Finite Group Databasebeta

← #62880

Subgroups — #62880

Order 256.

3681 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3562C218yesmaximal, complemented
2#3543C218yesmaximal, complemented
3#3542C218yesmaximal, complemented
4#3539C214yesmaximal, complemented
5#3537C218yesmaximal, complemented
6#3537C214yesmaximal, complemented
7#3534C218yesmaximal, complemented
8#3512C212yesmaximal, complemented
9#3506C212yesmaximal, complemented
10#3498C212yesmaximal, complemented
11#3496C212yesmaximal, complemented
12#3488C214yesmaximal, complemented
13#3455C211yesmaximal, characteristic, complemented
14C2×#540C211yesmaximal, characteristic, complemented
15#3449C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2214yescomplemented
17C22×#41C2212yescomplemented
18#562C22116yescomplemented
19#560C22116yescomplemented
20#560C22116yescomplemented
21#560C22116yescomplemented
22#559C2218yescomplemented
23#556C2214yescomplemented
24#555C2214yescomplemented
25#553C2218yescomplemented
26#553C22116yescomplemented
27#553C2218yescomplemented
28#553C22116yescomplemented
29#552C2218yescomplemented
30#551C22116yescomplemented
31#550C2218yescomplemented
32#550C2218yescomplemented
33#550C2214yescomplemented
34#548C2218yescomplemented
35#547C2214yescomplemented
36#547C2214yescomplemented
37#547C2214yescomplemented
38#546C22116yescomplemented
39#546C2218yescomplemented
40#546C2218yescomplemented
41#546C2218yescomplemented
42#546C22116yescomplemented
43D42C22116yescomplemented
44D42C2218yescomplemented
45D42C2214yescomplemented
46D42C2214yescomplemented
47D42C22116yescomplemented
48D42C2218yescomplemented
49D42C22116yescomplemented
50#542C2214yescomplemented
51#541C2212yescomplemented
52#541C2214yescomplemented
53#540C2214yescomplemented
54#540C2214yescomplemented
55#540C2218yescomplemented
56#540C2212yescomplemented
57#540C2214yescomplemented
58#540C22116yescomplemented
59#540C22116yescomplemented
60#539C2214yescomplemented
61#539C22116yescomplemented
62#538C2214yescomplemented
63#538C2218yescomplemented
64#538C22116yescomplemented
65#538C22116yescomplemented
66#537C2214yescomplemented
67#537C2214yescomplemented
68#537C2218yescomplemented
69#536C2214yescomplemented
70#535C2214yescomplemented
71#535C2214yescomplemented
72#535C2218yescomplemented
73#535C2218yescomplemented
74#535C22116yescomplemented
75#534C2214yescomplemented
76#534C2218yescomplemented
77#534C22116yescomplemented
78#534C22116yescomplemented
79#533C2211yescharacteristic, complemented
80#532C2212yescomplemented
81#532C2212yescomplemented
82#532C2211yescharacteristic, complemented
83#532C2214yescomplemented
84C2×#127C2211yescharacteristic, complemented
85C2×#127C2214yescomplemented
86#529C2218yescomplemented
87#529C2218yescomplemented
88C2×#125C2211yescharacteristic, complemented
89#525C2214yescomplemented
90#525C2214yescomplemented
91#525C2212yescomplemented
92#525C2218yescomplemented
93#525C2218yescomplemented
94C2×#123C2214yescomplemented
95C2×#121C2212yescomplemented
96C2×#121C2214yescomplemented
97C2×#121C2218yescomplemented
98C2×#121C2218yescomplemented
99#520C2214yescomplemented
100C4×#41C2212yescomplemented
101C2×C4×D4C2212yescomplemented
102C2×#117C2211yescharacteristic, complemented
103H(5,𝔽2)—416no—
104C2×#41—48no—
105C2×#41C2312yes—
106C2×#41C2314yes—
107C2×#41C2314yes—
108C2×#41C2318yes—
109C2×#41C2318yes—
110C2×#41C2314yes—
111C2×#41C2314yes—
112C2×#41C2318yes—
113C22×Q8C2312yes—
114C22×D4C2312yes—
115C22×D4C23116yes—
116C22×D4C2318yes—
117C22×D4C2318yes—
118C22×D4C2314yes—
119C22×D4C2312yes—
120C22×D4C2314yes—
121C22×D4C2314yes—
122C22×D4C2318yes—
123C22×D4C23116yes—
124C22×D4C23116yes—
125C23×C4C2311yescharacteristic
126C23×C4C2314yes—
127C23×C4C2314yes—
128C4⋊Q8C2312yes—
129C4⋊Q8C2312yes—
130C4⋊Q8C2311yescharacteristic
131C4⋊Q8C2312yes—
132#127C2314yes—
133#127C2312yes—
134#127C2318yes—
135#127C23116yes—
136#127C2314yes—
137#127C2312yes—
138#127C2311yescharacteristic
139#127C23116yes—
140#126C23116yes—
141#126C2318yes—
142#126C2318yes—
143#126C23116yes—
144#126C23116yes—
145#125C2314yes—
146#125C2314yes—
147#125C2318yes—
148#125C2312yes—
149#125C23116yes—
150#124C2314yes—
151#124C2318yes—
152#124C2318yes—
153#124C2314yes—
154#124C2312yes—
155#124C23116yes—
156#124C23116yes—
157#123C23116yes—
158#123C23116yes—
159#123C2318yes—
160#123C2318yes—
161#123C23116yes—
162#123C2314yes—
163#123C23116yes—
164#123C2318yes—
165#123C2318yes—
166#123C23116yes—
167#123C23116yes—
168#123C23116yes—
169#123C23116yes—
170#123C23116yes—
171C22⋊Q8C2314yes—
172C22⋊Q8C2314yes—
173C22⋊Q8C2318yes—
174C22⋊Q8C2318yes—
175C22⋊Q8C2318yes—
176C22⋊Q8C2318yes—
177C22⋊Q8C2318yes—
178C22⋊Q8C2318yes—
179C22⋊Q8C23116yes—
180#121C2314yes—
181#121C2318yes—
182#121C2314yes—
183#121C2314yes—
184#121C23116yes—
185#121C23116yes—
186#121C23116yes—
187#121C2318yes—
188#121C2318yes—
189#121C23116yes—
190#121C2318yes—
191#121C23116yes—
192#121C2318yes—
193#121C2318yes—
194#121C23116yes—
195#121C2314yes—
196#121C2318yes—
197#121C2318yes—
198#121C2318yes—
199#121C2318yes—
200#121C2318yes—
201#121C2318yes—
202#121C2318yes—
203#121C23116yes—
204#121C23116yes—
205#121C23116yes—
206#121C23116yes—
207#121C23116yes—
208#121C23116yes—
209#121C23116yes—
210#121C23116yes—
211#120C2318yes—
212#120C2318yes—
213#120C23116yes—
214#120C23116yes—
215#120C2314yes—
216#120C23116yes—
217#120C23116yes—
218#120C23116yes—
219#120C23116yes—
220#120C23116yes—
221#120C23116yes—
222C4×Q8C2314yes—
223C4×Q8C2318yes—
224C4×Q8C2314yes—
225C4×D4C2314yes—
226C4×D4C2314yes—
227C4×D4C2314yes—
228C4×D4C2314yes—
229C4×D4C2318yes—
230C4×D4C2314yes—
231C4×D4C23116yes—
232C4×D4C2318yes—
233C4×D4C2314yes—
234C4×D4C2318yes—
235C4×D4C2314yes—
236C4×D4C2318yes—
237C4×D4C23116yes—
238C4×D4C23116yes—
239C4×D4C23116yes—
240C4×D4C2314yes—
241C4×D4C23116yes—
242C4×D4C23116yes—
243#117C2312yes—
244#117C2312yes—
245#117C2314yes—
246#117C2312yes—
247#117C2314yes—
248#117C2318yes—
249#117C2318yes—
250#117C23116yes—
251C2×C4⋊C4C2312yes—
252C2×C4⋊C4C2312yes—
253C2×C4⋊C4C2318yes—
254C2×C4⋊C4C2314yes—
255C2×C4⋊C4C2318yes—
256C2×C22⋊C4C2312yes—
257C2×C22⋊C4C2314yes—
258C2×C22⋊C4C2314yes—
259C2×C22⋊C4C23116yes—
260C2×C22⋊C4C23116yes—
261C2×C42C2311yescharacteristic
262C2×C42C2311yescharacteristic
263C2×C42C2314yes—
264C24C2411yescharacteristic
265C24C2414yes—
266C24C24116yes—
267C24C2414yes—
268#41—216no—
269#41—28no—
270#41—48no—
271#41—48no—
272#41—48no—
273#41—416no—
274#41—416no—
275#41—88no—
276C2×Q8—44no—
277C2×Q8C2412yes—
278C2×Q8C2414yes—
279C2×Q8C2414yes—
280C2×Q8C2414yes—
281C2×D4—28no—
282C2×D4—44no—
283C2×D4—416no—
284C2×D4—416no—
285C2×D4—48no—
286C2×D4—432no—
287C2×D4—84no—
288C2×D4—88no—
289C2×D4—84no—
290C2×D4—816no—
291C2×D4—816no—
292C2×D4C2418yes—
293C2×D4C2414yes—
294C2×D4C2412yes—
295C2×D4C2414yes—
296C2×D4C2414yes—
297C2×D4C2414yes—
298C2×D4C2414yes—
299C2×D4C2414yes—
300C2×D4C24116yes—
301C2×D4C24116yes—
302C2×D4C2418yes—
303C2×D4C2414yes—
304C2×D4C24116yes—
305C2×D4C24116yes—
306C2×D4C2418yes—
307C2×D4C24116yes—
308C2×D4C2414yes—
309C2×D4C2418yes—
310C2×D4C24116yes—
311C2×D4C24116yes—
312C22×C4—24no—
313C22×C4—48no—
314C22×C4—84no—
315C22×C4C2414yes—
316C22×C4C2411yescharacteristic
317C22×C4C2418yes—
318C22×C4C2412yes—
319C22×C4C2412yes—
320C22×C4C2412yes—
321C22×C4C2412yes—
322C22×C4C2418yes—
323C22×C4C2414yes—
324C22×C4C2418yes—
325C22×C4C2418yes—
326C22×C4C24116yes—
327C22×C4C24116yes—
328C22×C4C2418yes—
329C22×C4C2414yes—
330C22×C4C2411yescharacteristic
331C4⋊C4C2414yes—
332C4⋊C4C2414yes—
333C4⋊C4C2414yes—
334C4⋊C4C2414yes—
335C4⋊C4C2414yes—
336C4⋊C4C2414yes—
337C4⋊C4C24116yes—
338C4⋊C4C24116yes—
339C4⋊C4C2414yes—
340C4⋊C4C24116yes—
341C4⋊C4C2414yes—
342C4⋊C4C2414yes—
343C4⋊C4C2418yes—
344C4⋊C4C2418yes—
345C4⋊C4C2418yes—
346C4⋊C4C2418yes—
347C22⋊C4C24116yes—
348C22⋊C4C2418yes—
349C22⋊C4C2418yes—
350C22⋊C4C24116yes—
351C22⋊C4C2418yes—
352C22⋊C4C2414yes—
353C22⋊C4C2414yes—
354C22⋊C4C24116yes—
355C22⋊C4C24116yes—
356C22⋊C4C2418yes—
357C22⋊C4C2418yes—
358C22⋊C4C2418yes—
359C22⋊C4C2414yes—
360C22⋊C4C2418yes—
361C22⋊C4C24116yes—
362C22⋊C4C2418yes—
363C22⋊C4C2418yes—
364C22⋊C4C24116yes—
365C22⋊C4C2418yes—
366C42C2412yes—
367C42C2411yescharacteristic
368C42C2411yescharacteristic
369C42C2412yes—
370C42C2412yes—
371C42C2414yes—
372C42C2418yes—
373C42C24116yes—
374C23—22no—
375C23—28no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—44no—
380C23—48no—
381C23—416no—
382C23—416no—
383C23—416no—
384C23C2514yes—
385C23C2512yes—
386C23C2518yes—
387C23C2518yes—
388C23C2511yescharacteristic
389Q8—28no—
390Q8—24no—
391Q8—44no—
392Q8—44no—
393D4—28no—
394D4—216no—
395D4—28no—
396D4—28no—
397D4—24no—
398D4—28no—
399D4—416no—
400D4—416no—
401D4—44no—
402D4—48no—
403D4—44no—
404D4—416no—
405D4—44no—
406D4—416no—
407D4—416no—
408D4—48no—
409D4—48no—
410D4—416no—
411D4—416no—
412D4—44no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—216no—
417C2×C4—28no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—416no—
425C2×C4—42no—
426C2×C4—48no—
427C2×C4—416no—
428C2×C4H(5,𝔽2)14yes—
429C2×C4C2518yes—
430C2×C4C2514yes—
431C2×C4C2512yes—
432C2×C4C2512yes—
433C2×C4C2518yes—
434C2×C4C2518yes—
435C2×C4C2514yes—
436C2×C4C2512yes—
437C2×C4C2511yescharacteristic
438C2×C4C2511yescharacteristic
439C22—22no—
440C22—22no—
441C22—28no—
442C22—24no—
443C22—28no—
444C22—28no—
445C22—28no—
446C22—28no—
447C22—24no—
448C22—28no—
449C22—28no—
450C22—82no—
451C22—88no—
452C22—88no—
453C22—1616no—
454C22C2×H(5,𝔽2)12yes—
455C22C2611yescharacteristic
456C4—22no—
457C4—24no—
458C4—28no—
459C4—22no—
460C4—28no—
461C4—22no—
462C4—24no—
463C4—28no—
464C4C2×H(5,𝔽2)12yes—
465C4C2×H(5,𝔽2)12yes—
466C2—22no—
467C2—42no—
468C2—48no—
469C2—48no—
470C2—44no—
471C2C22×H(5,𝔽2)11yescharacteristic
472C2H(7,𝔽2)12yes—
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. #3543C2semidirect
3. #3542C2semidirect
4. #3539C2semidirect
5. #3537C2semidirect
6. #3537C2semidirect
7. #3534C2semidirect
8. #3512C2semidirect
9. #3506C2semidirect
10. #3498C2semidirect
11. #3496C2semidirect
12. #3488C2semidirect
13. #3455C2semidirect
14. C2×#540C2semidirect
15. #3449C2semidirect
16. C2×H(5,𝔽2)C22semidirect
17. C22×#41C22semidirect
18. #562C22semidirect
19. #560C22semidirect
20. #560C22semidirect
21. #560C22semidirect
22. #559C22semidirect
23. #556C22semidirect
24. #555C22semidirect
25. #553C22semidirect
26. #553C22semidirect
27. #553C22semidirect
28. #553C22semidirect
29. #552C22semidirect
30. #551C22semidirect
31. #550C22semidirect
32. #550C22semidirect
33. #550C22semidirect
34. #548C22semidirect
35. #547C22semidirect
36. #547C22semidirect
37. #547C22semidirect
38. #546C22semidirect
39. #546C22semidirect
40. #546C22semidirect
41. #546C22semidirect
42. #546C22semidirect
43. D42C22semidirect
44. D42C22semidirect
45. D42C22semidirect
46. D42C22semidirect
47. D42C22semidirect
48. D42C22semidirect
49. D42C22semidirect
50. #542C22semidirect
51. #541C22semidirect
52. #541C22semidirect
53. #540C22semidirect
54. #540C22semidirect
55. #540C22semidirect
56. #540C22semidirect
57. #540C22semidirect
58. #540C22semidirect
59. #540C22semidirect
60. #539C22semidirect
61. #539C22semidirect
62. #538C22semidirect
63. #538C22semidirect
64. #538C22semidirect
65. #538C22semidirect
66. #537C22semidirect
67. #537C22semidirect
68. #537C22semidirect
69. #536C22semidirect
70. #535C22semidirect
71. #535C22semidirect
72. #535C22semidirect
73. #535C22semidirect
74. #535C22semidirect
75. #534C22semidirect
76. #534C22semidirect
77. #534C22semidirect
78. #534C22semidirect
79. #533C22semidirect
80. #532C22semidirect
81. #532C22semidirect
82. #532C22semidirect
83. #532C22semidirect
84. C2×#127C22semidirect
85. C2×#127C22semidirect
86. #529C22semidirect
87. #529C22semidirect
88. C2×#125C22semidirect
89. #525C22semidirect
90. #525C22semidirect
91. #525C22semidirect
92. #525C22semidirect
93. #525C22semidirect
94. C2×#123C22semidirect
95. C2×#121C22semidirect
96. C2×#121C22semidirect
97. C2×#121C22semidirect
98. C2×#121C22semidirect
99. #520C22semidirect
100. C4×#41C22semidirect
101. C2×C4×D4C22semidirect
102. C2×#117C22semidirect

Maximal subgroups

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

subgroupindex
#35622
#35432
#35422
#35392
#35372
#35342
#35122
#35062
#34982
#34962
#34882
#34552
C2×#5402
#34492