Finite Group Databasebeta

← #63014

Subgroups — #63014

Order 256.

3950 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3548C213yesmaximal, complemented
2#3546C216yesmaximal, complemented
3#3543C216yesmaximal, complemented
4#3542C2112yesmaximal, complemented
5#3522C212yesmaximal, complemented
6#3519C216yesmaximal, complemented
7#3517C212yesmaximal, complemented
8#3508C213yesmaximal, complemented
9#3498C213yesmaximal, complemented
10#3498C211yesmaximal, characteristic, complemented
11#3496C216yesmaximal, complemented
12#3488C214yesmaximal, complemented
13#3483C211yesmaximal, characteristic, complemented
14#3482C211yesmaximal, characteristic, complemented
15#3481C213yesmaximal, complemented
16D4×#41C212yesmaximal, complemented
17C2×#550C212yesmaximal, complemented
18C2×H(5,𝔽2)C2212yescomplemented
19C2×H(5,𝔽2)C2216yescomplemented
20C22×#41C2214yescomplemented
21#562C22112yescomplemented
22#560C22124yescomplemented
23#559C2213yescomplemented
24#559C2216yescomplemented
25Q82C2211yescharacteristic, complemented
26#557C2211yescharacteristic, complemented
27#556C2213yescomplemented
28#555C2213yescomplemented
29#555C2216yescomplemented
30#554C2211yescharacteristic, complemented
31#554C2213yescomplemented
32#554C2216yescomplemented
33#553C2216yescomplemented
34#553C2216yescomplemented
35#553C22112yescomplemented
36#552C2216yescomplemented
37#551C22112yescomplemented
38#550C2211yescharacteristic, complemented
39#550C2214yescomplemented
40#550C2212yescomplemented
41#550C2216yescomplemented
42#550C22112yescomplemented
43#550C2216yescomplemented
44#550C2212yescomplemented
45#550C2212yescomplemented
46#550C2213yescomplemented
47#550C2216yescomplemented
48#550C2216yescomplemented
49D4×Q8C2211yescharacteristic, complemented
50#548C2214yescomplemented
51#548C2212yescomplemented
52#547C2216yescomplemented
53#547C2213yescomplemented
54#547C2216yescomplemented
55#547C22112yescomplemented
56#546C22112yescomplemented
57#546C2216yescomplemented
58#546C2216yescomplemented
59#546C22112yescomplemented
60#546C22112yescomplemented
61#546C22112yescomplemented
62#546C22124yescomplemented
63D42C2216yescomplemented
64D42C2213yescomplemented
65D42C22112yescomplemented
66D42C22112yescomplemented
67D42C2216yescomplemented
68D42C2216yescomplemented
69D42C22112yescomplemented
70#542C22112yescomplemented
71#540C22112yescomplemented
72#540C22112yescomplemented
73#540C22112yescomplemented
74#538C22124yescomplemented
75#537C22112yescomplemented
76#537C22112yescomplemented
77#535C22112yescomplemented
78#535C22112yescomplemented
79#535C2216yescomplemented
80#535C2216yescomplemented
81#535C2216yescomplemented
82#534C22124yescomplemented
83#533C2213yescomplemented
84#532C2216yescomplemented
85#532C2213yescomplemented
86#532C2216yescomplemented
87#532C2213yescomplemented
88#532C2216yescomplemented
89#532C2216yescomplemented
90C2×#127C2216yescomplemented
91C2×#127C2213yescomplemented
92#529C2216yescomplemented
93#529C2216yescomplemented
94#529C2216yescomplemented
95#525C22112yescomplemented
96#525C22112yescomplemented
97C2×#121C22112yescomplemented
98C2×#121C22112yescomplemented
99#520C2214yescomplemented
100#518C2216yescomplemented
101C4×#41C2212yescomplemented
102C4×#41C2213yescomplemented
103C4×#41C2213yescomplemented
104C4×#41C2211yescharacteristic, complemented
105C2×C4×Q8C2212yescomplemented
106C2×C4×D4C2216yescomplemented
107C2×C4×D4C2216yescomplemented
108H(5,𝔽2)—424no—
109H(5,𝔽2)—48no—
110C2×#41—416no—
111C2×#41C2313yes—
112C2×#41C2311yescharacteristic
113C2×#41C2312yes—
114C2×#41C2314yes—
115C2×#41C2313yes—
116C2×#41C2316yes—
117C2×#41C23112yes—
118C2×#41C2316yes—
119C2×#41C23112yes—
120C2×#41C23112yes—
121C2×#41C23112yes—
122C22×Q8C2312yes—
123C22×D4C2316yes—
124C22×D4C2316yes—
125C22×D4C2316yes—
126C22×D4C23112yes—
127C22×D4C23112yes—
128C22×D4C23124yes—
129C22×D4C23112yes—
130C22×D4C2316yes—
131C22×D4C2316yes—
132C23×C4C23112yes—
133C4⋊Q8C2311yescharacteristic
134C4⋊Q8C2313yes—
135C4⋊Q8C2313yes—
136C4⋊Q8C2313yes—
137C4⋊Q8C2316yes—
138#127C2316yes—
139#127C2313yes—
140#127C2313yes—
141#127C2316yes—
142#127C2316yes—
143#127C23112yes—
144#127C23112yes—
145#127C23112yes—
146#127C2316yes—
147#126C23112yes—
148#126C23112yes—
149#126C23112yes—
150#125C2312yes—
151#125C2316yes—
152#125C2316yes—
153#125C2316yes—
154#124C2316yes—
155#124C2316yes—
156#124C2316yes—
157#124C23112yes—
158#124C2316yes—
159#124C2316yes—
160#124C23112yes—
161#124C23112yes—
162#123C23112yes—
163#123C23124yes—
164#123C23112yes—
165#123C23112yes—
166#123C23112yes—
167#123C23112yes—
168#123C23124yes—
169#123C23124yes—
170C22⋊Q8C2314yes—
171C22⋊Q8C2316yes—
172C22⋊Q8C23112yes—
173C22⋊Q8C23112yes—
174C22⋊Q8C2312yes—
175C22⋊Q8C2316yes—
176C22⋊Q8C2316yes—
177C22⋊Q8C2316yes—
178C22⋊Q8C23124yes—
179#121C23112yes—
180#121C2316yes—
181#121C2316yes—
182#121C23112yes—
183#121C2316yes—
184#121C23112yes—
185#121C23124yes—
186#121C23124yes—
187#121C23112yes—
188#121C23112yes—
189#121C23124yes—
190#121C23112yes—
191#121C23112yes—
192#121C2316yes—
193#121C2316yes—
194#121C23112yes—
195#121C23112yes—
196#121C23112yes—
197#121C23112yes—
198#121C2316yes—
199#121C2316yes—
200#121C2316yes—
201#121C2316yes—
202#121C23124yes—
203#121C23124yes—
204#121C23124yes—
205#120C23112yes—
206#120C23124yes—
207#120C23112yes—
208#120C23124yes—
209#120C23124yes—
210#120C23124yes—
211C4×Q8C2311yescharacteristic
212C4×Q8C2312yes—
213C4×Q8C2316yes—
214C4×Q8C2316yes—
215C4×Q8C2311yescharacteristic
216C4×Q8C2311yescharacteristic
217C4×Q8C2312yes—
218C4×Q8C2313yes—
219C4×Q8C2313yes—
220C4×Q8C2313yes—
221C4×D4C2312yes—
222C4×D4C2313yes—
223C4×D4C2313yes—
224C4×D4C2316yes—
225C4×D4C23112yes—
226C4×D4C23112yes—
227C4×D4C23112yes—
228C4×D4C2316yes—
229C4×D4C2316yes—
230C4×D4C2316yes—
231C4×D4C23112yes—
232C4×D4C2316yes—
233C4×D4C2311yescharacteristic
234C4×D4C2316yes—
235C4×D4C2316yes—
236C4×D4C2316yes—
237C4×D4C2313yes—
238C4×D4C2313yes—
239C4×D4C2316yes—
240C4×D4C2313yes—
241C4×D4C2316yes—
242C4×D4C2316yes—
243C4×D4C23112yes—
244C4×D4C23112yes—
245C4×D4C23112yes—
246C4×D4C23112yes—
247C4×D4C23112yes—
248#117C2313yes—
249#117C2316yes—
250#117C2316yes—
251#117C2313yes—
252#117C2313yes—
253#117C2316yes—
254#117C2316yes—
255#117C23112yes—
256C2×C4⋊C4C2312yes—
257C2×C4⋊C4C2316yes—
258C2×C4⋊C4C2316yes—
259C2×C4⋊C4C23112yes—
260C2×C22⋊C4C23112yes—
261C2×C22⋊C4C23112yes—
262C2×C22⋊C4C23124yes—
263C2×C42C2316yes—
264C2×C42C2316yes—
265C2×C42C2313yes—
266C24C24112yes—
267C24C24112yes—
268#41—28no—
269#41—216no—
270#41—24no—
271#41—212no—
272#41—412no—
273#41—412no—
274#41—424no—
275#41—424no—
276#41—424no—
277#41—46no—
278#41—424no—
279C2×Q8—28no—
280C2×Q8C2411yescharacteristic
281C2×Q8C2416yes—
282C2×Q8C2413yes—
283C2×Q8C2412yes—
284C2×Q8C2416yes—
285C2×Q8C2411yescharacteristic
286C2×D4—224no—
287C2×D4—412no—
288C2×D4—412no—
289C2×D4—424no—
290C2×D4—424no—
291C2×D4—412no—
292C2×D4—412no—
293C2×D4—424no—
294C2×D4—448no—
295C2×D4C2416yes—
296C2×D4C2413yes—
297C2×D4C24112yes—
298C2×D4C2416yes—
299C2×D4C2413yes—
300C2×D4C2416yes—
301C2×D4C24112yes—
302C2×D4C2416yes—
303C2×D4C2416yes—
304C2×D4C24112yes—
305C2×D4C24112yes—
306C2×D4C24112yes—
307C2×D4C24124yes—
308C2×D4C24112yes—
309C2×D4C2416yes—
310C2×D4C24112yes—
311C2×D4C24124yes—
312C2×D4C2416yes—
313C2×D4C2411yescharacteristic
314C22×C4—424no—
315C22×C4C2412yes—
316C22×C4C2416yes—
317C22×C4C2416yes—
318C22×C4C2416yes—
319C22×C4C2413yes—
320C22×C4C2416yes—
321C22×C4C2413yes—
322C22×C4C2413yes—
323C22×C4C24112yes—
324C22×C4C2416yes—
325C22×C4C2414yes—
326C22×C4C24112yes—
327C22×C4C24124yes—
328C22×C4C24112yes—
329C4⋊C4C2412yes—
330C4⋊C4C2416yes—
331C4⋊C4C2411yescharacteristic
332C4⋊C4C2413yes—
333C4⋊C4C2416yes—
334C4⋊C4C2413yes—
335C4⋊C4C24112yes—
336C4⋊C4C2412yes—
337C4⋊C4C2414yes—
338C4⋊C4C24112yes—
339C4⋊C4C24112yes—
340C4⋊C4C2413yes—
341C4⋊C4C2412yes—
342C4⋊C4C2411yescharacteristic
343C4⋊C4C2414yes—
344C4⋊C4C2413yes—
345C4⋊C4C2416yes—
346C4⋊C4C2416yes—
347C4⋊C4C2416yes—
348C4⋊C4C2416yes—
349C4⋊C4C2416yes—
350C22⋊C4C2416yes—
351C22⋊C4C2414yes—
352C22⋊C4C24112yes—
353C22⋊C4C2416yes—
354C22⋊C4C2416yes—
355C22⋊C4C24112yes—
356C22⋊C4C24112yes—
357C22⋊C4C24124yes—
358C22⋊C4C24112yes—
359C22⋊C4C24112yes—
360C22⋊C4C24112yes—
361C22⋊C4C2412yes—
362C22⋊C4C2416yes—
363C22⋊C4C2416yes—
364C22⋊C4C24112yes—
365C22⋊C4C24112yes—
366C22⋊C4C2416yes—
367C22⋊C4C24112yes—
368C42C2412yes—
369C42C2413yes—
370C42C2413yes—
371C42C2411yescharacteristic
372C42C2413yes—
373C42C2416yes—
374C42C2416yes—
375C42C2416yes—
376C42C2416yes—
377C42C2416yes—
378C23—224no—
379C23—412no—
380C23—412no—
381C23—412no—
382C23—412no—
383C23—412no—
384C23C2513yes—
385C23C25112yes—
386C23C2516yes—
387C23C2512yes—
388Q8—212no—
389Q8—26no—
390Q8—22no—
391Q8—42no—
392Q8—46no—
393Q8H(5,𝔽2)14yescomplemented
394D4—224no—
395D4—212no—
396D4—212no—
397D4—26no—
398D4—224no—
399D4—212no—
400D4—212no—
401D4—26no—
402D4—412no—
403D4—46no—
404D4—46no—
405D4—412no—
406D4—412no—
407D4—424no—
408D4—412no—
409D4—412no—
410D4—424no—
411D4—46no—
412D4H(5,𝔽2)14yescomplemented
413C2×C4—212no—
414C2×C4—212no—
415C2×C4—212no—
416C2×C4—224no—
417C2×C4—212no—
418C2×C4—26no—
419C2×C4—26no—
420C2×C4—24no—
421C2×C4—26no—
422C2×C4—44no—
423C2×C4—412no—
424C2×C4—412no—
425C2×C4—424no—
426C2×C4—46no—
427C2×C4C2512yes—
428C2×C4C2512yes—
429C2×C4C2513yes—
430C2×C4C2512yes—
431C2×C4C2511yescharacteristic
432C2×C4C2513yes—
433C2×C4C2516yes—
434C2×C4C2516yes—
435C2×C4C2516yes—
436C2×C4C2514yes—
437C2×C4C2513yes—
438C2×C4C2511yescharacteristic
439C2×C4C2511yescharacteristic
440C22—22no—
441C22—22no—
442C22—26no—
443C22—212no—
444C22—23no—
445C22—23no—
446C22—26no—
447C22—212no—
448C22—23no—
449C22—26no—
450C22—212no—
451C22—824no—
452C22—1612no—
453C22C2×H(5,𝔽2)14yes—
454C22C2611yescharacteristic
455C4—22no—
456C4—24no—
457C4—23no—
458C4—21no—
459C4—26no—
460C4—26no—
461C4—26no—
462C4—23no—
463C4—22no—
464C4—22no—
465C4C2×H(5,𝔽2)16yes—
466C4C2×H(5,𝔽2)12yes—
467C4#58512yes—
468C2—24no—
469C2—46no—
470C2—412no—
471C2—43no—
472C2C22×H(5,𝔽2)11yescharacteristic
473C2C22×H(5,𝔽2)11yescharacteristic
474C2C2×#58511yescharacteristic
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. #3543C2semidirect
4. #3542C2semidirect
5. #3522C2semidirect
6. #3519C2semidirect
7. #3517C2semidirect
8. #3508C2semidirect
9. #3498C2semidirect
10. #3498C2semidirect
11. #3496C2semidirect
12. #3488C2semidirect
13. #3483C2semidirect
14. #3482C2semidirect
15. #3481C2semidirect
16. D4×#41C2semidirect
17. C2×#550C2semidirect
18. C2×H(5,𝔽2)C22semidirect
19. C2×H(5,𝔽2)C22semidirect
20. C22×#41C22semidirect
21. #562C22semidirect
22. #560C22semidirect
23. #559C22semidirect
24. #559C22semidirect
25. Q82C22semidirect
26. #557C22semidirect
27. #556C22semidirect
28. #555C22semidirect
29. #555C22semidirect
30. #554C22semidirect
31. #554C22semidirect
32. #554C22semidirect
33. #553C22semidirect
34. #553C22semidirect
35. #553C22semidirect
36. #552C22semidirect
37. #551C22semidirect
38. #550C22semidirect
39. #550C22semidirect
40. #550C22semidirect
41. #550C22semidirect
42. #550C22semidirect
43. #550C22semidirect
44. #550C22semidirect
45. #550C22semidirect
46. #550C22semidirect
47. #550C22semidirect
48. #550C22semidirect
49. D4×Q8C22semidirect
50. #548C22semidirect
51. #548C22semidirect
52. #547C22semidirect
53. #547C22semidirect
54. #547C22semidirect
55. #547C22semidirect
56. #546C22semidirect
57. #546C22semidirect
58. #546C22semidirect
59. #546C22semidirect
60. #546C22semidirect
61. #546C22semidirect
62. #546C22semidirect
63. D42C22semidirect
64. D42C22semidirect
65. D42C22semidirect
66. D42C22semidirect
67. D42C22semidirect
68. D42C22semidirect
69. D42C22semidirect
70. #542C22semidirect
71. #540C22semidirect
72. #540C22semidirect
73. #540C22semidirect
74. #538C22semidirect
75. #537C22semidirect
76. #537C22semidirect
77. #535C22semidirect
78. #535C22semidirect
79. #535C22semidirect
80. #535C22semidirect
81. #535C22semidirect
82. #534C22semidirect
83. #533C22semidirect
84. #532C22semidirect
85. #532C22semidirect
86. #532C22semidirect
87. #532C22semidirect
88. #532C22semidirect
89. #532C22semidirect
90. C2×#127C22semidirect
91. C2×#127C22semidirect
92. #529C22semidirect
93. #529C22semidirect
94. #529C22semidirect
95. #525C22semidirect
96. #525C22semidirect
97. C2×#121C22semidirect
98. C2×#121C22semidirect
99. #520C22semidirect
100. #518C22semidirect
101. C4×#41C22semidirect
102. C4×#41C22semidirect
103. C4×#41C22semidirect
104. C4×#41C22semidirect
105. C2×C4×Q8C22semidirect
106. C2×C4×D4C22semidirect
107. C2×C4×D4C22semidirect
393. Q8H(5,𝔽2)semidirect
412. D4H(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
#35432
#35422
#35222
#35192
#35172
#35082
#34982
#34962
#34882
#34832
#34822
#34812
D4×#412
C2×#5502