318yes—
289C2×C4⋊C4C2314yes—
290C2×C22⋊C4C23116yes—
291C2×C22⋊C4C2318yes—
292C2×C22⋊C4C2318yes—
293C2×C22⋊C4C23116yes—
294C2×C22⋊C4C23116yes—
295C2×C42C2318yes—
296C2×C42C2312yes—
297C2×C42C2314yes—
298C2×C42C2314yes—
299C2×C42C2311yescharacteristic
300C24C2418yes—
301C24C2418yes—
302#41—216no—
303#41—24no—
304#41—48no—
305#41—48no—
306#41—44no—
307#41—416no—
308#41—416no—
309#41—48no—
310#41—48no—
311#41—44no—
312#41—416no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8C2412yes—
317C2×Q8C2411yescharacteristic
318C2×Q8C2414yes—
319C2×Q8C2414yes—
320C2×Q8C2414yes—
321C2×Q8C2418yes—
322C2×Q8C2414yes—
323C2×Q8C2412yes—
324C2×D4—216no—
325C2×D4—48no—
326C2×D4—416no—
327C2×D4—48no—
328C2×D4—416no—
329C2×D4—432no—
330C2×D4C2414yes—
331C2×D4C2412yes—
332C2×D4C2414yes—
333C2×D4C2412yes—
334C2×D4C2418yes—
335C2×D4C2418yes—
336C2×D4C2414yes—
337C2×D4C2414yes—
338C2×D4C2418yes—
339C2×D4C2418yes—
340C2×D4C24116yes—
341C2×D4C2418yes—
342C2×D4C2414yes—
343C2×D4C2412yes—
344C2×D4C2418yes—
345C2×D4C2414yes—
346C2×D4C24116yes—
347C2×D4C2418yes—
348C2×D4C2414yes—
349C2×D4C2411yescharacteristic
350C22×C4—416no—
351C22×C4C2412yes—
352C22×C4C2418yes—
353C22×C4C2414yes—
354C22×C4C2412yes—
355C22×C4C2414yes—
356C22×C4C2414yes—
357C22×C4C2411yescharacteristic
358C22×C4C2414yes—
359C22×C4C2418yes—
360C22×C4C2418yes—
361C22×C4C2418yes—
362C22×C4C2414yes—
363C22×C4C2418yes—
364C22×C4C2418yes—
365C22×C4C2418yes—
366C22×C4C24116yes—
367C22×C4C2414yes—
368C4⋊C4C2412yes—
369C4⋊C4C2418yes—
370C4⋊C4C2412yes—
371C4⋊C4C2414yes—
372C4⋊C4C2412yes—
373C4⋊C4C2414yes—
374C4⋊C4C2418yes—
375C4⋊C4C2414yes—
376C4⋊C4C24116yes—
377C4⋊C4C2418yes—
378C4⋊C4C24116yes—
379C4⋊C4C2418yes—
380C4⋊C4C2414yes—
381C4⋊C4C2418yes—
382C4⋊C4C2418yes—
383C4⋊C4C24116yes—
384C4⋊C4C2414yes—
385C4⋊C4C2418yes—
386C4⋊C4C2414yes—
387C22⋊C4C24116yes—
388C22⋊C4C2418yes—
389C22⋊C4C2418yes—
390C22⋊C4C2414yes—
391C22⋊C4C2418yes—
392C22⋊C4C2414yes—
393C22⋊C4C2418yes—
394C22⋊C4C2418yes—
395C22⋊C4C2418yes—
396C22⋊C4C24116yes—
397C22⋊C4C24116yes—
398C22⋊C4C24116yes—
399C22⋊C4C2418yes—
400C22⋊C4C2418yes—
401C22⋊C4C2418yes—
402C22⋊C4C24116yes—
403C22⋊C4C2418yes—
404C22⋊C4C2414yes—
405C22⋊C4C2414yes—
406C22⋊C4C2418yes—
407C22⋊C4C2418yes—
408C22⋊C4C2412yes—
409C42C2414yes—
410C42C2412yes—
411C42C2411yescharacteristic
412C42C2414yes—
413C42C2411yescharacteristic
414C42C2418yes—
415C42C2414yes—
416C42C24116yes—
417C42C2414yes—
418C42C2414yes—
419C42C2412yes—
420C42C2414yes—
421C23—216no—
422C23—48no—
423C23—48no—
424C23—48no—
425C23—416no—
426C23C2518yes—
427C23C2514yes—
428C23C2514yes—
429C23C2511yescharacteristic
430C23C2512yes—
431Q8—22no—
432Q8—28no—
433Q8—24no—
434Q8—42no—
435Q8—48no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439D4—216no—
440D4—216no—
441D4—24no—
442D4—28no—
443D4—28no—
444D4—24no—
445D4—28no—
446D4—44no—
447D4—48no—
448D4—48no—
449D4—416no—
450D4—48no—
451D4—416no—
452D4—48no—
453D4—44no—
454D4—48no—
455D4—44no—
456D4—44no—
457D4—42no—
458D4H(5,𝔽2)14yes—
459C2×C4—216no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—216no—
467C2×C4—24no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—416no—
472C2×C4—44no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4C2514yes—
477C2×C4C2512yes—
478C2×C4C2511yescharacteristic
479C2×C4C2518yes—
480C2×C4C2514yes—
481C2×C4C2514yes—
482C2×C4C2512yes—
483C2×C4C2514yes—
484C2×C4C2518yes—
485C2×C4C2514yes—
486C2×C4C2511yescharacteristic
487C2×C4C2512yes—
488C22—24no—
489C22—21no—
490C22—24no—
491C22—24no—
492C22—28no—
493C22—24no—
494C22—28no—
495C22—22no—
496C22—24no—
497C22—28no—
498C22—28no—
499C22—816no—
500C22—168no—
501C22C2×H(5,𝔽2)14yes—
502C22C2611yescharacteristic
503C4—24no—
504C4—21no—
505C4—24no—
506C4—28no—
507C4—22no—
508C4—24no—
509C4—24no—
510C4—28no—
511C4—22no—
512C4—24no—
513C4C2×H(5,𝔽2)12yes—
514C4#58514yes—
515C2—24no—
516C2—41no—
517C2—44no—
518C2—48no—
519C2—44no—
520C2C22×H(5,𝔽2)11yescharacteristic
521C2C2×#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. #3551C2semidirect
2. #3549C2semidirect
3. #3547C2semidirect
4. #3545C2semidirect
5. #3544C2semidirect
6. #3542C2semidirect
7. #3519C2semidirect
8. #3514C2semidirect
9. #3511C2semidirect
10. #3500C2semidirect
11. #3499C2semidirect
12. #3496C2semidirect
13. #3494C2semidirect
14. #3483C2semidirect
15. #3481C2semidirect
16. #3475C2semidirect
17. C2×#555C2semidirect
18. C2×2−1+4C22semidirect
19. C2×H(5,𝔽2)C22semidirect
20. #563C22semidirect
21. #562C22semidirect
22. H(3,𝔽4)C22semidirect
23. #560C22semidirect
24. #559C22semidirect
25. Q82C22semidirect
26. #556C22semidirect
27. #555C22semidirect
28. #555C22semidirect
29. #555C22semidirect
30. #555C22semidirect
31. #555C22semidirect
32. #555C22semidirect
33. #555C22semidirect
34. #555C22semidirect
35. #555C22semidirect
36. #554C22semidirect
37. #554C22semidirect
38. #554C22semidirect
39. #553C22semidirect
40. #553C22semidirect
41. #553C22semidirect
42. #553C22semidirect
43. #553C22semidirect
44. #552C22semidirect
45. #552C22semidirect
46. #551C22semidirect
47. #551C22semidirect
48. #551C22semidirect
49. #551C22semidirect
50. #550C22semidirect
51. #550C22semidirect
52. D4×Q8C22semidirect
53. #548C22semidirect
54. #548C22semidirect
55. #547C22semidirect
56. #547C22semidirect
57. #547C22semidirect
58. #547C22semidirect
59. #546C22semidirect
60. #546C22semidirect
61. #546C22semidirect
62. #546C22semidirect
63. #546C22semidirect
64. #546C22semidirect
65. #546C22semidirect
66. #546C22semidirect
68. D42C22semidirect
69. D42C22semidirect
70. D42C22semidirect
71. #544C22semidirect
72. #542C22semidirect
73. #542C22semidirect
74. #542C22semidirect
75. #540C22semidirect
76. #540C22semidirect
77. #538C22semidirect
78. #538C22semidirect
79. #537C22semidirect
80. #537C22semidirect
81. #536C22semidirect
82. #536C22semidirect
83. #536C22semidirect
84. #536C22semidirect
85. #535C22semidirect
86. #535C22semidirect
87. #535C22semidirect
88. #534C22semidirect
89. #533C22semidirect
90. #532C22semidirect
91. #532C22semidirect
92. #532C22semidirect
93. #532C22semidirect
94. #532C22semidirect
95. #532C22semidirect
96. #532C22semidirect
97. C2×C4⋊Q8C22semidirect
98. C2×C4⋊Q8C22semidirect
99. #529C22semidirect
100. #529C22semidirect
101. #529C22semidirect
102. #529C22semidirect
103. C2×#124C22semidirect
104. C2×#124C22semidirect
105. #525C22semidirect
106. #525C22semidirect
107. C2×#121C22semidirect
108. C2×#121C22semidirect
109. #520C22semidirect
110. #518C22semidirect
111. C4×#41C22semidirect
112. C4×#41C22semidirect
113. C4×#41C22semidirect
114. C4×#41C22semidirect
115. C2×C4×D4C22semidirect
116. C2×C4×D4C22semidirect
117. C2×#117C22semidirect

Maximal subgroups

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

subgroupindex
#35512
#35492
#35472
#35452
#35442
#35422
#35192
#35142
#35112
#35002
#34992
#34962
#34942
#34832
#34812
#34752
C2×#5552