Finite Group Databasebeta

← #62836

Subgroups — #62836

Order 256.

3627 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3562C218yesmaximal, complemented
2#3554C214yesmaximal, complemented
3#3546C214yesmaximal, complemented
4#3542C218yesmaximal, complemented
5#3517C214yesmaximal, complemented
6#3514C214yesmaximal, complemented
7#3513C218yesmaximal, complemented
8#3511C214yesmaximal, complemented
9#3510C212yesmaximal, complemented
10#3509C211yesmaximal, characteristic, complemented
11#3508C211yesmaximal, characteristic, complemented
12#3508C214yesmaximal, complemented
13#3500C212yesmaximal, complemented
14#3499C211yesmaximal, characteristic, complemented
15#3498C211yesmaximal, characteristic, complemented
16#3487C212yesmaximal, complemented
17#3485C212yesmaximal, complemented
18#3444C211yesmaximal, characteristic, complemented
19#3444C211yesmaximal, characteristic, complemented
20C2×#532C211yesmaximal, characteristic, complemented
21C2×2−1+4C2212yescomplemented
22C2×H(5,𝔽2)C2212yes—
23C22×#41C2212yescomplemented
24#562C22116yescomplemented
25#560C22116yescomplemented
26#559C2218yescomplemented
27#559C2218yescomplemented
28#559C2214yescomplemented
29#557C2214yescomplemented
30#556C2214yescomplemented
31#556C2218yescomplemented
32#555C2218yescomplemented
33#554C2218yescomplemented
34#553C2218yescomplemented
35#553C2218yescomplemented
36#553C22116yescomplemented
37#552C2218yescomplemented
38#551C22116yescomplemented
39#551C22116yescomplemented
40#550C2218yescomplemented
41#550C2218yescomplemented
42#550C2214yescomplemented
43#550C2218yescomplemented
44#550C2214yescomplemented
45D4×Q8C2214yescomplemented
46#548C2218yescomplemented
47#548C2218yescomplemented
48#548C2218yescomplemented
49#547C2218yescomplemented
50#547C2218yescomplemented
51#547C2218yescomplemented
52#547C2214yescomplemented
53#546C2218yescomplemented
54#546C2218yescomplemented
55#546C2218yescomplemented
56#546C22116yescomplemented
57#546C2218yescomplemented
58#546C2218yescomplemented
59#546C2218yescomplemented
60D42C2218yes—
61D42C2214yescomplemented
62#542C2218yescomplemented
63#542C2218yescomplemented
64#542C2218yescomplemented
65#541C2214yescomplemented
66#540C2218yescomplemented
67#540C2218yescomplemented
68#540C2214yescomplemented
69#540C22116yescomplemented
70#539C22116yescomplemented
71#538C22116yescomplemented
72#538C22116yescomplemented
73#537C2218yescomplemented
74#536C2212yes—
75#536C2212yescomplemented
76#536C2214yescomplemented
77#536C2214yescomplemented
78#535C2212yes—
79#535C2212yescomplemented
80#535C2218yescomplemented
81#535C2214yescomplemented
82#535C2214yescomplemented
83#535C2218yescomplemented
84#534C2218yescomplemented
85#533C2211yescharacteristic, complemented
86#533C2211yescharacteristic, complemented
87#533C2212yescomplemented
88#533C2212yescomplemented
89#533C2214yescomplemented
90#532C2211yescharacteristic
91#532C2211yescharacteristic
92#532C2214yescomplemented
93#532C2214yescomplemented
94#532C2214yescomplemented
95#532C2214yescomplemented
96#532C2212yescomplemented
97#532C2212yescomplemented
98#532C2212yescomplemented
99#532C2212yescomplemented
100#532C2214yescomplemented
101#532C2214yescomplemented
102C2×C4⋊Q8C2211yescharacteristic, complemented
103C2×C4⋊Q8C2212yescomplemented
104C2×#127C2211yescharacteristic
105C2×#127C2212yescomplemented
106#529C2214yescomplemented
107#529C2214yescomplemented
108#529C2212yescomplemented
109#529C2212yescomplemented
110#529C2214yescomplemented
111#529C2214yescomplemented
112#529C2214yescomplemented
113#529C2218yescomplemented
114C2×#124C2212yescomplemented
115C2×#124C2214yescomplemented
116#525C2211yescharacteristic
117#525C2212yescomplemented
118#525C2211yescharacteristic, complemented
119#525C2214yescomplemented
120#525C2214yescomplemented
121#525C2218yescomplemented
122C2×#123C2218yescomplemented
123C2×#121C2214yescomplemented
124#520C2218yescomplemented
125#519C2212yescomplemented
126#518C2212yescomplemented
127#518C2218yescomplemented
128C4×#41C2211yescharacteristic
129C4×#41C2211yescharacteristic, complemented
130C4×#41C2212yescomplemented
131C4×#41C2214yescomplemented
132C4×#41C2214yescomplemented
133C2×C4×D4C2214yescomplemented
134C22×C42C2211yescharacteristic, complemented
1352−1+4—48no—
136H(5,𝔽2)—48no—
137C2×#41—48no—
138C2×#41C2311yescharacteristic
139C2×#41C2312yes—
140C2×#41C2311yescharacteristic
141C2×#41C2314yes—
142C2×#41C2314yes—
143C2×#41C2318yes—
144C2×#41C2318yes—
145C2×#41C2318yes—
146C2×#41C2318yes—
147C2×#41C2314yes—
148C2×#41C2314yes—
149C2×#41C2314yes—
150C2×#41C2314yes—
151C22×Q8C2312yes—
152C22×Q8C2314yes—
153C22×Q8C2314yes—
154C22×D4C2312yes—
155C22×D4C2314yes—
156C22×D4C2318yes—
157C22×D4C2314yes—
158C22×D4C2318yes—
159C22×D4C2314yes—
160C22×D4C2314yes—
161C23×C4C2311yescharacteristic
162C23×C4C2314yes—
163C23×C4C2312yes—
164C4⋊Q8C2311yescharacteristic
165C4⋊Q8C2312yes—
166C4⋊Q8C2312yes—
167C4⋊Q8C2314yes—
168C4⋊Q8C2314yes—
169C4⋊Q8C2318yes—
170C4⋊Q8C2311yescharacteristic
171C4⋊Q8C2312yes—
172C4⋊Q8C2312yes—
173C4⋊Q8C2314yes—
174#127C2312yes—
175#127C2312yes—
176#127C2314yes—
177#127C2311yescharacteristic
178#127C2314yes—
179#127C2318yes—
180#127C2311yescharacteristic
181#127C2314yes—
182#127C23116yes—
183#126C2318yes—
184#126C2314yes—
185#126C2314yes—
186#126C23116yes—
187#126C23116yes—
188#126C23116yes—
189#125C2312yes—
190#125C2314yes—
191#125C2312yes—
192#125C2314yes—
193#125C2318yes—
194#125C23116yes—
195#124C2311yescharacteristic
196#124C2314yes—
197#124C2314yes—
198#124C2318yes—
199#124C2314yes—
200#124C2314yes—
201#124C23116yes—
202#124C2311yescharacteristic
203#124C2312yes—
204#124C2314yes—
205#124C2314yes—
206#124C2314yes—
207#124C2314yes—
208#124C2318yes—
209#124C23116yes—
210#124C23116yes—
211#123C2312yes—
212#123C2314yes—
213#123C2318yes—
214#123C2318yes—
215#123C23116yes—
216#123C23116yes—
217#123C2312yes—
218#123C2314yes—
219#123C2314yes—
220#123C2318yes—
221#123C23116yes—
222#123C23116yes—
223#123C23116yes—
224#123C23116yes—
225#123C23116yes—
226#123C23116yes—
227C22⋊Q8C2312yes—
228C22⋊Q8C2314yes—
229C22⋊Q8C2314yes—
230C22⋊Q8C2314yes—
231C22⋊Q8C2312yes—
232C22⋊Q8C2314yes—
233C22⋊Q8C2314yes—
234C22⋊Q8C23116yes—
235C22⋊Q8C2318yes—
236C22⋊Q8C2318yes—
237C22⋊Q8C2318yes—
238C22⋊Q8C23116yes—
239C22⋊Q8C23116yes—
240C22⋊Q8C23116yes—
241#121C2318yes—
242#121C2318yes—
243#121C2318yes—
244#121C2318yes—
245#121C2312yes—
246#121C2314yes—
247#121C2314yes—
248#121C2314yes—
249#121C2314yes—
250#121C2312yes—
251#121C2314yes—
252#121C23116yes—
253#121C2318yes—
254#121C2318yes—
255#121C2318yes—
256#121C2318yes—
257#121C2318yes—
258#121C23116yes—
259#121C23116yes—
260#121C23116yes—
261#121C23116yes—
262#121C23116yes—
263#121C23116yes—
264#121C23116yes—
265#121C23116yes—
266#120C2312yes—
267#120C2314yes—
268#120C2312yes—
269#120C2318yes—
270#120C2318yes—
271#120C23116yes—
272#120C23116yes—
273#120C23116yes—
274C4×Q8C2312yes—
275C4×Q8C2314yes—
276C4×Q8C2314yes—
277C4×Q8C2312yes—
278C4×Q8C2314yes—
279C4×Q8C2314yes—
280C4×Q8C2314yes—
281C4×Q8C2318yes—
282C4×Q8C2318yes—
283C4×Q8C2318yes—
284C4×D4C2318yes—
285C4×D4C2318yes—
286C4×D4C2318yes—
287C4×D4C2318yes—
288C4×D4C2314yes—
289C4×D4C2312yes—
290C4×D4C2314yes—
291C4×D4C2314yes—
292C4×D4C2318yes—
293C4×D4C2314yes—
294C4×D4C2318yes—
295C4×D4C2314yes—
296C4×D4C2318yes—
297C4×D4C2314yes—
298C4×D4C2314yes—
299C4×D4C2314yes—
300C4×D4C2314yes—
301C4×D4C2314yes—
302C4×D4C2314yes—
303C4×D4C2312yes—
304C4×D4C23116yes—
305C4×D4C23116yes—
306C4×D4C2318yes—
307C4×D4C2318yes—
308C4×D4C2318yes—
309C4×D4C2318yes—
310C4×D4C23116yes—
311C4×D4C23116yes—
312C4×D4C2318yes—
313#117C2311yescharacteristic
314#117C2312yes—
315#117C2311yescharacteristic
316#117C2314yes—
317#117C2312yes—
318#117C2314yes—
319#117C2314yes—
320#117C2314yes—
321#117C2314yes—
322#117C2312yes—
323#117C2318yes—
324#117C2318yes—
325#117C2318yes—
326#117C23116yes—
327C2×C4⋊C4C2314yes—
328C2×C4⋊C4C23116yes—
329C2×C4⋊C4C2318yes—
330C2×C22⋊C4C2318yes—
331C2×C22⋊C4C2318yes—
332C2×C22⋊C4C23116yes—
333C2×C22⋊C4C23116yes—
334C2×C42C2312yes—
335C2×C42C2312yes—
336C2×C42C2311yescharacteristic
337C2×C42C2314yes—
338C2×C42C2312yes—
339C2×C42C2311yescharacteristic
340C2×C42C2312yes—
341C2×C42C2312yes—
342C2×C42C2312yes—
343C2×C42C2312yes—
344C24C2414yes—
345C24C2411yescharacteristic
346C24C2418yes—
347#41—24no—
348#41—28no—
349#41—24no—
350#41—48no—
351#41—416no—
352#41—416no—
353#41—416no—
354#41—48no—
355#41—416no—
356#41—48no—
357#41—48no—
358#41—48no—
359#41—48no—
360C2×Q8—28no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8C2414yes—
364C2×Q8C2411yescharacteristic
365C2×Q8C2414yes—
366C2×Q8C2414yes—
367C2×Q8C2412yes—
368C2×Q8C2411yescharacteristic
369C2×Q8C2414yes—
370C2×Q8C2418yes—
371C2×D4—28no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4C2418yes—
379C2×D4C2414yes—
380C2×D4C2412yes—
381C2×D4C2414yes—
382C2×D4C2412yes—
383C2×D4C2411yescharacteristic
384C2×D4C2414yes—
385C2×D4C2414yes—
386C2×D4C2412yes—
387C2×D4C2411yescharacteristic
388C2×D4C2414yes—
389C2×D4C2414yes—
390C2×D4C2418yes—
391C2×D4C24116yes—
392C2×D4C2418yes—
393C2×D4C24116yes—
394C2×D4C2414yes—
395C2×D4C2418yes—
396C2×D4C2418yes—
397C2×D4C24116yes—
398C2×D4C2418yes—
399C22×C4—24no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4C2412yes—
403C22×C4C2412yes—
404C22×C4C2412yes—
405C22×C4C2412yes—
406C22×C4C2412yes—
407C22×C4C2411yescharacteristic
408C22×C4C2414yes—
409C22×C4C2414yes—
410C22×C4C2412yes—
411C22×C4C2414yes—
412C22×C4C2412yes—
413C22×C4C2414yes—
414C22×C4C2412yes—
415C22×C4C2412yes—
416C22×C4C2412yes—
417C22×C4C2418yes—
418C22×C4C24116yes—
419C22×C4C2418yes—
420C22×C4C24116yes—
421C22×C4C2418yes—
422C22×C4C2418yes—
423C22×C4C2411yescharacteristic
424C4⋊C4C2414yes—
425C4⋊C4C2418yes—
426C4⋊C4C2414yes—
427C4⋊C4C2418yes—
428C4⋊C4C24116yes—
429C4⋊C4C2418yes—
430C4⋊C4C24116yes—
431C4⋊C4C2414yes—
432C4⋊C4C2414yes—
433C4⋊C4C2412yes—
434C4⋊C4C2414yes—
435C4⋊C4C2414yes—
436C4⋊C4C2414yes—
437C4⋊C4C2412yes—
438C4⋊C4C2414yes—
439C4⋊C4C2418yes—
440C4⋊C4C2418yes—
441C4⋊C4C2414yes—
442C4⋊C4C2414yes—
443C4⋊C4C2414yes—
444C4⋊C4C24116yes—
445C22⋊C4C2412yes—
446C22⋊C4C2414yes—
447C22⋊C4C2414yes—
448C22⋊C4C2414yes—
449C22⋊C4C2418yes—
450C22⋊C4C24116yes—
451C22⋊C4C24116yes—
452C22⋊C4C24116yes—
453C22⋊C4C24116yes—
454C22⋊C4C24116yes—
455C22⋊C4C24116yes—
456C22⋊C4C2412yes—
457C22⋊C4C2414yes—
458C22⋊C4C2414yes—
459C22⋊C4C2414yes—
460C22⋊C4C24116yes—
461C22⋊C4C2418yes—
462C22⋊C4C2418yes—
463C22⋊C4C2414yes—
464C22⋊C4C2418yes—
465C22⋊C4C2414yes—
466C22⋊C4C2414yes—
467C42C2411yescharacteristic
468C42C2411yescharacteristic
469C42C2412yes—
470C42C2412yes—
471C42C2411yescharacteristic
472C42C2411yescharacteristic
473C42C2412yes—
474C42C2412yes—
475C42C2412yes—
476C42C2412yes—
477C42C2418yes—
478C42C2412yes—
479C42C2414yes—
480C42C2414yes—
481C42C2414yes—
482C42C2412yes—
483C42C24116yes—
484C23—22no—
485C23—28no—
486C23—42no—
487C23—48no—
488C23—48no—
489C23—416no—
490C23C2512yes—
491C23C2514yes—
492C23C2518yes—
493C23C2512yes—
494C23C2512yes—
495C23C2511yescharacteristic
496Q8—28no—
497Q8—28no—
498Q8—22no—
499Q8—24no—
500Q8—22no—
501Q8—28no—
502Q8—48no—
503Q8—44no—
504D4—24no—
505D4—28no—
506D4—24no—
507D4—28no—
508D4—28no—
509D4—22no—
510D4—24no—
511D4—22no—
512D4—216no—
513D4—28no—
514D4—48no—
515D4—48no—
516D4—48no—
517D4—416no—
518D4—416no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—48no—
523D4—416no—
524D4—48no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—416no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—416no—
546C2×C4H(5,𝔽2)14yes—
547C2×C4C2512yes—
548C2×C4C2514yes—
549C2×C4C2512yes—
550C2×C4C2518yes—
551C2×C4C2512yes—
552C2×C4C2512yes—
553C2×C4C2514yes—
554C2×C4C2512yes—
555C2×C4C2514yes—
556C2×C4C2518yes—
557C2×C4C2512yes—
558C2×C4C2511yescharacteristic
559C2×C4C2511yescharacteristic
560C2×C4C2512yes—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—24no—
565C22—22no—
566C22—28no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—216no—
572C22—28no—
573C22—44no—
574C22—88no—
575C22—168no—
576C22C2×H(5,𝔽2)12yes—
577C22C2611yescharacteristic
578C4—22no—
579C4—22no—
580C4—24no—
581C4—22no—
582C4—28no—
583C4—22no—
584C4—24no—
585C4—22no—
586C4—22no—
587C4—24no—
588C4—28no—
589C4C2×H(5,𝔽2)12yes—
590C4C2×H(5,𝔽2)12yes—
591C4#58514yes—
592C2—22no—
593C2—42no—
594C2—42no—
595C2—48no—
596C2—44no—
597C2—42no—
598C2C22×H(5,𝔽2)11yescharacteristic
599C2C2×#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. #3562C2semidirect
2. #3554C2semidirect
3. #3546C2semidirect
4. #3542C2semidirect
5. #3517C2semidirect
6. #3514C2semidirect
7. #3513C2semidirect
8. #3511C2semidirect
9. #3510C2semidirect
10. #3509C2semidirect
11. #3508C2semidirect
12. #3508C2semidirect
13. #3500C2semidirect
14. #3499C2semidirect
15. #3498C2semidirect
16. #3487C2semidirect
17. #3485C2semidirect
18. #3444C2semidirect
19. #3444C2semidirect
20. C2×#532C2semidirect
21. C2×2−1+4C22semidirect
23. C22×#41C22semidirect
24. #562C22semidirect
25. #560C22semidirect
26. #559C22semidirect
27. #559C22semidirect
28. #559C22semidirect
29. #557C22semidirect
30. #556C22semidirect
31. #556C22semidirect
32. #555C22semidirect
33. #554C22semidirect
34. #553C22semidirect
35. #553C22semidirect
36. #553C22semidirect
37. #552C22semidirect
38. #551C22semidirect
39. #551C22semidirect
40. #550C22semidirect
41. #550C22semidirect
42. #550C22semidirect
43. #550C22semidirect
44. #550C22semidirect
45. D4×Q8C22semidirect
46. #548C22semidirect
47. #548C22semidirect
48. #548C22semidirect
49. #547C22semidirect
50. #547C22semidirect
51. #547C22semidirect
52. #547C22semidirect
53. #546C22semidirect
54. #546C22semidirect
55. #546C22semidirect
56. #546C22semidirect
57. #546C22semidirect
58. #546C22semidirect
59. #546C22semidirect
61. D42C22semidirect
62. #542C22semidirect
63. #542C22semidirect
64. #542C22semidirect
65. #541C22semidirect
66. #540C22semidirect
67. #540C22semidirect
68. #540C22semidirect
69. #540C22semidirect
70. #539C22semidirect
71. #538C22semidirect
72. #538C22semidirect
73. #537C22semidirect
75. #536C22semidirect
76. #536C22semidirect
77. #536C22semidirect
79. #535C22semidirect
80. #535C22semidirect
81. #535C22semidirect
82. #535C22semidirect
83. #535C22semidirect
84. #534C22semidirect
85. #533C22semidirect
86. #533C22semidirect
87. #533C22semidirect
88. #533C22semidirect
89. #533C22semidirect
92. #532C22semidirect
93. #532C22semidirect
94. #532C22semidirect
95. #532C22semidirect
96. #532C22semidirect
97. #532C22semidirect
98. #532C22semidirect
99. #532C22semidirect
100. #532C22semidirect
101. #532C22semidirect
102. C2×C4⋊Q8C22semidirect
103. C2×C4⋊Q8C22semidirect
105. C2×#127C22semidirect
106. #529C22semidirect
107. #529C22semidirect
108. #529C22semidirect
109. #529C22semidirect
110. #529C22semidirect
111. #529C22semidirect
112. #529C22semidirect
113. #529C22semidirect
114. C2×#124C22semidirect
115. C2×#124C22semidirect
117. #525C22semidirect
118. #525C22semidirect
119. #525C22semidirect
120. #525C22semidirect
121. #525C22semidirect
122. C2×#123C22semidirect
123. C2×#121C22semidirect
124. #520C22semidirect
125. #519C22semidirect
126. #518C22semidirect
127. #518C22semidirect
129. C4×#41C22semidirect
130. C4×#41C22semidirect
131. C4×#41C22semidirect
132. C4×#41C22semidirect
133. C2×C4×D4C22semidirect
134. C22×C42C22semidirect

Maximal subgroups

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

subgroupindex
#35622
#35542
#35462
#35422
#35172
#35142
#35132
#35112
#35102
#35092
#35082
#35002
#34992
#34982
#34872
#34852
#34442
C2×#5322