Finite Group Databasebeta

← #62781

Subgroups — #62781

Order 256.

3615 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3521C218yesmaximal, complemented
2#3515C214yesmaximal, complemented
3#3514C214yesmaximal, complemented
4#3507C212yesmaximal, complemented
5#3505C214yesmaximal, complemented
6#3501C211yesmaximal, characteristic, complemented
7#3496C211yesmaximal, characteristic, complemented
8#3493C212yesmaximal, complemented
9#3491C214yesmaximal, complemented
10#3485C212yesmaximal, complemented
11#3482C214yesmaximal, complemented
12#3481C214yesmaximal, complemented
13#3475C218yesmaximal, complemented
14#3461C212yesmaximal, complemented
15#3459C214yesmaximal, complemented
16#3444C212yesmaximal, complemented
17C2×#529C214yesmaximal, complemented
18C4×H(5,𝔽2)C212yesmaximal, complemented
19C2×C4×#41C211yesmaximal, characteristic, complemented
20C2×H(5,𝔽2)C2212yes—
21C22×#41C2211yescharacteristic
22#559C2214yescomplemented
23#557C2214yescomplemented
24#556C2218yescomplemented
25#556C2218yescomplemented
26#556C2214yescomplemented
27#555C2218yescomplemented
28#554C2218yescomplemented
29#554C2214yescomplemented
30#554C2214yescomplemented
31#553C22116yescomplemented
32#553C2218yescomplemented
33#553C2218yescomplemented
34#552C22116yescomplemented
35#552C2218yescomplemented
36#552C2218yescomplemented
37#552C22116yescomplemented
38#552C2218yescomplemented
39#551C22116yescomplemented
40#551C22116yescomplemented
41#551C22116yescomplemented
42D4×Q8C2214yescomplemented
43#548C2218yescomplemented
44#547C2218yescomplemented
45#547C2218yescomplemented
46#547C2214yescomplemented
47#546C2218yescomplemented
48#546C2218yescomplemented
49#546C2218yescomplemented
50#546C2218yescomplemented
51D42C2214yescomplemented
52#544C2212yescomplemented
53#543C2214yescomplemented
54#542C2218yescomplemented
55#541C2218yescomplemented
56#539C2218yescomplemented
57#539C22116yescomplemented
58#539C2218yescomplemented
59#538C2218yescomplemented
60#538C22116yescomplemented
61#537C2212yes—
62#536C2214yescomplemented
63#534C2214yes—
64#534C2218yescomplemented
65#533C2212yescomplemented
66#533C2212yescomplemented
67#533C2212yescomplemented
68#533C2218yescomplemented
69#532C2212yes—
70#532C2212yescomplemented
71#532C2212yescomplemented
72#529C2214yescomplemented
73#529C2218yescomplemented
74#529C2218yescomplemented
75#529C2218yescomplemented
76#529C2218yescomplemented
77#529C2218yescomplemented
78#529C2218yescomplemented
79#529C2218yescomplemented
80#529C2214yescomplemented
81#529C2214yescomplemented
82#529C2214yescomplemented
83#529C2214yescomplemented
84#529C2214yescomplemented
85#529C22116yescomplemented
86#529C2218yescomplemented
87C2×#126C2218yescomplemented
88C2×#125C2214yescomplemented
89C2×#124C2214yescomplemented
90#525C2214yescomplemented
91#525C2214yescomplemented
92#525C2214yescomplemented
93#525C22116yescomplemented
94C2×#123C2218yescomplemented
95C2×C22⋊Q8C2214yescomplemented
96C2×#121C2214yes—
97#520C2214yescomplemented
98#520C2218yescomplemented
99#518C2212yes—
100#518C2218yescomplemented
101#518C2218yescomplemented
102C4×#41C2214yescomplemented
103C4×#41C2214yescomplemented
104C4×#41C2214yescomplemented
105C4×#41C2214yescomplemented
106C4×#41C2214yescomplemented
107C4×#41C2218yescomplemented
108C4×#41C2214yescomplemented
109C4×#41C2214yescomplemented
110C2×C4×Q8C2211yescharacteristic, complemented
111C2×C4×Q8C2211yescharacteristic, complemented
112C2×C4×Q8C2214yescomplemented
113C2×C4×D4C2211yescharacteristic
114C2×C4×D4C2211yescharacteristic
115C2×C4×D4C2212yescomplemented
116C2×C4×D4C2212yescomplemented
117C2×C4×D4C2218yescomplemented
118C2×C4×D4C2218yescomplemented
119C2×C4×D4C2214yescomplemented
120C2×#117C2212yescomplemented
121C2×#117C2211yescharacteristic, complemented
122C2×#117C2218yescomplemented
123C22×C42C2212yescomplemented
124C22×C42C2211yescharacteristic, complemented
125H(5,𝔽2)—216no—
126C2×#41—28no—
127C2×#41C2314yes—
128C2×#41C2314yes—
129C2×#41C2314yes—
130C2×#41C2318yes—
131C2×#41C2314yes—
132C2×#41C2314yes—
133C22×Q8C2311yescharacteristic
134C22×Q8C2314yes—
135C22×D4C2312yes—
136C22×D4C2318yes—
137C22×D4C2318yes—
138C22×D4C2314yes—
139C22×D4C2311yescharacteristic
140C23×C4C2312yes—
141C23×C4C2311yescharacteristic
142C23×C4C2318yes—
143C23×C4C2312yes—
144C23×C4C2312yes—
145C23×C4C2311yescharacteristic
146C23×C4C2311yescharacteristic
147C4⋊Q8C2312yes—
148C4⋊Q8C2312yes—
149C4⋊Q8C2312yes—
150C4⋊Q8C2318yes—
151C4⋊Q8C2318yes—
152#127C2312yes—
153#126C2318yes—
154#126C23116yes—
155#126C23116yes—
156#126C23116yes—
157#126C2318yes—
158#126C23116yes—
159#126C23116yes—
160#126C23116yes—
161#125C2314yes—
162#125C2318yes—
163#125C2318yes—
164#125C2318yes—
165#125C2314yes—
166#125C2314yes—
167#125C23116yes—
168#125C2318yes—
169#125C23116yes—
170#124C2314yes—
171#124C2318yes—
172#124C2318yes—
173#124C2318yes—
174#124C2314yes—
175#124C2314yes—
176#124C2318yes—
177#124C23116yes—
178#123C2318yes—
179#123C23116yes—
180#123C23116yes—
181#123C23116yes—
182#123C2318yes—
183#123C2318yes—
184#123C23116yes—
185#123C23116yes—
186#123C23116yes—
187#123C23116yes—
188#123C23116yes—
189C22⋊Q8C2314yes—
190C22⋊Q8C2318yes—
191C22⋊Q8C2318yes—
192C22⋊Q8C2318yes—
193C22⋊Q8C2314yes—
194C22⋊Q8C2314yes—
195C22⋊Q8C2314yes—
196C22⋊Q8C2314yes—
197C22⋊Q8C2314yes—
198C22⋊Q8C23116yes—
199C22⋊Q8C23116yes—
200C22⋊Q8C2318yes—
201C22⋊Q8C23116yes—
202C22⋊Q8C2318yes—
203C22⋊Q8C2318yes—
204C22⋊Q8C23116yes—
205#121C2314yes—
206#121C2318yes—
207#121C2318yes—
208#121C2318yes—
209#121C2314yes—
210#121C2314yes—
211#121C2314yes—
212#121C2314yes—
213#121C2314yes—
214#121C23116yes—
215#121C23116yes—
216#121C23116yes—
217#121C2318yes—
218#120C2318yes—
219#120C23116yes—
220#120C23116yes—
221C4×Q8C2314yes—
222C4×Q8C2314yes—
223C4×Q8C2314yes—
224C4×Q8C2314yes—
225C4×Q8C2314yes—
226C4×Q8C2318yes—
227C4×Q8C2318yes—
228C4×Q8C2318yes—
229C4×Q8C2314yes—
230C4×Q8C2314yes—
231C4×Q8C2318yes—
232C4×Q8C2314yes—
233C4×Q8C2314yes—
234C4×Q8C2314yes—
235C4×D4C2314yes—
236C4×D4C2314yes—
237C4×D4C2314yes—
238C4×D4C2314yes—
239C4×D4C2318yes—
240C4×D4C23116yes—
241C4×D4C2314yes—
242C4×D4C2314yes—
243C4×D4C2318yes—
244C4×D4C2318yes—
245C4×D4C2318yes—
246C4×D4C2318yes—
247C4×D4C2318yes—
248C4×D4C23116yes—
249C4×D4C23116yes—
250C4×D4C23116yes—
251C4×D4C2318yes—
252C4×D4C2318yes—
253C4×D4C2318yes—
254C4×D4C2318yes—
255C4×D4C2318yes—
256C4×D4C2318yes—
257C4×D4C2314yes—
258C4×D4C23116yes—
259C4×D4C2318yes—
260C4×D4C2318yes—
261C4×D4C2318yes—
262C4×D4C2318yes—
263C4×D4C2314yes—
264C4×D4C2314yes—
265C4×D4C2314yes—
266#117C2318yes—
267#117C2318yes—
268#117C2318yes—
269#117C2314yes—
270#117C2314yes—
271#117C23116yes—
272#117C23116yes—
273#117C23116yes—
274#117C2314yes—
275#117C2318yes—
276#117C23116yes—
277#117C2318yes—
278#117C2314yes—
279#117C2314yes—
280#117C2318yes—
281C2×C4⋊C4C2311yescharacteristic
282C2×C4⋊C4C2311yescharacteristic
283C2×C4⋊C4C2312yes—
284C2×C4⋊C4C2312yes—
285C2×C4⋊C4C2318yes—
286C2×C4⋊C4C2318yes—
287C2×C4⋊C4C2318yes—
288C2×C4⋊C4C2318yes—
289C2×C4⋊C4C2314yes—
290C2×C4⋊C4C2314yes—
291C2×C4⋊C4C2318yes—
292C2×C22⋊C4C2312yes—
293C2×C22⋊C4C2312yes—
294C2×C22⋊C4C2312yes—
295C2×C22⋊C4C2318yes—
296C2×C22⋊C4C2318yes—
297C2×C22⋊C4C2318yes—
298C2×C22⋊C4C2318yes—
299C2×C22⋊C4C23116yes—
300C2×C22⋊C4C2318yes—
301C2×C42C2314yes—
302C2×C42C2311yescharacteristic
303C2×C42C2312yes—
304C2×C42C2318yes—
305C2×C42C2311yescharacteristic
306C2×C42C2311yescharacteristic
307C2×C42C2314yes—
308C2×C42C2312yes—
309C2×C42C2312yes—
310C2×C42C2318yes—
311C2×C42C2312yes—
312C2×C42C2312yes—
313C2×C42C2314yes—
314C2×C42C2314yes—
315C2×C42C2312yes—
316C2×C42C2312yes—
317C2×C42C2311yescharacteristic
318C24C2418yes—
319C24C2411yescharacteristic
320C24C2412yes—
321#41—216no—
322#41—216no—
323#41—232no—
324#41—216no—
325#41—48no—
326#41—48no—
327C2×Q8—24no—
328C2×Q8—48no—
329C2×Q8C2414yes—
330C2×Q8C2414yes—
331C2×Q8C2414yes—
332C2×Q8C2414yes—
333C2×Q8C2418yes—
334C2×D4—28no—
335C2×D4—232no—
336C2×D4—232no—
337C2×D4—48no—
338C2×D4#4118yescomplemented
339C2×D4C2418yes—
340C2×D4C2418yes—
341C2×D4C24116yes—
342C2×D4C2414yes—
343C2×D4C2414yes—
344C2×D4C2414yes—
345C2×D4C2418yes—
346C2×D4C2418yes—
347C2×D4C2418yes—
348C2×D4C2414yes—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—44no—
352C22×C4—416no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4C2411yescharacteristic
357C22×C4C2411yescharacteristic
358C22×C4C2414yes—
359C22×C4C2414yes—
360C22×C4C2414yes—
361C22×C4C2412yes—
362C22×C4C2412yes—
363C22×C4C2412yes—
364C22×C4C2412yes—
365C22×C4C2418yes—
366C22×C4C2418yes—
367C22×C4C2414yes—
368C22×C4C2411yescharacteristic
369C22×C4C2412yes—
370C22×C4C2418yes—
371C22×C4C2411yescharacteristic
372C22×C4C2412yes—
373C22×C4C2414yes—
374C22×C4C2414yes—
375C22×C4C2411yescharacteristic
376C22×C4C2412yes—
377C22×C4C2412yes—
378C22×C4C2412yes—
379C22×C4C2418yes—
380C22×C4C2418yes—
381C22×C4C2418yes—
382C22×C4C24116yes—
383C22×C4C2411yescharacteristic
384C4⋊C4C2414yes—
385C4⋊C4C2418yes—
386C4⋊C4C2418yes—
387C4⋊C4C2414yes—
388C4⋊C4C2414yes—
389C4⋊C4C2418yes—
390C4⋊C4C24116yes—
391C4⋊C4C2414yes—
392C4⋊C4C2418yes—
393C4⋊C4C2414yes—
394C4⋊C4C2418yes—
395C4⋊C4C2418yes—
396C4⋊C4C24116yes—
397C4⋊C4C24116yes—
398C4⋊C4C2418yes—
399C4⋊C4C2418yes—
400C4⋊C4C24116yes—
401C4⋊C4C2414yes—
402C4⋊C4C2418yes—
403C4⋊C4C2414yes—
404C4⋊C4C2414yes—
405C4⋊C4C2418yes—
406C22⋊C4C2418yes—
407C22⋊C4C2418yes—
408C22⋊C4C2418yes—
409C22⋊C4C2418yes—
410C22⋊C4C24116yes—
411C22⋊C4C2418yes—
412C22⋊C4C2418yes—
413C22⋊C4C2418yes—
414C22⋊C4C24116yes—
415C22⋊C4C2418yes—
416C22⋊C4C2418yes—
417C22⋊C4C24116yes—
418C22⋊C4C24116yes—
419C22⋊C4C24116yes—
420C22⋊C4C2418yes—
421C22⋊C4C2418yes—
422C22⋊C4C2418yes—
423C42C2411yescharacteristic
424C42C2411yescharacteristic
425C42C2414yes—
426C42C2412yes—
427C42C2412yes—
428C42C2414yes—
429C42C2414yes—
430C42C2412yes—
431C42C2412yes—
432C42C2414yes—
433C42C2412yes—
434C42C2414yes—
435C42C2418yes—
436C42C24116yes—
437C42C2418yes—
438C42C2411yescharacteristic
439C42C2412yes—
440C42C2411yescharacteristic
441C42C2418yes—
442C42C2414yes—
443C23—22no—
444C23—216no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—48no—
450C23—48no—
451C23C2×#4118yes—
452C23C2518yes—
453C23C2512yes—
454C23C2514yes—
455C23C2511yescharacteristic
456Q8—28no—
457Q8—28no—
458Q8—28no—
459Q8—48no—
460Q8—44no—
461D4—232no—
462D4—216no—
463D4—216no—
464D4—28no—
465D4—28no—
466D4—48no—
467D4—48no—
468D4—44no—
469D4—48no—
470D4C2×#41116yes—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—216no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—216no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—216no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—416no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4C2×#4114yes—
499C2×C4C2511yescharacteristic
500C2×C4C2511yescharacteristic
501C2×C4C2511yescharacteristic
502C2×C4C2511yescharacteristic
503C2×C4C2512yes—
504C2×C4C2514yes—
505C2×C4C2518yes—
506C2×C4C2514yes—
507C2×C4C2512yes—
508C2×C4C2514yes—
509C2×C4C2518yes—
510C2×C4C2518yes—
511C2×C4C2512yes—
512C2×C4C2511yescharacteristic
513C2×C4C2511yescharacteristic
514C22—21no—
515C22—21no—
516C22—24no—
517C22—24no—
518C22—22no—
519C22—28no—
520C22—28no—
521C22—22no—
522C22—22no—
523C22—28no—
524C22—48no—
525C22—44no—
526C22—816no—
527C22C22×#4118yes—
528C22C22×#4112yes—
529C22C2611yescharacteristic
530C4—21no—
531C4—21no—
532C4—28no—
533C4—24no—
534C4—21no—
535C4—24no—
536C4—21no—
537C4—22no—
538C4—28no—
539C4—24no—
540C4—22no—
541C4—28no—
542C4C22×#4114yes—
543C4#58512yes—
544C4#58512yes—
545C2—22no—
546C2—28no—
547C2—42no—
548C2—48no—
549C2C23×#4111yescharacteristic
550C2C2×#58511yescharacteristic
551C2C2×#58511yescharacteristic
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. #3521C2semidirect
2. #3515C2semidirect
3. #3514C2semidirect
4. #3507C2semidirect
5. #3505C2semidirect
6. #3501C2semidirect
7. #3496C2semidirect
8. #3493C2semidirect
9. #3491C2semidirect
10. #3485C2semidirect
11. #3482C2semidirect
12. #3481C2semidirect
13. #3475C2semidirect
14. #3461C2semidirect
15. #3459C2semidirect
16. #3444C2semidirect
17. C2×#529C2semidirect
18. C4×H(5,𝔽2)C2semidirect
19. C2×C4×#41C2semidirect
22. #559C22semidirect
23. #557C22semidirect
24. #556C22semidirect
25. #556C22semidirect
26. #556C22semidirect
27. #555C22semidirect
28. #554C22semidirect
29. #554C22semidirect
30. #554C22semidirect
31. #553C22semidirect
32. #553C22semidirect
33. #553C22semidirect
34. #552C22semidirect
35. #552C22semidirect
36. #552C22semidirect
37. #552C22semidirect
38. #552C22semidirect
39. #551C22semidirect
40. #551C22semidirect
41. #551C22semidirect
42. D4×Q8C22semidirect
43. #548C22semidirect
44. #547C22semidirect
45. #547C22semidirect
46. #547C22semidirect
47. #546C22semidirect
48. #546C22semidirect
49. #546C22semidirect
50. #546C22semidirect
51. D42C22semidirect
52. #544C22semidirect
53. #543C22semidirect
54. #542C22semidirect
55. #541C22semidirect
56. #539C22semidirect
57. #539C22semidirect
58. #539C22semidirect
59. #538C22semidirect
60. #538C22semidirect
62. #536C22semidirect
64. #534C22semidirect
65. #533C22semidirect
66. #533C22semidirect
67. #533C22semidirect
68. #533C22semidirect
70. #532C22semidirect
71. #532C22semidirect
72. #529C22semidirect
73. #529C22semidirect
74. #529C22semidirect
75. #529C22semidirect
76. #529C22semidirect
77. #529C22semidirect
78. #529C22semidirect
79. #529C22semidirect
80. #529C22semidirect
81. #529C22semidirect
82. #529C22semidirect
83. #529C22semidirect
84. #529C22semidirect
85. #529C22semidirect
86. #529C22semidirect
87. C2×#126C22semidirect
88. C2×#125C22semidirect
89. C2×#124C22semidirect
90. #525C22semidirect
91. #525C22semidirect
92. #525C22semidirect
93. #525C22semidirect
94. C2×#123C22semidirect
95. C2×C22⋊Q8C22semidirect
97. #520C22semidirect
98. #520C22semidirect
100. #518C22semidirect
101. #518C22semidirect
102. C4×#41C22semidirect
103. C4×#41C22semidirect
104. C4×#41C22semidirect
105. C4×#41C22semidirect
106. C4×#41C22semidirect
107. C4×#41C22semidirect
108. C4×#41C22semidirect
109. C4×#41C22semidirect
110. C2×C4×Q8C22semidirect
111. C2×C4×Q8C22semidirect
112. C2×C4×Q8C22semidirect
115. C2×C4×D4C22semidirect
116. C2×C4×D4C22semidirect
117. C2×C4×D4C22semidirect
118. C2×C4×D4C22semidirect
119. C2×C4×D4C22semidirect
120. C2×#117C22semidirect
121. C2×#117C22semidirect
122. C2×#117C22semidirect
123. C22×C42C22semidirect
124. C22×C42C22semidirect
338. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
#35212
#35152
#35142
#35072
#35052
#35012
#34962
#34932
#34912
#34852
#34822
#34812
#34752
#34612
#34592
#34442
C2×#5292
C4×H(5,𝔽2)2
C2×C4×#412