Finite Group Databasebeta

← #62897

Subgroups — #62897

Order 256.

3547 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3562C216yesmaximal, complemented
2#3561C213yesmaximal, complemented
3#3558C213yesmaximal, complemented
4#3554C216yesmaximal, complemented
5#3542C216yesmaximal, complemented
6#3541C216yesmaximal, complemented
7#3540C216yesmaximal, complemented
8#3524C212yesmaximal, complemented
9#3523C211yesmaximal, characteristic, complemented
10#3522C211yesmaximal, characteristic, complemented
11#3520C216yesmaximal, complemented
12#3518C212yesmaximal, complemented
13#3506C213yesmaximal, complemented
14#3489C212yesmaximal, complemented
15#3487C212yesmaximal, complemented
16#3486C212yesmaximal, complemented
17D4×#41C212yesmaximal, complemented
18C2×#548C211yesmaximal, characteristic, complemented
19#3459C213yesmaximal, complemented
20C2×2−1+4C2212yescomplemented
21C22×#41C2211yescharacteristic
22C22×#41C2212yescomplemented
23C23×Q8C2211yescharacteristic, complemented
24#563C2216yescomplemented
25#562C22112yescomplemented
26#562C2216yescomplemented
27#562C2216yescomplemented
28H(3,𝔽4)C2216yescomplemented
29#560C22112yescomplemented
30#559C22112yescomplemented
31#559C2213yescomplemented
32#559C2216yescomplemented
33Q82C2211yescharacteristic, complemented
34#557C2211yescharacteristic, complemented
35#557C2213yescomplemented
36#556C2216yescomplemented
37#556C2216yescomplemented
38#556C2216yescomplemented
39#556C2213yescomplemented
40#555C2216yescomplemented
41#555C2216yescomplemented
42#554C22112yescomplemented
43#554C2213yescomplemented
44#553C22112yescomplemented
45#553C2216yescomplemented
46#552C2216yescomplemented
47#552C2216yescomplemented
48#551C22112yescomplemented
49#551C22112yescomplemented
50#550C2212yescomplemented
51#550C2211yescharacteristic, complemented
52#550C2216yescomplemented
53#550C2216yescomplemented
54D4×Q8C2212yescomplemented
55D4×Q8C2212yescomplemented
56D4×Q8C2216yescomplemented
57D4×Q8C2216yescomplemented
58D4×Q8C2211yescharacteristic, complemented
59#548C2212yescomplemented
60#548C2212yescomplemented
61#548C2216yescomplemented
62#548C2216yescomplemented
63#548C2212yescomplemented
64#548C2212yescomplemented
65#548C2212yescomplemented
66#548C2216yescomplemented
67#548C2216yescomplemented
68#548C2216yescomplemented
69#548C2216yescomplemented
70#547C2216yescomplemented
71#547C2216yescomplemented
72#547C2216yescomplemented
73#547C2216yescomplemented
74#546C2216yescomplemented
75#546C2216yescomplemented
76#546C2216yescomplemented
77#546C2216yescomplemented
78#546C22112yescomplemented
79D42C2216yescomplemented
80#543C2213yescomplemented
81#542C2216yescomplemented
82#542C2216yescomplemented
83#542C22112yescomplemented
84#542C2216yescomplemented
85#542C2216yescomplemented
86#542C22112yescomplemented
87#541C2216yescomplemented
88#540C2216yescomplemented
89#540C22112yescomplemented
90#539C2216yescomplemented
91#539C2216yescomplemented
92#539C2216yescomplemented
93#539C22112yescomplemented
94#538C2211yescharacteristic
95#538C2213yescomplemented
96#538C2216yescomplemented
97#538C2216yescomplemented
98#538C2216yescomplemented
99#538C22112yescomplemented
100#538C22112yescomplemented
101#537C2216yescomplemented
102#536C2216yescomplemented
103#536C2216yescomplemented
104#536C2216yescomplemented
105#535C2216yescomplemented
106#535C2216yescomplemented
107#535C2216yescomplemented
108#534C2213yes—
109#534C2216yescomplemented
110#533C2213yescomplemented
111#533C2216yescomplemented
112#532C2216yescomplemented
113#532C2213yescomplemented
114#529C2213yes—
115#529C2216yescomplemented
116#529C2213yescomplemented
117#529C2216yescomplemented
118#529C2216yescomplemented
119C2×#124C2213yescomplemented
120#525C2216yescomplemented
121#525C2216yescomplemented
122#525C2216yescomplemented
123#525C2216yescomplemented
124#525C2216yescomplemented
125#525C2216yescomplemented
126#525C2216yescomplemented
127C2×C22⋊Q8C2213yescomplemented
128C2×C22⋊Q8C2216yescomplemented
129C2×#121C2213yescomplemented
130#520C2212yescomplemented
131#520C2216yescomplemented
132#519C2212yescomplemented
133#519C2212yescomplemented
134C4×#41C2212yescomplemented
135C2×C4×Q8C2211yescharacteristic, complemented
136C2×C4×D4C2213yescomplemented
137C2×C4×D4C2216yescomplemented
1382−1+4—48no—
139C2×#41—44no—
140C2×#41—48no—
141C2×#41C2312yes—
142C2×#41C2312yes—
143C2×#41C2316yes—
144C2×#41C2316yes—
145C2×#41C2316yes—
146C2×#41C2316yes—
147C2×#41C2316yes—
148C2×#41C2316yes—
149C2×#41C2316yes—
150C22×Q8—44no—
151C22×Q8C2311yescharacteristic
152C22×Q8C2312yes—
153C22×Q8C2312yes—
154C22×Q8C2313yes—
155C22×Q8C2316yes—
156C22×Q8C2316yes—
157C22×D4C2313yes—
158C22×D4C2316yes—
159C22×D4C2313yes—
160C22×D4C2316yes—
161C23×C4C2313yes—
162C23×C4C2313yes—
163C23×C4C2316yes—
164C4⋊Q8C2316yes—
165C4⋊Q8C2313yes—
166C4⋊Q8C2313yes—
167C4⋊Q8C2316yes—
168C4⋊Q8C2316yes—
169C4⋊Q8C2313yes—
170C4⋊Q8C23112yes—
171#127C2313yes—
172#127C2316yes—
173#127C23112yes—
174#126C2316yes—
175#126C2312yes—
176#126C2316yes—
177#126C2316yes—
178#126C2316yes—
179#126C2316yes—
180#126C23112yes—
181#126C23112yes—
182#126C23112yes—
183#126C23112yes—
184#125C2313yes—
185#125C2316yes—
186#125C2316yes—
187#125C2313yes—
188#125C2316yes—
189#125C2316yes—
190#125C23112yes—
191#124C2316yes—
192#124C2316yes—
193#124C2316yes—
194#124C2316yes—
195#124C2313yes—
196#124C2312yes—
197#124C2316yes—
198#124C2316yes—
199#124C2313yes—
200#124C2316yes—
201#124C2316yes—
202#124C2316yes—
203#124C23112yes—
204#124C23112yes—
205#124C23112yes—
206#124C23112yes—
207#124C23112yes—
208#123C2312yes—
209#123C2316yes—
210#123C2316yes—
211#123C2316yes—
212#123C23112yes—
213#123C2316yes—
214#123C2316yes—
215#123C23112yes—
216#123C23112yes—
217#123C2316yes—
218#123C2316yes—
219#123C2316yes—
220#123C2316yes—
221#123C2316yes—
222#123C2316yes—
223#123C23112yes—
224#123C23112yes—
225#123C23112yes—
226#123C23112yes—
227#123C23112yes—
228C22⋊Q8C2316yes—
229C22⋊Q8C2316yes—
230C22⋊Q8C2316yes—
231C22⋊Q8C2316yes—
232C22⋊Q8C23112yes—
233C22⋊Q8C2316yes—
234C22⋊Q8C2316yes—
235C22⋊Q8C23112yes—
236C22⋊Q8C2316yes—
237C22⋊Q8C2316yes—
238C22⋊Q8C2316yes—
239C22⋊Q8C2316yes—
240C22⋊Q8C2316yes—
241C22⋊Q8C2316yes—
242C22⋊Q8C2316yes—
243C22⋊Q8C2316yes—
244C22⋊Q8C2316yes—
245C22⋊Q8C2316yes—
246C22⋊Q8C2313yes—
247C22⋊Q8C2311yescharacteristic
248C22⋊Q8C2316yes—
249C22⋊Q8C2316yes—
250C22⋊Q8C23112yes—
251C22⋊Q8C23112yes—
252C22⋊Q8C23112yes—
253C22⋊Q8C23112yes—
254#121C2316yes—
255#121C2316yes—
256#121C23112yes—
257#121C2312yes—
258#121C2316yes—
259#121C2316yes—
260#121C2316yes—
261#121C2316yes—
262#121C2316yes—
263#121C2316yes—
264#121C2316yes—
265#121C2316yes—
266#121C2316yes—
267#121C2316yes—
268#121C2316yes—
269#121C2311yescharacteristic
270#121C2313yes—
271#121C2316yes—
272#121C2316yes—
273#121C23112yes—
274#121C23112yes—
275#121C23112yes—
276#121C23112yes—
277#121C23112yes—
278#121C23112yes—
279#121C23112yes—
280#120C2312yes—
281#120C2316yes—
282#120C2316yes—
283#120C2316yes—
284#120C2316yes—
285#120C2316yes—
286#120C23112yes—
287#120C23112yes—
288#120C23112yes—
289C4×Q8C2311yescharacteristic
290C4×Q8C2311yescharacteristic
291C4×Q8C2313yes—
292C4×Q8C2313yes—
293C4×Q8C2312yes—
294C4×Q8C2312yes—
295C4×Q8C2316yes—
296C4×Q8C2316yes—
297C4×Q8C2316yes—
298C4×Q8C2316yes—
299C4×Q8C23112yes—
300C4×Q8C2313yes—
301C4×Q8C2313yes—
302C4×Q8C2316yes—
303C4×D4C2312yes—
304C4×D4C2316yes—
305C4×D4C2316yes—
306C4×D4C2316yes—
307C4×D4C2316yes—
308C4×D4C2316yes—
309C4×D4C2316yes—
310C4×D4C2316yes—
311C4×D4C2316yes—
312C4×D4C2316yes—
313C4×D4C2316yes—
314C4×D4C2316yes—
315C4×D4C2316yes—
316C4×D4C2316yes—
317C4×D4C23112yes—
318C4×D4C23112yes—
319C4×D4C2316yes—
320C4×D4C2316yes—
321C4×D4C2313yes—
322C4×D4C2316yes—
323C4×D4C2313yes—
324C4×D4C2316yes—
325C4×D4C23112yes—
326C4×D4C23112yes—
327C4×D4C23112yes—
328C4×D4C23112yes—
329C4×D4C2316yes—
330#117C2316yes—
331#117C2316yes—
332#117C2316yes—
333#117C2316yes—
334#117C2316yes—
335#117C2316yes—
336#117C23112yes—
337#117C2316yes—
338#117C23112yes—
339C2×C4⋊C4C2313yes—
340C2×C4⋊C4C2316yes—
341C2×C4⋊C4C2313yes—
342C2×C4⋊C4C2316yes—
343C2×C4⋊C4C23112yes—
344C2×C22⋊C4C2311yescharacteristic
345C2×C22⋊C4C2313yes—
346C2×C22⋊C4C2316yes—
347C2×C22⋊C4C2316yes—
348C2×C22⋊C4C2316yes—
349C2×C22⋊C4C23112yes—
350C2×C42C2313yes—
351C2×C42C2316yes—
352C24C2413yes—
353C24C2411yescharacteristic
354C24C2416yes—
355#41—28no—
356#41—28no—
357#41—412no—
358#41—412no—
359#41—412no—
360#41—412no—
361#41—412no—
362#41—412no—
363#41—412no—
364C2×Q8—24no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—46no—
368C2×Q8—412no—
369C2×Q8—412no—
370C2×Q8C2416yes—
371C2×Q8C2413yes—
372C2×Q8C2416yes—
373C2×Q8C2411yescharacteristic
374C2×Q8C2413yes—
375C2×Q8C2416yes—
376C2×Q8C2416yes—
377C2×Q8C2416yes—
378C2×Q8C2411yescharacteristic
379C2×D4—46no—
380C2×D4—412no—
381C2×D4—46no—
382C2×D4—412no—
383C2×D4C2416yes—
384C2×D4C2412yes—
385C2×D4C2416yes—
386C2×D4C2416yes—
387C2×D4C2411yescharacteristic
388C2×D4C2412yes—
389C2×D4C24112yes—
390C2×D4C2413yes—
391C2×D4C2416yes—
392C2×D4C2416yes—
393C2×D4C2416yes—
394C2×D4C2416yes—
395C2×D4C2416yes—
396C2×D4C2416yes—
397C2×D4C2412yes—
398C2×D4C2416yes—
399C2×D4C2416yes—
400C2×D4C2416yes—
401C2×D4C24112yes—
402C22×C4—46no—
403C22×C4—46no—
404C22×C4—412no—
405C22×C4C2413yes—
406C22×C4C2413yes—
407C22×C4C2416yes—
408C22×C4C2416yes—
409C22×C4C2416yes—
410C22×C4C2416yes—
411C22×C4C2411yescharacteristic
412C22×C4C2416yes—
413C22×C4C2416yes—
414C22×C4C2413yes—
415C22×C4C2411yescharacteristic
416C22×C4C24112yes—
417C22×C4C2416yes—
418C22×C4C24112yes—
419C22×C4C2416yes—
420C22×C4C2416yes—
421C22×C4C2416yes—
422C22×C4C2412yes—
423C4⋊C4C2416yes—
424C4⋊C4C2412yes—
425C4⋊C4C2413yes—
426C4⋊C4C2416yes—
427C4⋊C4C2416yes—
428C4⋊C4C2416yes—
429C4⋊C4C2416yes—
430C4⋊C4C2413yes—
431C4⋊C4C2416yes—
432C4⋊C4C2416yes—
433C4⋊C4C2416yes—
434C4⋊C4C24112yes—
435C4⋊C4C2416yes—
436C4⋊C4C2416yes—
437C4⋊C4C2416yes—
438C4⋊C4C2412yes—
439C4⋊C4C2416yes—
440C4⋊C4C2416yes—
441C4⋊C4C2416yes—
442C4⋊C4C2416yes—
443C4⋊C4C2416yes—
444C4⋊C4C2416yes—
445C4⋊C4C2411yescharacteristic
446C4⋊C4C2416yes—
447C4⋊C4C24112yes—
448C4⋊C4C2416yes—
449C4⋊C4C2413yes—
450C4⋊C4C2413yes—
451C4⋊C4C2411yescharacteristic
452C22⋊C4C2412yes—
453C22⋊C4C2412yes—
454C22⋊C4C2416yes—
455C22⋊C4C2416yes—
456C22⋊C4C24112yes—
457C22⋊C4C2416yes—
458C22⋊C4C2416yes—
459C22⋊C4C24112yes—
460C22⋊C4C24112yes—
461C22⋊C4C2412yes—
462C22⋊C4C2412yes—
463C22⋊C4C2416yes—
464C22⋊C4C2416yes—
465C22⋊C4C2416yes—
466C22⋊C4C2416yes—
467C22⋊C4C2416yes—
468C22⋊C4C2416yes—
469C22⋊C4C2412yes—
470C22⋊C4C2416yes—
471C22⋊C4C2412yes—
472C22⋊C4C2416yes—
473C22⋊C4C24112yes—
474C22⋊C4C24112yes—
475C22⋊C4C2416yes—
476C22⋊C4C2416yes—
477C22⋊C4C2412yes—
478C22⋊C4C2416yes—
479C22⋊C4C2416yes—
480C22⋊C4C2416yes—
481C22⋊C4C24112yes—
482C42C2412yes—
483C42C2416yes—
484C42C2413yes—
485C42C2416yes—
486C42C2416yes—
487C42C2413yes—
488C42C2413yes—
489C42C2413yes—
490C42C24112yes—
491C42C24112yes—
492C23—22no—
493C23—26no—
494C23—41no—
495C23—41no—
496C23—43no—
497C23—46no—
498C23—46no—
499C23—43no—
500C23—46no—
501C23C2512yes—
502C23C2516yes—
503C23C2512yes—
504C23C2516yes—
505C23C2511yescharacteristic
506Q8—212no—
507Q8—26no—
508Q8—212no—
509Q8—46no—
510Q8—46no—
511Q8—41no—
512Q8—43no—
513Q8—46no—
514Q8H(5,𝔽2)14yes—
515D4—24no—
516D4—212no—
517D4—212no—
518D4—212no—
519D4—46no—
520D4—46no—
521D4—46no—
522D4—46no—
523D4—412no—
524D4—46no—
525D4—42no—
526D4—412no—
527D4—46no—
528D4—43no—
529D4—41no—
530D4—46no—
531D4—42no—
532D4—46no—
533D4—46no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—212no—
537C2×C4—212no—
538C2×C4—212no—
539C2×C4—212no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—46no—
543C2×C4—43no—
544C2×C4—46no—
545C2×C4—46no—
546C2×C4—46no—
547C2×C4—46no—
548C2×C4—41no—
549C2×C4—46no—
550C2×C4—412no—
551C2×C4—412no—
552C2×C4C2511yescharacteristic
553C2×C4C2516yes—
554C2×C4C2511yescharacteristic
555C2×C4C2513yes—
556C2×C4C2512yes—
557C2×C4C2516yes—
558C2×C4C2512yes—
559C2×C4C2512yes—
560C2×C4C2516yes—
561C2×C4C2513yes—
562C2×C4C2513yes—
563C2×C4C2516yes—
564C2×C4C2512yes—
565C2×C4C2513yes—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—26no—
571C22—26no—
572C22—26no—
573C22—26no—
574C22—22no—
575C22—22no—
576C22—26no—
577C22—22no—
578C22—26no—
579C22—22no—
580C22—44no—
581C22—86no—
582C22—812no—
583C22#58512yes—
584C22C2611yescharacteristic
585C4—22no—
586C4—22no—
587C4—26no—
588C4—23no—
589C4—26no—
590C4—23no—
591C4—23no—
592C4—22no—
593C4—26no—
594C4—22no—
595C4—26no—
596C4—21no—
597C4—21no—
598C4C2×H(5,𝔽2)16yes—
599C2—22no—
600C2—42no—
601C2—46no—
602C2—46no—
603C2—42no—
604C2C22×H(5,𝔽2)11yescharacteristic
605C2C2×#58511yescharacteristic
606C22−1+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. #3562C2semidirect
2. #3561C2semidirect
3. #3558C2semidirect
4. #3554C2semidirect
5. #3542C2semidirect
6. #3541C2semidirect
7. #3540C2semidirect
8. #3524C2semidirect
9. #3523C2semidirect
10. #3522C2semidirect
11. #3520C2semidirect
12. #3518C2semidirect
13. #3506C2semidirect
14. #3489C2semidirect
15. #3487C2semidirect
16. #3486C2semidirect
17. D4×#41C2semidirect
18. C2×#548C2semidirect
19. #3459C2semidirect
20. C2×2−1+4C22semidirect
22. C22×#41C22semidirect
23. C23×Q8C22semidirect
24. #563C22semidirect
25. #562C22semidirect
26. #562C22semidirect
27. #562C22semidirect
28. H(3,𝔽4)C22semidirect
29. #560C22semidirect
30. #559C22semidirect
31. #559C22semidirect
32. #559C22semidirect
33. Q82C22semidirect
34. #557C22semidirect
35. #557C22semidirect
36. #556C22semidirect
37. #556C22semidirect
38. #556C22semidirect
39. #556C22semidirect
40. #555C22semidirect
41. #555C22semidirect
42. #554C22semidirect
43. #554C22semidirect
44. #553C22semidirect
45. #553C22semidirect
46. #552C22semidirect
47. #552C22semidirect
48. #551C22semidirect
49. #551C22semidirect
50. #550C22semidirect
51. #550C22semidirect
52. #550C22semidirect
53. #550C22semidirect
54. D4×Q8C22semidirect
55. D4×Q8C22semidirect
56. D4×Q8C22semidirect
57. D4×Q8C22semidirect
58. D4×Q8C22semidirect
59. #548C22semidirect
60. #548C22semidirect
61. #548C22semidirect
62. #548C22semidirect
63. #548C22semidirect
64. #548C22semidirect
65. #548C22semidirect
66. #548C22semidirect
67. #548C22semidirect
68. #548C22semidirect
69. #548C22semidirect
70. #547C22semidirect
71. #547C22semidirect
72. #547C22semidirect
73. #547C22semidirect
74. #546C22semidirect
75. #546C22semidirect
76. #546C22semidirect
77. #546C22semidirect
78. #546C22semidirect
79. D42C22semidirect
80. #543C22semidirect
81. #542C22semidirect
82. #542C22semidirect
83. #542C22semidirect
84. #542C22semidirect
85. #542C22semidirect
86. #542C22semidirect
87. #541C22semidirect
88. #540C22semidirect
89. #540C22semidirect
90. #539C22semidirect
91. #539C22semidirect
92. #539C22semidirect
93. #539C22semidirect
95. #538C22semidirect
96. #538C22semidirect
97. #538C22semidirect
98. #538C22semidirect
99. #538C22semidirect
100. #538C22semidirect
101. #537C22semidirect
102. #536C22semidirect
103. #536C22semidirect
104. #536C22semidirect
105. #535C22semidirect
106. #535C22semidirect
107. #535C22semidirect
109. #534C22semidirect
110. #533C22semidirect
111. #533C22semidirect
112. #532C22semidirect
113. #532C22semidirect
115. #529C22semidirect
116. #529C22semidirect
117. #529C22semidirect
118. #529C22semidirect
119. C2×#124C22semidirect
120. #525C22semidirect
121. #525C22semidirect
122. #525C22semidirect
123. #525C22semidirect
124. #525C22semidirect
125. #525C22semidirect
126. #525C22semidirect
127. C2×C22⋊Q8C22semidirect
128. C2×C22⋊Q8C22semidirect
129. C2×#121C22semidirect
130. #520C22semidirect
131. #520C22semidirect
132. #519C22semidirect
133. #519C22semidirect
134. C4×#41C22semidirect
135. C2×C4×Q8C22semidirect
136. C2×C4×D4C22semidirect
137. C2×C4×D4C22semidirect

Maximal subgroups

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

subgroupindex
#35622
#35612
#35582
#35542
#35422
#35412
#35402
#35242
#35232
#35222
#35202
#35182
#35062
#34892
#34872
#34862
D4×#412
C2×#5482
#34592