Finite Group Databasebeta

← #62759

Subgroups — #62759

Order 256.

3635 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3551C214yesmaximal, complemented
2#3550C211yesmaximal, characteristic, complemented
3#3549C211yesmaximal, characteristic, complemented
4#3548C212yesmaximal, complemented
5#3547C214yesmaximal, complemented
6#3546C214yesmaximal, complemented
7#3545C214yesmaximal, complemented
8#3544C212yesmaximal, complemented
9#3543C212yesmaximal, complemented
10#3542C218yesmaximal, complemented
11#3529C214yesmaximal, complemented
12#3483C212yesmaximal, complemented
13#3481C212yesmaximal, complemented
14#3475C214yesmaximal, complemented
15D4×#41C214yesmaximal, complemented
16#3460C218yesmaximal, complemented
17#3449C214yesmaximal, complemented
18#3444C212yesmaximal, complemented
19C2×#532C211yesmaximal, characteristic, complemented
20C22×#41C2212yescomplemented
21C22×#41C2214yescomplemented
22#563C2218yescomplemented
23#563C2214yescomplemented
24#562C22116yescomplemented
25#562C2218yescomplemented
26H(3,𝔽4)C2214yescomplemented
27#560C22116yescomplemented
28#560C2218yescomplemented
29#559C2212yescomplemented
30Q82C2212yescomplemented
31#556C2214yescomplemented
32#555C2214yescomplemented
33#555C2218yescomplemented
34#555C2212yescomplemented
35#555C2218yescomplemented
36#554C2214yescomplemented
37#554C2214yescomplemented
38#553C2218yescomplemented
39#553C2218yescomplemented
40#552C2218yescomplemented
41#552C2218yescomplemented
42#551C22116yescomplemented
43#551C2218yescomplemented
44#550C2214yescomplemented
45#550C2218yescomplemented
46#550C2214yescomplemented
47D4×Q8C2214yescomplemented
48#548C2218yescomplemented
49#547C2218yescomplemented
50#547C2214yescomplemented
51#547C2214yescomplemented
52#546C22116yescomplemented
53#546C2218yescomplemented
54#546C2218yescomplemented
55#546C2218yescomplemented
56D42C2218yescomplemented
57D42C2214yescomplemented
58D42C2212yescomplemented
59#544C2218yescomplemented
60#542C2218yescomplemented
61#542C2218yescomplemented
62#542C22116yescomplemented
63#540C2218yescomplemented
64#540C22116yescomplemented
65#540C2218yescomplemented
66#538C22116yescomplemented
67#538C22116yescomplemented
68#537C2218yescomplemented
69#537C2218yescomplemented
70#537C2218yescomplemented
71#536C2218yescomplemented
72#536C2214yescomplemented
73#536C2214yescomplemented
74#535C2218yescomplemented
75#535C2214yescomplemented
76#535C2214yescomplemented
77#535C2218yescomplemented
78#535C2218yescomplemented
79#534C2218yescomplemented
80#534C2218yescomplemented
81#533C2212yescomplemented
82#533C2212yescomplemented
83#533C2212yescomplemented
84#532C2211yescharacteristic
85#532C2211yescharacteristic
86#532C2212yes—
87#532C2214yescomplemented
88#532C2211yescharacteristic, complemented
89#532C2212yescomplemented
90#532C2214yescomplemented
91#532C2211yescharacteristic, complemented
92#532C2212yescomplemented
93#532C2212yescomplemented
94#532C2212yescomplemented
95#532C2218yescomplemented
96#532C2218yescomplemented
97#532C2218yescomplemented
98C2×C4⋊Q8C2211yescharacteristic, complemented
99C2×#127C2211yescharacteristic
100#529C2214yescomplemented
101#529C2214yescomplemented
102#529C2214yescomplemented
103#529C2214yescomplemented
104#529C2214yescomplemented
105#529C2214yescomplemented
106#529C2218yescomplemented
107#529C22116yescomplemented
108C2×#124C2212yescomplemented
109#525C2214yescomplemented
110#525C2214yescomplemented
111#525C2214yescomplemented
112#525C2218yescomplemented
113#525C2218yescomplemented
114#525C2218yescomplemented
115#525C22116yescomplemented
116C2×#121C2214yescomplemented
117C4×#41C2214yescomplemented
118C4×#41C2214yescomplemented
119C4×#41C2214yescomplemented
120C4×#41C2218yescomplemented
121C2×C4×D4C2214yescomplemented
122C2×C4×D4C2218yescomplemented
123C2×#117C2214yescomplemented
124C22×C42C2211yescharacteristic, complemented
125C2×#41—48no—
126C2×#41—416no—
127C2×#41C2314yes—
128C2×#41C2314yes—
129C2×#41C2314yes—
130C2×#41C2318yes—
131C2×#41C2312yes—
132C2×#41C2314yes—
133C2×#41C2318yes—
134C2×#41C2312yes—
135C2×#41C2312yes—
136C2×#41C2314yes—
137C2×#41C2312yes—
138C2×#41C2318yes—
139C2×#41C2318yes—
140C22×Q8C2312yes—
141C22×Q8C2314yes—
142C22×D4C2314yes—
143C22×D4C2318yes—
144C22×D4C2314yes—
145C22×D4C2312yes—
146C22×D4C2318yes—
147C22×D4C2314yes—
148C23×C4C2314yes—
149C23×C4C2311yescharacteristic
150C23×C4C2318yes—
151C23×C4C2312yes—
152C4⋊Q8C2311yescharacteristic
153C4⋊Q8C2311yescharacteristic
154C4⋊Q8C2314yes—
155C4⋊Q8C2312yes—
156C4⋊Q8C2312yes—
157C4⋊Q8C2312yes—
158C4⋊Q8C2312yes—
159C4⋊Q8C2318yes—
160C4⋊Q8C2318yes—
161C4⋊Q8C2318yes—
162#127C2311yescharacteristic
163#127C2314yes—
164#127C2311yescharacteristic
165#127C2312yes—
166#127C2312yes—
167#127C2318yes—
168#127C2318yes—
169#127C2318yes—
170#126C2318yes—
171#126C2318yes—
172#126C23116yes—
173#126C23116yes—
174#126C23116yes—
175#125C2314yes—
176#125C2314yes—
177#125C2314yes—
178#125C23116yes—
179#125C2318yes—
180#124C2312yes—
181#124C2312yes—
182#124C2318yes—
183#124C2312yes—
184#124C2312yes—
185#124C2314yes—
186#124C2314yes—
187#124C2314yes—
188#124C2318yes—
189#124C23116yes—
190#124C2318yes—
191#124C2318yes—
192#124C2318yes—
193#124C2318yes—
194#124C23116yes—
195#123C2318yes—
196#123C2318yes—
197#123C2318yes—
198#123C2318yes—
199#123C2318yes—
200#123C23116yes—
201#123C2318yes—
202#123C2318yes—
203#123C23116yes—
204#123C23116yes—
205#123C23116yes—
206#123C23116yes—
207#123C23116yes—
208C22⋊Q8C2314yes—
209C22⋊Q8C2314yes—
210C22⋊Q8C2314yes—
211C22⋊Q8C2314yes—
212C22⋊Q8C2314yes—
213C22⋊Q8C2314yes—
214C22⋊Q8C23116yes—
215C22⋊Q8C2318yes—
216C22⋊Q8C2318yes—
217C22⋊Q8C23116yes—
218C22⋊Q8C23116yes—
219C22⋊Q8C23116yes—
220C22⋊Q8C2318yes—
221C22⋊Q8C23116yes—
222#121C2314yes—
223#121C2314yes—
224#121C2314yes—
225#121C2318yes—
226#121C2314yes—
227#121C2318yes—
228#121C2314yes—
229#121C2314yes—
230#121C2314yes—
231#121C2314yes—
232#121C2314yes—
233#121C2314yes—
234#121C23116yes—
235#121C2318yes—
236#121C2318yes—
237#121C23116yes—
238#121C23116yes—
239#121C23116yes—
240#121C2318yes—
241#121C23116yes—
242#121C23116yes—
243#121C23116yes—
244#121C23116yes—
245#121C23116yes—
246#121C2318yes—
247#121C2318yes—
248#121C23116yes—
249#120C2318yes—
250#120C2318yes—
251#120C2318yes—
252#120C23116yes—
253#120C2318yes—
254#120C2318yes—
255#120C23116yes—
256#120C23116yes—
257C4×Q8C2314yes—
258C4×Q8C2314yes—
259C4×Q8C2314yes—
260C4×Q8C2314yes—
261C4×Q8C2314yes—
262C4×Q8C2314yes—
263C4×Q8C2318yes—
264C4×D4C2314yes—
265C4×D4C2314yes—
266C4×D4C2314yes—
267C4×D4C23116yes—
268C4×D4C2318yes—
269C4×D4C2314yes—
270C4×D4C2318yes—
271C4×D4C2314yes—
272C4×D4C2318yes—
273C4×D4C2318yes—
274C4×D4C23116yes—
275C4×D4C2314yes—
276C4×D4C2318yes—
277C4×D4C2318yes—
278C4×D4C23116yes—
279C4×D4C2318yes—
280C4×D4C2318yes—
281C4×D4C2318yes—
282C4×D4C2318yes—
283C4×D4C2314yes—
284C4×D4C2314yes—
285C4×D4C2314yes—
286C4×D4C2314yes—
287C4×D4C23116yes—
288C4×D4C2318yes—
289#117C2314yes—
290#117C2318yes—
291#117C2318yes—
292#117C2314yes—
293#117C2314yes—
294#117C2318yes—
295#117C2314yes—
296#117C2314yes—
297#117C2318yes—
298#117C2318yes—
299#117C2318yes—
300#117C23116yes—
301C2×C4⋊C4C2318yes—
302C2×C4⋊C4C2314yes—
303C2×C4⋊C4C2318yes—
304C2×C22⋊C4C2318yes—
305C2×C22⋊C4C2318yes—
306C2×C22⋊C4C23116yes—
307C2×C42C2318yes—
308C2×C42C2312yes—
309C2×C42C2318yes—
310C2×C42C2311yescharacteristic
311C2×C42C2312yes—
312C2×C42C2311yescharacteristic
313C2×C42C2311yescharacteristic
314C2×C42C2311yescharacteristic
315C2×C42C2311yescharacteristic
316C2×C42C2312yes—
317C2×C42C2311yescharacteristic
318C24C2414yes—
319C24C2411yescharacteristic
320C24C2418yes—
321#41—216no—
322#41—44no—
323#41—48no—
324#41—44no—
325#41—48no—
326#41—44no—
327#41—44no—
328#41—416no—
329#41—48no—
330#41—416no—
331#41—416no—
332#41—48no—
333#41—416no—
334C2×Q8—44no—
335C2×Q8—48no—
336C2×Q8C2414yes—
337C2×Q8C2414yes—
338C2×Q8C2414yes—
339C2×Q8C2412yes—
340C2×Q8C2414yes—
341C2×Q8C2418yes—
342C2×Q8C2412yes—
343C2×D4—48no—
344C2×D4—416no—
345C2×D4—48no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—416no—
349C2×D4C2414yes—
350C2×D4C2414yes—
351C2×D4C2414yes—
352C2×D4C2418yes—
353C2×D4C2414yes—
354C2×D4C2414yes—
355C2×D4C24116yes—
356C2×D4C2418yes—
357C2×D4C2418yes—
358C2×D4C24116yes—
359C2×D4C2418yes—
360C2×D4C2412yes—
361C2×D4C2414yes—
362C2×D4C2414yes—
363C2×D4C2414yes—
364C2×D4C2418yes—
365C2×D4C24116yes—
366C2×D4C2414yes—
367C2×D4C2414yes—
368C2×D4C2412yes—
369C22×C4—24no—
370C22×C4—216no—
371C22×C4—44no—
372C22×C4—416no—
373C22×C4C2412yes—
374C22×C4C2412yes—
375C22×C4C2412yes—
376C22×C4C2411yescharacteristic
377C22×C4C2411yescharacteristic
378C22×C4C2411yescharacteristic
379C22×C4C2418yes—
380C22×C4C2414yes—
381C22×C4C2414yes—
382C22×C4C2414yes—
383C22×C4C2414yes—
384C22×C4C2411yescharacteristic
385C22×C4C2411yescharacteristic
386C22×C4C2418yes—
387C22×C4C2414yes—
388C22×C4C2412yes—
389C22×C4C2412yes—
390C22×C4C2412yes—
391C22×C4C2418yes—
392C22×C4C24116yes—
393C22×C4C24116yes—
394C22×C4C2418yes—
395C22×C4C2411yescharacteristic
396C4⋊C4C2414yes—
397C4⋊C4C2414yes—
398C4⋊C4C2414yes—
399C4⋊C4C2414yes—
400C4⋊C4C2414yes—
401C4⋊C4C24116yes—
402C4⋊C4C2418yes—
403C4⋊C4C24116yes—
404C4⋊C4C2414yes—
405C4⋊C4C2418yes—
406C4⋊C4C2418yes—
407C4⋊C4C2418yes—
408C4⋊C4C2418yes—
409C4⋊C4C2418yes—
410C4⋊C4C24116yes—
411C4⋊C4C2418yes—
412C4⋊C4C2414yes—
413C4⋊C4C2414yes—
414C22⋊C4C2418yes—
415C22⋊C4C2418yes—
416C22⋊C4C2418yes—
417C22⋊C4C2418yes—
418C22⋊C4C2418yes—
419C22⋊C4C2418yes—
420C22⋊C4C24116yes—
421C22⋊C4C2418yes—
422C22⋊C4C24116yes—
423C22⋊C4C2418yes—
424C22⋊C4C2418yes—
425C22⋊C4C2418yes—
426C22⋊C4C2418yes—
427C22⋊C4C24116yes—
428C22⋊C4C2418yes—
429C22⋊C4C2418yes—
430C22⋊C4C2418yes—
431C22⋊C4C2418yes—
432C22⋊C4C24116yes—
433C42C2411yescharacteristic
434C42C2412yes—
435C42C2412yes—
436C42C2412yes—
437C42C2411yescharacteristic
438C42C2412yes—
439C42C2411yescharacteristic
440C42C2411yescharacteristic
441C42C2418yes—
442C42C2418yes—
443C42C2418yes—
444C42C2418yes—
445C42C2412yes—
446C42C2412yes—
447C42C2418yes—
448C23—22no—
449C23—28no—
450C23—42no—
451C23—48no—
452C23—48no—
453C23—416no—
454C23C2514yes—
455C23C2518yes—
456C23C2511yescharacteristic
457C23C2511yescharacteristic
458C23C2514yes—
459C23C2511yescharacteristic
460Q8—28no—
461Q8—28no—
462Q8—42no—
463Q8—44no—
464Q8—42no—
465Q8—48no—
466Q8—44no—
467D4—216no—
468D4—28no—
469D4—28no—
470D4—28no—
471D4—28no—
472D4—48no—
473D4—416no—
474D4—44no—
475D4—44no—
476D4—42no—
477D4—44no—
478D4—44no—
479D4—42no—
480D4—44no—
481D4—48no—
482D4—416no—
483D4—48no—
484D4—44no—
485D4—48no—
486D4—416no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—216no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—216no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—416no—
507C2×C4—48no—
508C2×C4—416no—
509C2×C4H(5,𝔽2)14yes—
510C2×C4C2512yes—
511C2×C4C2512yes—
512C2×C4C2518yes—
513C2×C4C2518yes—
514C2×C4C2512yes—
515C2×C4C2511yescharacteristic
516C2×C4C2511yescharacteristic
517C2×C4C2514yes—
518C2×C4C2518yes—
519C2×C4C2514yes—
520C2×C4C2511yescharacteristic
521C2×C4C2512yes—
522C2×C4C2511yescharacteristic
523C22—22no—
524C22—24no—
525C22—21no—
526C22—21no—
527C22—24no—
528C22—28no—
529C22—22no—
530C22—22no—
531C22—28no—
532C22—28no—
533C22—28no—
534C22—28no—
535C22—44no—
536C22—88no—
537C22—816no—
538C22C2×H(5,𝔽2)12yes—
539C22C2611yescharacteristic
540C4—24no—
541C4—21no—
542C4—21no—
543C4—24no—
544C4—28no—
545C4—22no—
546C4—22no—
547C4—28no—
548C4—22no—
549C4—28no—
550C4C2×H(5,𝔽2)12yes—
551C4C2×H(5,𝔽2)12yes—
552C4#58514yes—
553C2—22no—
554C2—41no—
555C2—41no—
556C2—44no—
557C2—48no—
558C2—44no—
559C2C22×H(5,𝔽2)11yescharacteristic
560C2C2×#58512yes—
subgroupG/H
centerC2×C4C25
commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3551C2semidirect
2. #3550C2semidirect
3. #3549C2semidirect
4. #3548C2semidirect
5. #3547C2semidirect
6. #3546C2semidirect
7. #3545C2semidirect
8. #3544C2semidirect
9. #3543C2semidirect
10. #3542C2semidirect
11. #3529C2semidirect
12. #3483C2semidirect
13. #3481C2semidirect
14. #3475C2semidirect
15. D4×#41C2semidirect
16. #3460C2semidirect
17. #3449C2semidirect
18. #3444C2semidirect
19. C2×#532C2semidirect
20. C22×#41C22semidirect
21. C22×#41C22semidirect
22. #563C22semidirect
23. #563C22semidirect
24. #562C22semidirect
25. #562C22semidirect
26. H(3,𝔽4)C22semidirect
27. #560C22semidirect
28. #560C22semidirect
29. #559C22semidirect
30. Q82C22semidirect
31. #556C22semidirect
32. #555C22semidirect
33. #555C22semidirect
34. #555C22semidirect
35. #555C22semidirect
36. #554C22semidirect
37. #554C22semidirect
38. #553C22semidirect
39. #553C22semidirect
40. #552C22semidirect
41. #552C22semidirect
42. #551C22semidirect
43. #551C22semidirect
44. #550C22semidirect
45. #550C22semidirect
46. #550C22semidirect
47. D4×Q8C22semidirect
48. #548C22semidirect
49. #547C22semidirect
50. #547C22semidirect
51. #547C22semidirect
52. #546C22semidirect
53. #546C22semidirect
54. #546C22semidirect
55. #546C22semidirect
56. D42C22semidirect
57. D42C22semidirect
58. D42C22semidirect
59. #544C22semidirect
60. #542C22semidirect
61. #542C22semidirect
62. #542C22semidirect
63. #540C22semidirect
64. #540C22semidirect
65. #540C22semidirect
66. #538C22semidirect
67. #538C22semidirect
68. #537C22semidirect
69. #537C22semidirect
70. #537C22semidirect
71. #536C22semidirect
72. #536C22semidirect
73. #536C22semidirect
74. #535C22semidirect
75. #535C22semidirect
76. #535C22semidirect
77. #535C22semidirect
78. #535C22semidirect
79. #534C22semidirect
80. #534C22semidirect
81. #533C22semidirect
82. #533C22semidirect
83. #533C22semidirect
87. #532C22semidirect
88. #532C22semidirect
89. #532C22semidirect
90. #532C22semidirect
91. #532C22semidirect
92. #532C22semidirect
93. #532C22semidirect
94. #532C22semidirect
95. #532C22semidirect
96. #532C22semidirect
97. #532C22semidirect
98. C2×C4⋊Q8C22semidirect
100. #529C22semidirect
101. #529C22semidirect
102. #529C22semidirect
103. #529C22semidirect
104. #529C22semidirect
105. #529C22semidirect
106. #529C22semidirect
107. #529C22semidirect
108. C2×#124C22semidirect
109. #525C22semidirect
110. #525C22semidirect
111. #525C22semidirect
112. #525C22semidirect
113. #525C22semidirect
114. #525C22semidirect
115. #525C22semidirect
116. C2×#121C22semidirect
117. C4×#41C22semidirect
118. C4×#41C22semidirect
119. C4×#41C22semidirect
120. C4×#41C22semidirect
121. C2×C4×D4C22semidirect
122. C2×C4×D4C22semidirect
123. C2×#117C22semidirect
124. C22×C42C22semidirect

Maximal subgroups

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

subgroupindex
#35512
#35502
#35492
#35482
#35472
#35462
#35452
#35442
#35432
#35422
#35292
#34832
#34812
#34752
D4×#412
#34602
#34492
#34442
C2×#5322