Finite Group Databasebeta

← #62914

Subgroups — #62914

Order 256.

3869 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3548C214yesmaximal, complemented
2#3546C214yesmaximal, complemented
3#3542C218yesmaximal, complemented
4#3536C218yesmaximal, complemented
5#3536C214yesmaximal, complemented
6#3534C218yesmaximal, complemented
7#3520C214yesmaximal, complemented
8#3511C212yesmaximal, complemented
9#3496C211yesmaximal, characteristic, complemented
10#3496C214yesmaximal, complemented
11#3492C212yesmaximal, complemented
12#3488C214yesmaximal, complemented
13#3488C214yesmaximal, complemented
14D4×#41C212yesmaximal, complemented
15C2×#547C211yesmaximal, characteristic, complemented
16#3460C211yesmaximal, characteristic, complemented
17C2×#537C212yesmaximal, complemented
18C2×H(5,𝔽2)C2212yescomplemented
19C2×H(5,𝔽2)C2214yescomplemented
20C22×#41C2214yescomplemented
21C22×#41C2212yescomplemented
22#562C22116yescomplemented
23#562C22116yescomplemented
24#560C22116yescomplemented
25#556C2214yescomplemented
26#555C2214yescomplemented
27#554C2218yescomplemented
28#554C2214yescomplemented
29#554C2218yescomplemented
30#553C2218yescomplemented
31#553C2218yescomplemented
32#552C2218yescomplemented
33#551C22116yescomplemented
34#550C2212yescomplemented
35#550C2218yescomplemented
36#550C2218yescomplemented
37D4×Q8C2212yescomplemented
38#548C2214yescomplemented
39#548C2218yescomplemented
40#548C2218yescomplemented
41#547C2214yescomplemented
42#547C2218yescomplemented
43#547C2214yescomplemented
44#547C2214yescomplemented
45#547C2212yescomplemented
46#547C2218yescomplemented
47#547C2214yescomplemented
48#547C2214yescomplemented
49#546C2218yescomplemented
50#546C2214yescomplemented
51#546C2218yescomplemented
52#546C22116yescomplemented
53#546C22116yescomplemented
54#546C2218yescomplemented
55#546C22116yescomplemented
56#546C2218yescomplemented
57#546C2218yescomplemented
58D42C2214yescomplemented
59D42C2212yescomplemented
60D42C22116yescomplemented
61D42C2218yescomplemented
62D42C2214yescomplemented
63#544C2211yescharacteristic, complemented
64#543C2218yescomplemented
65#542C2214yescomplemented
66#542C2218yescomplemented
67#542C2218yescomplemented
68#540C2214yescomplemented
69#540C22116yescomplemented
70#539C2218yescomplemented
71#539C22116yescomplemented
72#538C2214yescomplemented
73#538C22116yescomplemented
74#538C2218yescomplemented
75#537C2211yescharacteristic, complemented
76#537C2214yescomplemented
77#537C2212yescomplemented
78#537C2218yescomplemented
79#537C2218yescomplemented
80#537C2218yescomplemented
81#537C2218yescomplemented
82#537C2218yescomplemented
83#537C2214yescomplemented
84#537C2218yescomplemented
85#535C2212yescomplemented
86#535C2218yescomplemented
87#535C2218yescomplemented
88#535C2214yescomplemented
89#534C2214yescomplemented
90#534C22116yescomplemented
91#533C2214yescomplemented
92#532C2214yescomplemented
93#532C2212yescomplemented
94#532C2214yescomplemented
95#532C2214yescomplemented
96C2×C4⋊Q8C2211yescharacteristic, complemented
97#529C2214yescomplemented
98#525C2214yescomplemented
99#525C2214yescomplemented
100#525C2218yescomplemented
101#525C2218yescomplemented
102#525C2218yescomplemented
103C2×#123C2214yescomplemented
104C2×C22⋊Q8C2218yescomplemented
105C2×C22⋊Q8C2212yescomplemented
106C2×#121C2218yescomplemented
107C2×#121C2218yescomplemented
108C2×#121C2212yescomplemented
109C2×#121C2214yescomplemented
110C2×#121C2218yescomplemented
111#520C2214yescomplemented
112#520C2214yescomplemented
113C4×#41C2212yescomplemented
114C2×C4×D4C2211yescharacteristic, complemented
115C2×C4×D4C2211yescharacteristic, complemented
116C2×C4×D4C2214yescomplemented
117C22×C4⋊C4C2212yescomplemented
118H(5,𝔽2)—416no—
119H(5,𝔽2)—48no—
120C2×#41—416no—
121C2×#41—48no—
122C2×#41C2312yes—
123C2×#41C2314yes—
124C2×#41C2314yes—
125C2×#41C2314yes—
126C2×#41C2318yes—
127C2×#41C2318yes—
128C2×#41C2318yes—
129C2×#41C2318yes—
130C2×#41C2318yes—
131C2×#41C2314yes—
132C2×#41C2318yes—
133C2×#41C2314yes—
134C2×#41C2314yes—
135C22×Q8C2314yes—
136C22×Q8C2312yes—
137C22×D4C2311yescharacteristic
138C22×D4C2311yescharacteristic
139C22×D4C2314yes—
140C22×D4C2318yes—
141C22×D4C2314yes—
142C22×D4C2318yes—
143C22×D4C2314yes—
144C22×D4C23116yes—
145C22×D4C2318yes—
146C22×D4C2318yes—
147C22×D4C2314yes—
148C23×C4C2312yes—
149C23×C4C2314yes—
150C23×C4C2312yes—
151C23×C4C2318yes—
152C4⋊Q8C2314yes—
153C4⋊Q8C2314yes—
154C4⋊Q8C2312yes—
155C4⋊Q8C2314yes—
156C4⋊Q8C2314yes—
157#127C2312yes—
158#127C2314yes—
159#127C2314yes—
160#127C23116yes—
161#126C2318yes—
162#126C23116yes—
163#126C23116yes—
164#125C2314yes—
165#125C2318yes—
166#125C23116yes—
167#124C2314yes—
168#124C2318yes—
169#124C2318yes—
170#124C2318yes—
171#124C2314yes—
172#124C23116yes—
173#124C23116yes—
174#123C23116yes—
175#123C23116yes—
176#123C23116yes—
177#123C2318yes—
178#123C2318yes—
179#123C23116yes—
180#123C2318yes—
181#123C2318yes—
182#123C2318yes—
183#123C23116yes—
184#123C23116yes—
185#123C23116yes—
186C22⋊Q8C2314yes—
187C22⋊Q8C2318yes—
188C22⋊Q8C23116yes—
189C22⋊Q8C2318yes—
190C22⋊Q8C2318yes—
191C22⋊Q8C23116yes—
192C22⋊Q8C2318yes—
193C22⋊Q8C2318yes—
194C22⋊Q8C2314yes—
195C22⋊Q8C2314yes—
196C22⋊Q8C2314yes—
197C22⋊Q8C2314yes—
198C22⋊Q8C2318yes—
199C22⋊Q8C2318yes—
200C22⋊Q8C2318yes—
201C22⋊Q8C2318yes—
202C22⋊Q8C23116yes—
203C22⋊Q8C23116yes—
204C22⋊Q8C23116yes—
205#121C2318yes—
206#121C2314yes—
207#121C2318yes—
208#121C2314yes—
209#121C2318yes—
210#121C2314yes—
211#121C2314yes—
212#121C2314yes—
213#121C23116yes—
214#121C23116yes—
215#121C2318yes—
216#121C2318yes—
217#121C2318yes—
218#121C2318yes—
219#121C2318yes—
220#121C23116yes—
221#121C2318yes—
222#121C23116yes—
223#121C23116yes—
224#121C23116yes—
225#121C2318yes—
226#121C2314yes—
227#121C2314yes—
228#121C2314yes—
229#121C2318yes—
230#121C2318yes—
231#121C2318yes—
232#121C2318yes—
233#121C2314yes—
234#121C2318yes—
235#121C2314yes—
236#121C2314yes—
237#121C2318yes—
238#121C23116yes—
239#121C23116yes—
240#121C23116yes—
241#120C2318yes—
242#120C2318yes—
243#120C23116yes—
244#120C2318yes—
245#120C2318yes—
246#120C23116yes—
247#120C23116yes—
248#120C23116yes—
249C4×Q8C2312yes—
250C4×Q8C2312yes—
251C4×Q8C2318yes—
252C4×Q8C2318yes—
253C4×D4C2312yes—
254C4×D4C2314yes—
255C4×D4C2318yes—
256C4×D4C2314yes—
257C4×D4C2314yes—
258C4×D4C2314yes—
259C4×D4C2312yes—
260C4×D4C2318yes—
261C4×D4C2314yes—
262C4×D4C2314yes—
263C4×D4C2314yes—
264C4×D4C2312yes—
265C4×D4C2314yes—
266C4×D4C2312yes—
267C4×D4C23116yes—
268C4×D4C23116yes—
269C4×D4C2318yes—
270C4×D4C2314yes—
271C4×D4C2318yes—
272C4×D4C2318yes—
273C4×D4C23116yes—
274C4×D4C23116yes—
275C4×D4C2318yes—
276#117C2312yes—
277#117C2314yes—
278#117C2314yes—
279#117C2318yes—
280#117C2318yes—
281#117C2318yes—
282C2×C4⋊C4C2311yescharacteristic
283C2×C4⋊C4C2312yes—
284C2×C4⋊C4C2318yes—
285C2×C4⋊C4C2314yes—
286C2×C4⋊C4C2318yes—
287C2×C4⋊C4C2311yescharacteristic
288C2×C4⋊C4C2314yes—
289C2×C4⋊C4C2318yes—
290C2×C4⋊C4C2318yes—
291C2×C4⋊C4C2314yes—
292C2×C4⋊C4C2312yes—
293C2×C22⋊C4C2312yes—
294C2×C22⋊C4C2312yes—
295C2×C22⋊C4C2318yes—
296C2×C22⋊C4C23116yes—
297C2×C22⋊C4C2314yes—
298C2×C22⋊C4C2318yes—
299C2×C22⋊C4C23116yes—
300C2×C42C2311yescharacteristic
301C2×C42C2314yes—
302C24C2412yes—
303C24C2412yes—
304C24C2418yes—
305C24C2418yes—
306#41—216no—
307#41—28no—
308#41—216no—
309#41—416no—
310#41—48no—
311#41—416no—
312#41—416no—
313#41—416no—
314#41—48no—
315#41—48no—
316#41—48no—
317#41—416no—
318#41—416no—
319C2×Q8—48no—
320C2×Q8—44no—
321C2×Q8C2412yes—
322C2×Q8C2414yes—
323C2×Q8C2414yes—
324C2×Q8C2418yes—
325C2×Q8C2414yes—
326C2×Q8C2414yes—
327C2×D4—24no—
328C2×D4—216no—
329C2×D4—24no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—48no—
335C2×D4—416no—
336C2×D4—432no—
337C2×D4—416no—
338C2×D4C2412yes—
339C2×D4C2414yes—
340C2×D4C2418yes—
341C2×D4C2418yes—
342C2×D4C2418yes—
343C2×D4C2414yes—
344C2×D4C2414yes—
345C2×D4C2412yes—
346C2×D4C2414yes—
347C2×D4C2411yescharacteristic
348C2×D4C2414yes—
349C2×D4C2418yes—
350C2×D4C2418yes—
351C2×D4C2418yes—
352C2×D4C2418yes—
353C2×D4C2418yes—
354C2×D4C24116yes—
355C2×D4C2418yes—
356C2×D4C2418yes—
357C2×D4C24116yes—
358C2×D4C2418yes—
359C2×D4C2414yes—
360C2×D4C2411yescharacteristic
361C22×C4—28no—
362C22×C4—416no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4C2411yescharacteristic
366C22×C4C2414yes—
367C22×C4C2414yes—
368C22×C4C2412yes—
369C22×C4C2418yes—
370C22×C4C2414yes—
371C22×C4C2414yes—
372C22×C4C2418yes—
373C22×C4C2414yes—
374C22×C4C2411yescharacteristic
375C22×C4C2414yes—
376C22×C4C2418yes—
377C22×C4C2414yes—
378C22×C4C2418yes—
379C22×C4C2414yes—
380C22×C4C2412yes—
381C22×C4C2418yes—
382C22×C4C24116yes—
383C22×C4C2418yes—
384C22×C4C2418yes—
385C22×C4C2411yescharacteristic
386C4⋊C4C2414yes—
387C4⋊C4C2418yes—
388C4⋊C4C2412yes—
389C4⋊C4C2412yes—
390C4⋊C4C2414yes—
391C4⋊C4C24116yes—
392C4⋊C4C2414yes—
393C4⋊C4C2418yes—
394C4⋊C4C2418yes—
395C4⋊C4C2418yes—
396C4⋊C4C2412yes—
397C4⋊C4C24116yes—
398C4⋊C4C24116yes—
399C4⋊C4C2412yes—
400C4⋊C4C2414yes—
401C4⋊C4C2414yes—
402C4⋊C4C2414yes—
403C4⋊C4C2414yes—
404C4⋊C4C2418yes—
405C4⋊C4C2418yes—
406C22⋊C4C2414yes—
407C22⋊C4C24116yes—
408C22⋊C4C24116yes—
409C22⋊C4C2418yes—
410C22⋊C4C24116yes—
411C22⋊C4C2418yes—
412C22⋊C4C2418yes—
413C22⋊C4C2418yes—
414C22⋊C4C2418yes—
415C22⋊C4C24116yes—
416C22⋊C4C2418yes—
417C22⋊C4C2418yes—
418C22⋊C4C2418yes—
419C22⋊C4C2414yes—
420C22⋊C4C2414yes—
421C22⋊C4C24116yes—
422C22⋊C4C2418yes—
423C22⋊C4C2414yes—
424C22⋊C4C2414yes—
425C22⋊C4C2418yes—
426C42C2412yes—
427C42C2412yes—
428C42C2414yes—
429C42C2414yes—
430C42C2414yes—
431C42C24116yes—
432C23—24no—
433C23—24no—
434C23—216no—
435C23—24no—
436C23—42no—
437C23—48no—
438C23—42no—
439C23—42no—
440C23—48no—
441C23—48no—
442C23—48no—
443C23—48no—
444C23C2518yes—
445C23C2514yes—
446C23C2514yes—
447C23C2512yes—
448C23C2512yes—
449C23C2511yescharacteristic
450Q8—24no—
451Q8—28no—
452Q8—28no—
453Q8—48no—
454Q8—44no—
455Q8—44no—
456D4—216no—
457D4—24no—
458D4—216no—
459D4—216no—
460D4—28no—
461D4—28no—
462D4—28no—
463D4—22no—
464D4—28no—
465D4—24no—
466D4—44no—
467D4—48no—
468D4—48no—
469D4—416no—
470D4—416no—
471D4—48no—
472D4—48no—
473D4—48no—
474D4—48no—
475D4—44no—
476D4—48no—
477D4—48no—
478D4H(5,𝔽2)14yescomplemented
479C2×C4—24no—
480C2×C4—216no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—216no—
487C2×C4—28no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—416no—
498C2×C4—48no—
499C2×C4H(5,𝔽2)14yescomplemented
500C2×C4C2514yes—
501C2×C4C2512yes—
502C2×C4C2511yescharacteristic
503C2×C4C2514yes—
504C2×C4C2514yes—
505C2×C4C2518yes—
506C2×C4C2518yes—
507C2×C4C2514yes—
508C2×C4C2514yes—
509C2×C4C2511yescharacteristic
510C2×C4C2512yes—
511C22—22no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—24no—
517C22—24no—
518C22—28no—
519C22—24no—
520C22—22no—
521C22—28no—
522C22—24no—
523C22—24no—
524C22—22no—
525C22—28no—
526C22—24no—
527C22—48no—
528C22—84no—
529C22—816no—
530C22—168no—
531C22C2×H(5,𝔽2)14yes—
532C22C2×H(5,𝔽2)12yes—
533C22C2611yescharacteristic
534C4—24no—
535C4—24no—
536C4—24no—
537C4—21no—
538C4—28no—
539C4—28no—
540C4—24no—
541C4—22no—
542C4—24no—
543C4C2×H(5,𝔽2)14yes—
544C4C2×H(5,𝔽2)12yes—
545C2—22no—
546C2—24no—
547C2—42no—
548C2—44no—
549C2—44no—
550C2—48no—
551C2C22×H(5,𝔽2)11yescharacteristic
552C2C22×H(5,𝔽2)11yescharacteristic
553C22−1+611yescharacteristic
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. #3548C2semidirect
2. #3546C2semidirect
3. #3542C2semidirect
4. #3536C2semidirect
5. #3536C2semidirect
6. #3534C2semidirect
7. #3520C2semidirect
8. #3511C2semidirect
9. #3496C2semidirect
10. #3496C2semidirect
11. #3492C2semidirect
12. #3488C2semidirect
13. #3488C2semidirect
14. D4×#41C2semidirect
15. C2×#547C2semidirect
16. #3460C2semidirect
17. C2×#537C2semidirect
18. C2×H(5,𝔽2)C22semidirect
19. C2×H(5,𝔽2)C22semidirect
20. C22×#41C22semidirect
21. C22×#41C22semidirect
22. #562C22semidirect
23. #562C22semidirect
24. #560C22semidirect
25. #556C22semidirect
26. #555C22semidirect
27. #554C22semidirect
28. #554C22semidirect
29. #554C22semidirect
30. #553C22semidirect
31. #553C22semidirect
32. #552C22semidirect
33. #551C22semidirect
34. #550C22semidirect
35. #550C22semidirect
36. #550C22semidirect
37. D4×Q8C22semidirect
38. #548C22semidirect
39. #548C22semidirect
40. #548C22semidirect
41. #547C22semidirect
42. #547C22semidirect
43. #547C22semidirect
44. #547C22semidirect
45. #547C22semidirect
46. #547C22semidirect
47. #547C22semidirect
48. #547C22semidirect
49. #546C22semidirect
50. #546C22semidirect
51. #546C22semidirect
52. #546C22semidirect
53. #546C22semidirect
54. #546C22semidirect
55. #546C22semidirect
56. #546C22semidirect
57. #546C22semidirect
58. D42C22semidirect
59. D42C22semidirect
60. D42C22semidirect
61. D42C22semidirect
62. D42C22semidirect
63. #544C22semidirect
64. #543C22semidirect
65. #542C22semidirect
66. #542C22semidirect
67. #542C22semidirect
68. #540C22semidirect
69. #540C22semidirect
70. #539C22semidirect
71. #539C22semidirect
72. #538C22semidirect
73. #538C22semidirect
74. #538C22semidirect
75. #537C22semidirect
76. #537C22semidirect
77. #537C22semidirect
78. #537C22semidirect
79. #537C22semidirect
80. #537C22semidirect
81. #537C22semidirect
82. #537C22semidirect
83. #537C22semidirect
84. #537C22semidirect
85. #535C22semidirect
86. #535C22semidirect
87. #535C22semidirect
88. #535C22semidirect
89. #534C22semidirect
90. #534C22semidirect
91. #533C22semidirect
92. #532C22semidirect
93. #532C22semidirect
94. #532C22semidirect
95. #532C22semidirect
96. C2×C4⋊Q8C22semidirect
97. #529C22semidirect
98. #525C22semidirect
99. #525C22semidirect
100. #525C22semidirect
101. #525C22semidirect
102. #525C22semidirect
103. C2×#123C22semidirect
104. C2×C22⋊Q8C22semidirect
105. C2×C22⋊Q8C22semidirect
106. C2×#121C22semidirect
107. C2×#121C22semidirect
108. C2×#121C22semidirect
109. C2×#121C22semidirect
110. C2×#121C22semidirect
111. #520C22semidirect
112. #520C22semidirect
113. C4×#41C22semidirect
114. C2×C4×D4C22semidirect
115. C2×C4×D4C22semidirect
116. C2×C4×D4C22semidirect
117. C22×C4⋊C4C22semidirect
478. D4H(5,𝔽2)semidirect
499. C2×C4H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#35482
#35462
#35422
#35362
#35342
#35202
#35112
#34962
#34922
#34882
D4×#412
C2×#5472
#34602
C2×#5372