Finite Group Databasebeta

← #62800

Subgroups — #62800

Order 256.

3928 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3522C212yesmaximal, complemented
2#3519C216yesmaximal, complemented
3#3517C212yesmaximal, complemented
4#3514C216yesmaximal, complemented
5#3511C213yesmaximal, complemented
6#3508C213yesmaximal, complemented
7#3503C211yesmaximal, characteristic, complemented
8#3498C211yesmaximal, characteristic, complemented
9#3494C216yesmaximal, complemented
10#3488C212yesmaximal, complemented
11#3483C212yesmaximal, complemented
12#3481C216yesmaximal, complemented
13#3480C212yesmaximal, complemented
14D4×#41C216yesmaximal, complemented
15#3460C216yesmaximal, complemented
16#3444C213yesmaximal, complemented
17C2×#532C213yesmaximal, complemented
18C4×H(5,𝔽2)C212yesmaximal, complemented
19C2×C4×#41C211yesmaximal, characteristic, complemented
20C2×H(5,𝔽2)C2212yes—
21C22×#41C2211yescharacteristic
22C22×#41C2216yescomplemented
23#559C2216yescomplemented
24#559C2216yescomplemented
25Q82C2212yescomplemented
26#557C2212yescomplemented
27#556C2216yescomplemented
28#555C22112yescomplemented
29#555C2216yescomplemented
30#554C2216yescomplemented
31#554C2216yescomplemented
32#553C22112yescomplemented
33#553C22112yescomplemented
34#553C22112yescomplemented
35#551C22124yescomplemented
36#550C2216yescomplemented
37#550C2212yescomplemented
38#550C2214yescomplemented
39#550C2212yescomplemented
40#550C2216yescomplemented
41D4×Q8C2216yescomplemented
42#548C22112yescomplemented
43#548C2214yescomplemented
44#547C22112yescomplemented
45#547C2216yescomplemented
46#546C22124yescomplemented
47#546C22112yescomplemented
48#546C22112yescomplemented
49#546C22112yescomplemented
50D42C22112yescomplemented
51D42C2216yescomplemented
52D42C2216yescomplemented
53#544C2216yescomplemented
54#542C22112yescomplemented
55#542C22112yescomplemented
56#540C22112yescomplemented
57#540C22112yescomplemented
58#538C22124yescomplemented
59#537C2216yescomplemented
60#535C2216yescomplemented
61#535C2216yescomplemented
62#533C2211yescharacteristic, complemented
63#533C2213yescomplemented
64#533C2213yescomplemented
65#532C2213yescomplemented
66#532C2216yescomplemented
67#532C2216yescomplemented
68#532C2216yescomplemented
69#532C2216yescomplemented
70#532C2216yescomplemented
71#532C2216yescomplemented
72#532C2216yescomplemented
73#532C2213yescomplemented
74#532C2213yescomplemented
75#532C22112yescomplemented
76C2×C4⋊Q8C2213yescomplemented
77C2×#127C2213yescomplemented
78#529C2216yescomplemented
79#529C2216yescomplemented
80#529C2216yescomplemented
81#529C2216yescomplemented
82#529C2216yescomplemented
83#529C2216yescomplemented
84#529C22112yescomplemented
85C2×#124C2216yescomplemented
86#525C2216yescomplemented
87#525C2216yescomplemented
88#525C2216yescomplemented
89#525C22124yescomplemented
90C2×#121C2216yescomplemented
91C2×#121C2216yescomplemented
92#520C2212yes—
93#520C2216yescomplemented
94#520C2214yescomplemented
95#518C2216yescomplemented
96#518C22112yescomplemented
97C4×#41C2211yescharacteristic
98C4×#41C2211yescharacteristic
99C4×#41C2213yescomplemented
100C4×#41C2213yescomplemented
101C4×#41C2211yescharacteristic, complemented
102C4×#41C2213yescomplemented
103C4×#41C2213yescomplemented
104C4×#41C2211yescharacteristic, complemented
105C4×#41C2216yescomplemented
106C4×#41C2214yescomplemented
107C4×#41C2216yescomplemented
108C4×#41C2216yescomplemented
109C2×C4×Q8C2211yescharacteristic, complemented
110C2×C4×Q8C2211yescharacteristic, complemented
111C2×C4×D4C2213yescomplemented
112C2×C4×D4C2213yescomplemented
113C2×C4×D4C2216yescomplemented
114C2×C4×D4C22112yescomplemented
115C2×C4×D4C2216yescomplemented
116C2×#117C2213yescomplemented
117C22×C42C2213yescomplemented
118H(5,𝔽2)—216no—
119C2×#41—28no—
120C2×#41—424no—
121C2×#41C2313yes—
122C2×#41C2313yes—
123C2×#41C2311yescharacteristic
124C2×#41C2316yes—
125C2×#41C2314yes—
126C2×#41C2316yes—
127C2×#41C2316yes—
128C2×#41C23112yes—
129C2×#41C23112yes—
130C2×#41C23112yes—
131C2×#41C2311yescharacteristic
132C22×Q8C2311yescharacteristic
133C22×Q8C2316yes—
134C22×D4C2313yes—
135C22×D4C2316yes—
136C22×D4C23112yes—
137C22×D4C2316yes—
138C22×D4C23112yes—
139C22×D4C2316yes—
140C23×C4C2313yes—
141C23×C4C2313yes—
142C23×C4C23112yes—
143C23×C4C2313yes—
144C4⋊Q8C2313yes—
145C4⋊Q8C2316yes—
146C4⋊Q8C2316yes—
147C4⋊Q8C2316yes—
148C4⋊Q8C2311yescharacteristic
149C4⋊Q8C2311yescharacteristic
150C4⋊Q8C23112yes—
151#127C2313yes—
152#127C2316yes—
153#127C2316yes—
154#127C2316yes—
155#127C23112yes—
156#126C23112yes—
157#126C23112yes—
158#126C23124yes—
159#125C2312yes—
160#125C2316yes—
161#125C2316yes—
162#125C23112yes—
163#124C2316yes—
164#124C23112yes—
165#124C23112yes—
166#124C23112yes—
167#124C2316yes—
168#124C2316yes—
169#124C23112yes—
170#124C23124yes—
171#123C23112yes—
172#123C23112yes—
173#123C23112yes—
174#123C23124yes—
175#123C23124yes—
176#123C23124yes—
177C22⋊Q8C2312yes—
178C22⋊Q8C2312yes—
179C22⋊Q8C2316yes—
180C22⋊Q8C2316yes—
181C22⋊Q8C2316yes—
182C22⋊Q8C2316yes—
183C22⋊Q8C23124yes—
184C22⋊Q8C23124yes—
185C22⋊Q8C23112yes—
186#121C2316yes—
187#121C2316yes—
188#121C23112yes—
189#121C23112yes—
190#121C23112yes—
191#121C23112yes—
192#121C23112yes—
193#121C23112yes—
194#121C2316yes—
195#121C2316yes—
196#121C2316yes—
197#121C2316yes—
198#121C23124yes—
199#121C23124yes—
200#121C23112yes—
201#121C23112yes—
202#121C23112yes—
203#121C23124yes—
204#120C23112yes—
205#120C23124yes—
206#120C23124yes—
207C4×Q8C2311yescharacteristic
208C4×Q8C2311yescharacteristic
209C4×Q8C2311yescharacteristic
210C4×Q8C2311yescharacteristic
211C4×Q8C2313yes—
212C4×Q8C2313yes—
213C4×Q8C2313yes—
214C4×Q8C2313yes—
215C4×Q8C2312yes—
216C4×Q8C2312yes—
217C4×Q8C2314yes—
218C4×Q8C2316yes—
219C4×Q8C2316yes—
220C4×Q8C2316yes—
221C4×Q8C2316yes—
222C4×D4C2313yes—
223C4×D4C23112yes—
224C4×D4C2316yes—
225C4×D4C23112yes—
226C4×D4C2316yes—
227C4×D4C2313yes—
228C4×D4C2313yes—
229C4×D4C2313yes—
230C4×D4C2316yes—
231C4×D4C2313yes—
232C4×D4C2316yes—
233C4×D4C2313yes—
234C4×D4C2316yes—
235C4×D4C2316yes—
236C4×D4C2313yes—
237C4×D4C2313yes—
238C4×D4C23112yes—
239C4×D4C23112yes—
240C4×D4C23112yes—
241C4×D4C23124yes—
242C4×D4C23112yes—
243C4×D4C23112yes—
244C4×D4C23112yes—
245C4×D4C23112yes—
246C4×D4C23112yes—
247C4×D4C2316yes—
248C4×D4C23112yes—
249C4×D4C23112yes—
250C4×D4C23112yes—
251C4×D4C23112yes—
252C4×D4C23112yes—
253C4×D4C2316yes—
254C4×D4C2316yes—
255C4×D4C2316yes—
256#117C2313yes—
257#117C2313yes—
258#117C2316yes—
259#117C2316yes—
260#117C2313yes—
261#117C2313yes—
262#117C2312yes—
263#117C2316yes—
264#117C23112yes—
265#117C23112yes—
266#117C23112yes—
267#117C2316yes—
268#117C23112yes—
269C2×C4⋊C4C2313yes—
270C2×C4⋊C4C2313yes—
271C2×C4⋊C4C2316yes—
272C2×C4⋊C4C23112yes—
273C2×C4⋊C4C2316yes—
274C2×C22⋊C4C2316yes—
275C2×C22⋊C4C23112yes—
276C2×C22⋊C4C23124yes—
277C2×C22⋊C4C23112yes—
278C2×C42C2313yes—
279C2×C42C2313yes—
280C2×C42C2313yes—
281C2×C42C2313yes—
282C2×C42C2313yes—
283C2×C42C23112yes—
284C2×C42C2313yes—
285C2×C42C2313yes—
286C2×C42C2313yes—
287C2×C42C2313yes—
288C2×C42C2313yes—
289C2×C42C2313yes—
290C2×C42C2311yescharacteristic
291C24C2413yes—
292C24C24112yes—
293#41—24no—
294#41—216no—
295#41—212no—
296#41—212no—
297#41—224no—
298#41—224no—
299#41—424no—
300#41—424no—
301#41—424no—
302#41—412no—
303#41#4118yescomplemented
304C2×Q8—24no—
305C2×Q8—412no—
306C2×Q8C2413yes—
307C2×Q8C2412yes—
308C2×Q8C2412yes—
309C2×Q8C2416yes—
310C2×Q8C24112yes—
311C2×Q8C2411yescharacteristic
312C2×D4—212no—
313C2×D4—248no—
314C2×D4—224no—
315C2×D4—412no—
316C2×D4—412no—
317C2×D4—424no—
318C2×D4C2416yes—
319C2×D4C24112yes—
320C2×D4C2416yes—
321C2×D4C24112yes—
322C2×D4C2416yes—
323C2×D4C2413yes—
324C2×D4C24112yes—
325C2×D4C2416yes—
326C2×D4C24112yes—
327C2×D4C24112yes—
328C2×D4C24112yes—
329C2×D4C24124yes—
330C2×D4C2413yes—
331C22×C4—212no—
332C22×C4—212no—
333C22×C4—46no—
334C22×C4—424no—
335C22×C4C2411yescharacteristic
336C22×C4C2411yescharacteristic
337C22×C4C2413yes—
338C22×C4C2413yes—
339C22×C4C2413yes—
340C22×C4C2413yes—
341C22×C4C2413yes—
342C22×C4C2413yes—
343C22×C4C2413yes—
344C22×C4C2413yes—
345C22×C4C2413yes—
346C22×C4C24112yes—
347C22×C4C2413yes—
348C22×C4C2413yes—
349C22×C4C24112yes—
350C22×C4C2413yes—
351C22×C4C2413yes—
352C22×C4C2413yes—
353C22×C4C24112yes—
354C22×C4C24112yes—
355C22×C4C24124yes—
356C22×C4C2413yes—
357C22×C4C2411yescharacteristic
358C4⋊C4C2413yes—
359C4⋊C4C2416yes—
360C4⋊C4C24112yes—
361C4⋊C4C2416yes—
362C4⋊C4C24112yes—
363C4⋊C4C2413yes—
364C4⋊C4C2413yes—
365C4⋊C4C2413yes—
366C4⋊C4C2416yes—
367C4⋊C4C2416yes—
368C4⋊C4C24112yes—
369C4⋊C4C2412yes—
370C4⋊C4C2412yes—
371C4⋊C4C2414yes—
372C4⋊C4C2414yes—
373C4⋊C4C2412yes—
374C4⋊C4C2412yes—
375C4⋊C4C24112yes—
376C4⋊C4C24112yes—
377C4⋊C4C24112yes—
378C22⋊C4C2416yes—
379C22⋊C4C24112yes—
380C22⋊C4C24112yes—
381C22⋊C4C24112yes—
382C22⋊C4C24124yes—
383C22⋊C4C24112yes—
384C22⋊C4C2416yes—
385C22⋊C4C2416yes—
386C22⋊C4C2416yes—
387C22⋊C4C24124yes—
388C22⋊C4C2414yes—
389C22⋊C4C24112yes—
390C22⋊C4C24112yes—
391C22⋊C4C24112yes—
392C22⋊C4C24112yes—
393C42C2411yescharacteristic
394C42C2411yescharacteristic
395C42C2413yes—
396C42C2413yes—
397C42C2413yes—
398C42C2413yes—
399C42C2413yes—
400C42C2413yes—
401C42C2413yes—
402C42C2413yes—
403C42C2413yes—
404C42C2413yes—
405C42C24112yes—
406C42C24112yes—
407C42C2411yescharacteristic
408C42C2413yes—
409C42C2411yescharacteristic
410C42C2413yes—
411C42C2414yes—
412C23—26no—
413C23—26no—
414C23—224no—
415C23—43no—
416C23—412no—
417C23—412no—
418C23C25112yes—
419C23C2513yes—
420C23C2513yes—
421C23C2511yescharacteristic
422Q8—26no—
423Q8—24no—
424Q8—212no—
425Q8—24no—
426Q8—412no—
427Q8C2×#4114yescomplemented
428D4—224no—
429D4—212no—
430D4—224no—
431D4—224no—
432D4—212no—
433D4—26no—
434D4—212no—
435D4—212no—
436D4—412no—
437D4—412no—
438D4—424no—
439D4—412no—
440D4C2×#41112yescomplemented
441C2×C4—26no—
442C2×C4—26no—
443C2×C4—22no—
444C2×C4—26no—
445C2×C4—26no—
446C2×C4—26no—
447C2×C4—224no—
448C2×C4—26no—
449C2×C4—26no—
450C2×C4—26no—
451C2×C4—26no—
452C2×C4—224no—
453C2×C4—22no—
454C2×C4—26no—
455C2×C4—26no—
456C2×C4—43no—
457C2×C4—43no—
458C2×C4—43no—
459C2×C4—412no—
460C2×C4—424no—
461C2×C4—412no—
462C2×C4C2×#41112yescomplemented
463C2×C4H(5,𝔽2)14yescomplemented
464C2×C4C2511yescharacteristic
465C2×C4C2511yescharacteristic
466C2×C4C2513yes—
467C2×C4C2513yes—
468C2×C4C2514yes—
469C2×C4C2514yes—
470C2×C4C2513yes—
471C2×C4C2513yes—
472C2×C4C2513yes—
473C2×C4C25112yes—
474C2×C4C2513yes—
475C2×C4C2511yescharacteristic
476C2×C4C2511yescharacteristic
477C2×C4C2511yescharacteristic
478C2×C4C2511yescharacteristic
479C22—21no—
480C22—23no—
481C22—21no—
482C22—23no—
483C22—23no—
484C22—212no—
485C22—23no—
486C22—212no—
487C22—23no—
488C22—212no—
489C22—412no—
490C22—824no—
491C22C22×#4116yes—
492C22C2×H(5,𝔽2)12yes—
493C22C2611yescharacteristic
494C4—21no—
495C4—24no—
496C4—23no—
497C4—23no—
498C4—23no—
499C4—212no—
500C4—21no—
501C4—23no—
502C4—23no—
503C4—24no—
504C4C22×#4116yes—
505C4C22×#4112yes—
506C4C2×H(5,𝔽2)12yes—
507C4C2×H(5,𝔽2)12yes—
508C4#58512yes—
509C2—22no—
510C2—26no—
511C2—43no—
512C2—412no—
513C2C23×#4111yescharacteristic
514C2C22×H(5,𝔽2)11yescharacteristic
515C2C2×#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. #3522C2semidirect
2. #3519C2semidirect
3. #3517C2semidirect
4. #3514C2semidirect
5. #3511C2semidirect
6. #3508C2semidirect
7. #3503C2semidirect
8. #3498C2semidirect
9. #3494C2semidirect
10. #3488C2semidirect
11. #3483C2semidirect
12. #3481C2semidirect
13. #3480C2semidirect
14. D4×#41C2semidirect
15. #3460C2semidirect
16. #3444C2semidirect
17. C2×#532C2semidirect
18. C4×H(5,𝔽2)C2semidirect
19. C2×C4×#41C2semidirect
22. C22×#41C22semidirect
23. #559C22semidirect
24. #559C22semidirect
25. Q82C22semidirect
26. #557C22semidirect
27. #556C22semidirect
28. #555C22semidirect
29. #555C22semidirect
30. #554C22semidirect
31. #554C22semidirect
32. #553C22semidirect
33. #553C22semidirect
34. #553C22semidirect
35. #551C22semidirect
36. #550C22semidirect
37. #550C22semidirect
38. #550C22semidirect
39. #550C22semidirect
40. #550C22semidirect
41. D4×Q8C22semidirect
42. #548C22semidirect
43. #548C22semidirect
44. #547C22semidirect
45. #547C22semidirect
46. #546C22semidirect
47. #546C22semidirect
48. #546C22semidirect
49. #546C22semidirect
50. D42C22semidirect
51. D42C22semidirect
52. D42C22semidirect
53. #544C22semidirect
54. #542C22semidirect
55. #542C22semidirect
56. #540C22semidirect
57. #540C22semidirect
58. #538C22semidirect
59. #537C22semidirect
60. #535C22semidirect
61. #535C22semidirect
62. #533C22semidirect
63. #533C22semidirect
64. #533C22semidirect
65. #532C22semidirect
66. #532C22semidirect
67. #532C22semidirect
68. #532C22semidirect
69. #532C22semidirect
70. #532C22semidirect
71. #532C22semidirect
72. #532C22semidirect
73. #532C22semidirect
74. #532C22semidirect
75. #532C22semidirect
76. C2×C4⋊Q8C22semidirect
77. C2×#127C22semidirect
78. #529C22semidirect
79. #529C22semidirect
80. #529C22semidirect
81. #529C22semidirect
82. #529C22semidirect
83. #529C22semidirect
84. #529C22semidirect
85. C2×#124C22semidirect
86. #525C22semidirect
87. #525C22semidirect
88. #525C22semidirect
89. #525C22semidirect
90. C2×#121C22semidirect
91. C2×#121C22semidirect
93. #520C22semidirect
94. #520C22semidirect
95. #518C22semidirect
96. #518C22semidirect
99. C4×#41C22semidirect
100. C4×#41C22semidirect
101. C4×#41C22semidirect
102. C4×#41C22semidirect
103. C4×#41C22semidirect
104. C4×#41C22semidirect
105. C4×#41C22semidirect
106. C4×#41C22semidirect
107. C4×#41C22semidirect
108. C4×#41C22semidirect
109. C2×C4×Q8C22semidirect
110. C2×C4×Q8C22semidirect
111. C2×C4×D4C22semidirect
112. C2×C4×D4C22semidirect
113. C2×C4×D4C22semidirect
114. C2×C4×D4C22semidirect
115. C2×C4×D4C22semidirect
116. C2×#117C22semidirect
117. C22×C42C22semidirect
303. #41#41semidirect
427. Q8C2×#41semidirect
440. D4C2×#41semidirect
462. C2×C4C2×#41semidirect
463. 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
#35222
#35192
#35172
#35142
#35112
#35082
#35032
#34982
#34942
#34882
#34832
#34812
#34802
D4×#412
#34602
#34442
C2×#5322
C4×H(5,𝔽2)2
C2×C4×#412