Finite Group Databasebeta

← #63052

Subgroups — #63052

Order 256.

3562 conjugacy classes of subgroups.

The subgroup lattice has 607 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#3548C213yesmaximal, complemented
5#3546C216yesmaximal, complemented
6#3546C216yesmaximal, complemented
7#3542C216yesmaximal, complemented
8#3541C216yesmaximal, complemented
9#3536C216yesmaximal, complemented
10#3529C213yesmaximal, complemented
11#3522C212yesmaximal, complemented
12#3518C211yesmaximal, characteristic, complemented
13#3517C211yesmaximal, characteristic, complemented
14#3516C212yesmaximal, complemented
15#3500C211yesmaximal, characteristic, complemented
16#3497C212yesmaximal, complemented
17#3489C212yesmaximal, complemented
18#3488C212yesmaximal, complemented
19#3487C212yesmaximal, complemented
20C2×2−1+4C2212yescomplemented
21C2×2−1+4C2211yescharacteristic, complemented
22C2×H(5,𝔽2)C2211yescharacteristic
23C22×#41C2212yescomplemented
24#563C2216yescomplemented
25#563C2213yescomplemented
26#562C2211yescharacteristic, complemented
27#562C2216yescomplemented
28#562C2213yescomplemented
29#562C2216yescomplemented
30#562C22112yescomplemented
31#562C2216yescomplemented
32H(3,𝔽4)C2213yescomplemented
33#560C2216yescomplemented
34#560C2216yescomplemented
35#560C2216yescomplemented
36#560C22112yescomplemented
37#559C2216yescomplemented
38Q82C2211yescharacteristic, complemented
39#557C2211yescharacteristic, complemented
40#556C2216yescomplemented
41#556C2216yescomplemented
42#556C2213yescomplemented
43#555C2213yescomplemented
44#554C2216yescomplemented
45#554C2216yescomplemented
46#554C2216yescomplemented
47#553C2216yescomplemented
48#553C2216yescomplemented
49#553C22112yescomplemented
50#552C2216yescomplemented
51#551C22112yescomplemented
52#550C2212yes—
53#550C2211yescharacteristic, complemented
54#550C2216yescomplemented
55#550C2216yescomplemented
56#550C2212yescomplemented
57#550C2216yescomplemented
58#550C2213yescomplemented
59#550C2216yescomplemented
60D4×Q8C2211yescharacteristic, complemented
61D4×Q8C2212yescomplemented
62D4×Q8C2216yescomplemented
63#548C2212yes—
64#548C2212yescomplemented
65#548C2212yescomplemented
66#548C2216yescomplemented
67#548C2216yescomplemented
68#548C2212yescomplemented
69#548C2212yescomplemented
70#548C2216yescomplemented
71#548C2216yescomplemented
72#547C2216yescomplemented
73#547C2216yescomplemented
74#547C2213yescomplemented
75#547C2216yescomplemented
76#547C2216yescomplemented
77#546C2216yescomplemented
78#546C2216yescomplemented
79#546C2216yescomplemented
80#546C2216yescomplemented
81#546C2216yescomplemented
82#546C2216yescomplemented
83#546C22112yescomplemented
84#546C22112yescomplemented
85D42C2216yescomplemented
86D42C2216yescomplemented
87#543C2216yescomplemented
88#542C2216yescomplemented
89#542C2216yescomplemented
90#542C2216yescomplemented
91#542C2216yescomplemented
92#542C22112yescomplemented
93#541C2216yescomplemented
94#540C2216yescomplemented
95#540C22112yescomplemented
96#539C22112yescomplemented
97#538C22112yescomplemented
98#538C22112yescomplemented
99#538C22112yescomplemented
100#538C2216yescomplemented
101#538C2216yescomplemented
102#537C2216yescomplemented
103#537C2216yescomplemented
104#537C2216yescomplemented
105#537C2216yescomplemented
106#537C22112yescomplemented
107#536C2216yescomplemented
108#536C2213yescomplemented
109#535C2216yescomplemented
110#535C2216yescomplemented
111#535C2216yescomplemented
112#535C2213yescomplemented
113#535C22112yescomplemented
114#534C2216yescomplemented
115#533C2213yescomplemented
116#532C2216yescomplemented
117#532C2216yescomplemented
118#532C2213yescomplemented
119#532C2213yescomplemented
120#532C2216yescomplemented
121#529C2216yescomplemented
122#529C2216yescomplemented
123#529C2213yescomplemented
124#529C2213yescomplemented
125#529C2216yescomplemented
126C2×#124C2213yescomplemented
127#525C2216yescomplemented
128#525C2216yescomplemented
129#525C2216yescomplemented
130#525C2216yescomplemented
131C2×C22⋊Q8C2216yescomplemented
132C2×#121C2216yescomplemented
133#520C2212yescomplemented
134#520C2212yescomplemented
135#519C2212yescomplemented
136C4×#41C2211yescharacteristic, complemented
137C2×C4×Q8C2212yescomplemented
1382−1+4—48no—
1392−1+4—44no—
140H(5,𝔽2)—44no—
141C2×#41—48no—
142C2×#41C2311yescharacteristic
143C2×#41C2312yes—
144C2×#41C2312yes—
145C2×#41C2316yes—
146C2×#41C2313yes—
147C2×#41C2316yes—
148C2×#41C2313yes—
149C2×#41C2316yes—
150C2×#41C2316yes—
151C2×#41C2316yes—
152C2×#41C2316yes—
153C2×#41C23112yes—
154C2×#41C2316yes—
155C22×Q8C2312yes—
156C22×Q8C2312yes—
157C22×Q8C2316yes—
158C22×Q8C2313yes—
159C22×D4C2316yes—
160C22×D4C2313yes—
161C22×D4C2316yes—
162C22×D4C23112yes—
163C23×C4C2316yes—
164C4⋊Q8C2313yes—
165C4⋊Q8C2316yes—
166C4⋊Q8C2313yes—
167C4⋊Q8C2316yes—
168C4⋊Q8C2316yes—
169C4⋊Q8C2313yes—
170C4⋊Q8C2313yes—
171#127C2316yes—
172#127C2313yes—
173#127C2313yes—
174#127C2316yes—
175#127C2316yes—
176#127C23112yes—
177#126C23112yes—
178#126C2316yes—
179#126C2316yes—
180#126C2316yes—
181#126C2316yes—
182#126C2316yes—
183#126C2316yes—
184#126C23112yes—
185#126C23112yes—
186#125C2311yescharacteristic
187#125C2313yes—
188#125C2316yes—
189#125C2316yes—
190#125C2316yes—
191#125C23112yes—
192#124C2313yes—
193#124C2316yes—
194#124C2316yes—
195#124C2316yes—
196#124C2311yescharacteristic
197#124C2313yes—
198#124C2316yes—
199#124C2316yes—
200#124C2316yes—
201#124C2311yescharacteristic
202#124C23112yes—
203#124C2316yes—
204#124C2316yes—
205#124C2316yes—
206#124C2316yes—
207#124C23112yes—
208#124C23112yes—
209#123C2316yes—
210#123C2312yes—
211#123C2316yes—
212#123C2316yes—
213#123C2316yes—
214#123C2316yes—
215#123C2312yes—
216#123C2316yes—
217#123C2316yes—
218#123C2316yes—
219#123C2316yes—
220#123C2316yes—
221#123C23112yes—
222#123C23112yes—
223#123C23112yes—
224#123C23112yes—
225#123C23112yes—
226#123C23112yes—
227#123C23112yes—
228C22⋊Q8C2316yes—
229C22⋊Q8C23112yes—
230C22⋊Q8C2316yes—
231C22⋊Q8C2316yes—
232C22⋊Q8C2316yes—
233C22⋊Q8C2316yes—
234C22⋊Q8C2312yes—
235C22⋊Q8C2316yes—
236C22⋊Q8C2316yes—
237C22⋊Q8C2316yes—
238C22⋊Q8C2316yes—
239C22⋊Q8C2316yes—
240C22⋊Q8C2316yes—
241C22⋊Q8C2312yes—
242C22⋊Q8C2316yes—
243C22⋊Q8C2316yes—
244C22⋊Q8C2313yes—
245C22⋊Q8C2313yes—
246C22⋊Q8C2316yes—
247C22⋊Q8C2316yes—
248C22⋊Q8C23112yes—
249C22⋊Q8C23112yes—
250C22⋊Q8C23112yes—
251C22⋊Q8C23112yes—
252C22⋊Q8C23112yes—
253#121C2316yes—
254#121C23112yes—
255#121C2316yes—
256#121C2316yes—
257#121C2316yes—
258#121C2316yes—
259#121C2312yes—
260#121C2316yes—
261#121C2316yes—
262#121C2316yes—
263#121C2316yes—
264#121C2316yes—
265#121C2316yes—
266#121C2316yes—
267#121C2313yes—
268#121C2316yes—
269#121C2316yes—
270#121C2313yes—
271#121C2312yes—
272#121C2316yes—
273#121C2316yes—
274#121C23112yes—
275#121C2316yes—
276#121C2316yes—
277#121C2316yes—
278#121C2316yes—
279#121C2316yes—
280#121C2316yes—
281#121C2316yes—
282#121C23112yes—
283#121C23112yes—
284#121C23112yes—
285#121C23112yes—
286#121C23112yes—
287#121C23112yes—
288#121C23112yes—
289#121C23112yes—
290#121C23112yes—
291#120C2316yes—
292#120C23112yes—
293#120C2316yes—
294#120C2316yes—
295#120C2316yes—
296#120C2316yes—
297#120C2316yes—
298#120C23112yes—
299#120C23112yes—
300#120C23112yes—
301C4×Q8C2312yes—
302C4×Q8C2312yes—
303C4×Q8C2316yes—
304C4×Q8C2316yes—
305C4×Q8C2311yescharacteristic
306C4×Q8C2311yescharacteristic
307C4×Q8C2316yes—
308C4×Q8C2316yes—
309C4×Q8C2313yes—
310C4×Q8C2313yes—
311C4×Q8C2316yes—
312C4×D4C2313yes—
313C4×D4C2313yes—
314C4×D4C2316yes—
315C4×D4C2316yes—
316C4×D4C2316yes—
317C4×D4C2316yes—
318C4×D4C2316yes—
319C4×D4C2313yes—
320C4×D4C2316yes—
321C4×D4C2313yes—
322C4×D4C2316yes—
323C4×D4C2316yes—
324C4×D4C2316yes—
325C4×D4C2316yes—
326C4×D4C2316yes—
327C4×D4C2316yes—
328C4×D4C23112yes—
329C4×D4C23112yes—
330C4×D4C23112yes—
331C4×D4C23112yes—
332C4×D4C23112yes—
333C4×D4C2316yes—
334C4×D4C23112yes—
335#117C2313yes—
336#117C2316yes—
337#117C2316yes—
338#117C2316yes—
339#117C2316yes—
340#117C23112yes—
341#117C2316yes—
342C2×C4⋊C4C2316yes—
343C2×C4⋊C4C2316yes—
344C2×C4⋊C4C2316yes—
345C2×C4⋊C4C23112yes—
346C2×C22⋊C4C23112yes—
347C2×C22⋊C4C23112yes—
348C2×C22⋊C4C2316yes—
349C2×C22⋊C4C2316yes—
350C2×C42C2316yes—
351C2×C42C2313yes—
352C24C2416yes—
353C24C2416yes—
354#41—28no—
355#41—28no—
356#41—24no—
357#41—46no—
358#41—412no—
359#41—46no—
360#41—412no—
361#41—412no—
362#41—412no—
363#41—412no—
364#41—412no—
365#41—812no—
366#41—86no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—412no—
370C2×Q8—46no—
371C2×Q8C2416yes—
372C2×Q8C2416yes—
373C2×Q8C2411yescharacteristic
374C2×Q8C2413yes—
375C2×Q8C2416yes—
376C2×Q8C2416yes—
377C2×Q8C2411yescharacteristic
378C2×Q8C2413yes—
379C2×Q8C2411yescharacteristic
380C2×D4—412no—
381C2×D4—412no—
382C2×D4—46no—
383C2×D4—812no—
384C2×D4C2416yes—
385C2×D4C2413yes—
386C2×D4C2416yes—
387C2×D4C2411yescharacteristic
388C2×D4C2416yes—
389C2×D4C2416yes—
390C2×D4C2412yes—
391C2×D4C2416yes—
392C2×D4C2416yes—
393C2×D4C2416yes—
394C2×D4C2416yes—
395C2×D4C2413yes—
396C2×D4C24112yes—
397C2×D4C24112yes—
398C2×D4C2416yes—
399C2×D4C2416yes—
400C2×D4C2416yes—
401C2×D4C2412yes—
402C2×D4C2416yes—
403C2×D4C24112yes—
404C2×D4C2413yes—
405C2×D4C2416yes—
406C2×D4C2416yes—
407C2×D4C2411yescharacteristic
408C22×C4—412no—
409C22×C4C2416yes—
410C22×C4C2413yes—
411C22×C4C2416yes—
412C22×C4C2416yes—
413C22×C4C2416yes—
414C22×C4C2412yes—
415C22×C4C2413yes—
416C22×C4C2413yes—
417C22×C4C2416yes—
418C22×C4C2416yes—
419C22×C4C2416yes—
420C22×C4C24112yes—
421C22×C4C24112yes—
422C22×C4C24112yes—
423C22×C4C2416yes—
424C22×C4C2412yes—
425C4⋊C4C2416yes—
426C4⋊C4C2413yes—
427C4⋊C4C2416yes—
428C4⋊C4C2416yes—
429C4⋊C4C2416yes—
430C4⋊C4C2416yes—
431C4⋊C4C2416yes—
432C4⋊C4C2416yes—
433C4⋊C4C24112yes—
434C4⋊C4C24112yes—
435C4⋊C4C2413yes—
436C4⋊C4C2412yes—
437C4⋊C4C2412yes—
438C4⋊C4C2413yes—
439C4⋊C4C2416yes—
440C4⋊C4C2413yes—
441C4⋊C4C2416yes—
442C4⋊C4C2416yes—
443C4⋊C4C2416yes—
444C4⋊C4C2416yes—
445C4⋊C4C2412yes—
446C4⋊C4C2416yes—
447C4⋊C4C2416yes—
448C4⋊C4C2411yescharacteristic
449C4⋊C4C2412yes—
450C4⋊C4C2416yes—
451C4⋊C4C2416yes—
452C4⋊C4C2411yescharacteristic
453C22⋊C4C24112yes—
454C22⋊C4C2416yes—
455C22⋊C4C2416yes—
456C22⋊C4C2416yes—
457C22⋊C4C2416yes—
458C22⋊C4C24112yes—
459C22⋊C4C2416yes—
460C22⋊C4C2416yes—
461C22⋊C4C24112yes—
462C22⋊C4C24112yes—
463C22⋊C4C2416yes—
464C22⋊C4C2412yes—
465C22⋊C4C2416yes—
466C22⋊C4C2416yes—
467C22⋊C4C2416yes—
468C22⋊C4C2416yes—
469C22⋊C4C2416yes—
470C22⋊C4C2416yes—
471C22⋊C4C2412yes—
472C22⋊C4C2412yes—
473C22⋊C4C2416yes—
474C22⋊C4C2416yes—
475C22⋊C4C2416yes—
476C22⋊C4C24112yes—
477C22⋊C4C2416yes—
478C22⋊C4C2416yes—
479C22⋊C4C2412yes—
480C22⋊C4C2416yes—
481C22⋊C4C2412yes—
482C22⋊C4C2412yes—
483C42C2411yescharacteristic
484C42C2413yes—
485C42C2416yes—
486C42C2416yes—
487C42C2413yes—
488C42C2416yes—
489C42C2413yes—
490C42C2416yes—
491C42C24112yes—
492C23—46no—
493C23—46no—
494C23—46no—
495C23—46no—
496C23—46no—
497C23—46no—
498C23C2512yes—
499C23C2516yes—
500C23C2513yes—
501C23C2516yes—
502C23C2512yes—
503Q8—212no—
504Q8—212no—
505Q8—46no—
506Q8—43no—
507Q8—41no—
508Q8—41no—
509Q8—43no—
510Q8—46no—
511Q8H(5,𝔽2)14yes—
512D4—212no—
513D4—26no—
514D4—212no—
515D4—46no—
516D4—43no—
517D4—412no—
518D4—46no—
519D4—46no—
520D4—412no—
521D4—412no—
522D4—41no—
523D4—42no—
524D4—46no—
525D4—46no—
526D4—46no—
527D4—46no—
528D4—42no—
529D4—41no—
530D4—46no—
531D4—46no—
532D4—46no—
533D4—46no—
534D4—46no—
535D4—43no—
536C2×C4—212no—
537C2×C4—212no—
538C2×C4—212no—
539C2×C4—26no—
540C2×C4—212no—
541C2×C4—42no—
542C2×C4—46no—
543C2×C4—46no—
544C2×C4—43no—
545C2×C4—42no—
546C2×C4—412no—
547C2×C4—412no—
548C2×C4—46no—
549C2×C4—412no—
550C2×C4—46no—
551C2×C4—43no—
552C2×C4C2512yes—
553C2×C4C2512yes—
554C2×C4C2513yes—
555C2×C4C2511yescharacteristic
556C2×C4C2511yescharacteristic
557C2×C4C2516yes—
558C2×C4C2511yescharacteristic
559C2×C4C2513yes—
560C2×C4C2516yes—
561C2×C4C2516yes—
562C2×C4C2516yes—
563C2×C4C2512yes—
564C2×C4C2512yes—
565C2×C4C2513yes—
566C22—22no—
567C22—22no—
568C22—26no—
569C22—23no—
570C22—26no—
571C22—22no—
572C22—26no—
573C22—26no—
574C22—23no—
575C22—22no—
576C22—22no—
577C22—23no—
578C22—26no—
579C22—22no—
580C22—26no—
581C22—812no—
582C22—166no—
583C22C2611yescharacteristic
584C4—22no—
585C4—22no—
586C4—21no—
587C4—26no—
588C4—26no—
589C4—26no—
590C4—23no—
591C4—26no—
592C4—21no—
593C4—21no—
594C4—23no—
595C4—22no—
596C4—22no—
597C4C2×H(5,𝔽2)16yes—
598C2—42no—
599C2—46no—
600C2—42no—
601C2—46no—
602C2—43no—
603C2C22×H(5,𝔽2)11yescharacteristic
604C2H(7,𝔽2)11yescharacteristic
605C22−1+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. #3548C2semidirect
5. #3546C2semidirect
6. #3546C2semidirect
7. #3542C2semidirect
8. #3541C2semidirect
9. #3536C2semidirect
10. #3529C2semidirect
11. #3522C2semidirect
12. #3518C2semidirect
13. #3517C2semidirect
14. #3516C2semidirect
15. #3500C2semidirect
16. #3497C2semidirect
17. #3489C2semidirect
18. #3488C2semidirect
19. #3487C2semidirect
20. C2×2−1+4C22semidirect
21. C2×2−1+4C22semidirect
23. C22×#41C22semidirect
24. #563C22semidirect
25. #563C22semidirect
26. #562C22semidirect
27. #562C22semidirect
28. #562C22semidirect
29. #562C22semidirect
30. #562C22semidirect
31. #562C22semidirect
32. H(3,𝔽4)C22semidirect
33. #560C22semidirect
34. #560C22semidirect
35. #560C22semidirect
36. #560C22semidirect
37. #559C22semidirect
38. Q82C22semidirect
39. #557C22semidirect
40. #556C22semidirect
41. #556C22semidirect
42. #556C22semidirect
43. #555C22semidirect
44. #554C22semidirect
45. #554C22semidirect
46. #554C22semidirect
47. #553C22semidirect
48. #553C22semidirect
49. #553C22semidirect
50. #552C22semidirect
51. #551C22semidirect
53. #550C22semidirect
54. #550C22semidirect
55. #550C22semidirect
56. #550C22semidirect
57. #550C22semidirect
58. #550C22semidirect
59. #550C22semidirect
60. D4×Q8C22semidirect
61. D4×Q8C22semidirect
62. D4×Q8C22semidirect
64. #548C22semidirect
65. #548C22semidirect
66. #548C22semidirect
67. #548C22semidirect
68. #548C22semidirect
69. #548C22semidirect
70. #548C22semidirect
71. #548C22semidirect
72. #547C22semidirect
73. #547C22semidirect
74. #547C22semidirect
75. #547C22semidirect
76. #547C22semidirect
77. #546C22semidirect
78. #546C22semidirect
79. #546C22semidirect
80. #546C22semidirect
81. #546C22semidirect
82. #546C22semidirect
83. #546C22semidirect
84. #546C22semidirect
85. D42C22semidirect
86. D42C22semidirect
87. #543C22semidirect
88. #542C22semidirect
89. #542C22semidirect
90. #542C22semidirect
91. #542C22semidirect
92. #542C22semidirect
93. #541C22semidirect
94. #540C22semidirect
95. #540C22semidirect
96. #539C22semidirect
97. #538C22semidirect
98. #538C22semidirect
99. #538C22semidirect
100. #538C22semidirect
101. #538C22semidirect
102. #537C22semidirect
103. #537C22semidirect
104. #537C22semidirect
105. #537C22semidirect
106. #537C22semidirect
107. #536C22semidirect
108. #536C22semidirect
109. #535C22semidirect
110. #535C22semidirect
111. #535C22semidirect
112. #535C22semidirect
113. #535C22semidirect
114. #534C22semidirect
115. #533C22semidirect
116. #532C22semidirect
117. #532C22semidirect
118. #532C22semidirect
119. #532C22semidirect
120. #532C22semidirect
121. #529C22semidirect
122. #529C22semidirect
123. #529C22semidirect
124. #529C22semidirect
125. #529C22semidirect
126. C2×#124C22semidirect
127. #525C22semidirect
128. #525C22semidirect
129. #525C22semidirect
130. #525C22semidirect
131. C2×C22⋊Q8C22semidirect
132. C2×#121C22semidirect
133. #520C22semidirect
134. #520C22semidirect
135. #519C22semidirect
136. C4×#41C22semidirect
137. C2×C4×Q8C22semidirect

Maximal subgroups

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

subgroupindex
#35622
#35612
#35582
#35482
#35462
#35422
#35412
#35362
#35292
#35222
#35182
#35172
#35162
#35002
#34972
#34892
#34882
#34872