Finite Group Databasebeta

← #62887

Subgroups — #62887

Order 256.

3455 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3562C218yesmaximal, complemented
2#3561C218yesmaximal, complemented
3#3554C218yesmaximal, complemented
4#3552C218yesmaximal, complemented
5#3535C218yesmaximal, complemented
6#3533C218yesmaximal, complemented
7#3510C212yesmaximal, complemented
8#3509C212yesmaximal, complemented
9#3500C212yesmaximal, complemented
10#3499C212yesmaximal, complemented
11#3487C211yesmaximal, characteristic, complemented
12#3486C212yesmaximal, complemented
13#3485C211yesmaximal, characteristic, complemented
14#3455C212yesmaximal, complemented
15C2×#536C211yesmaximal, characteristic, complemented
16C2×2−1+4C2212yes—
17C2×2−1+4C2211yescharacteristic, complemented
18C2×H(5,𝔽2)C2211yescharacteristic
19C22×#41C2211yescharacteristic
20C23×Q8C2211yescharacteristic, complemented
21#563C2218yescomplemented
22#563C22116yescomplemented
23#562C22116yescomplemented
24#562C22116yescomplemented
25#560C2218yescomplemented
26#559C2218yescomplemented
27#559C2218yescomplemented
28#557C2218yescomplemented
29#556C2218yescomplemented
30#556C2218yescomplemented
31#554C22116yescomplemented
32#553C22116yescomplemented
33#552C22116yescomplemented
34#551C22116yescomplemented
35#551C2218yescomplemented
36#551C22116yescomplemented
37#550C2218yescomplemented
38D4×Q8C2218yescomplemented
39D4×Q8C2218yescomplemented
40D4×Q8C2218yescomplemented
41#548C2218yescomplemented
42#548C2218yescomplemented
43#548C2218yescomplemented
44#548C2218yescomplemented
45#547C2218yes—
46#547C22116yescomplemented
47#547C2218yescomplemented
48#546C2218yes—
49#546C22116yes—
50#546C2218yescomplemented
51#544C2218yescomplemented
52#543C2218yescomplemented
53#542C2214yescomplemented
54#542C2214yescomplemented
55#542C2218yescomplemented
56#542C22116yescomplemented
57#542C22116yescomplemented
58#541C2214yescomplemented
59#541C2218yescomplemented
60#540C2214yescomplemented
61#540C22116yescomplemented
62#539C2218yescomplemented
63#539C22116yescomplemented
64#539C22116yescomplemented
65#539C2218yescomplemented
66#538C2218yescomplemented
67#538C22116yescomplemented
68#538C2218yescomplemented
69#536C2212yes—
70#536C2212yes—
71#536C2218yescomplemented
72#536C2218yescomplemented
73#536C2212yescomplemented
74#536C2218yescomplemented
75#536C2214yescomplemented
76#536C2214yescomplemented
77#536C2214yescomplemented
78#535C2212yes—
79#535C2214yescomplemented
80#535C2214yescomplemented
81#534C2214yes—
82#533C2212yescomplemented
83#533C2218yescomplemented
84#532C2212yes—
85#532C2214yescomplemented
86#532C2214yescomplemented
87C2×C4⋊Q8C2212yescomplemented
88C2×C4⋊Q8C2214yescomplemented
89#529C2218yescomplemented
90#529C2218yescomplemented
91#529C2218yescomplemented
92C2×#124C2212yes—
93C2×#124C2214yescomplemented
94#525C2212yes—
95#525C2212yescomplemented
96#525C2218yescomplemented
97#525C2214yescomplemented
98C2×#123C2214yes—
99C2×#123C2214yescomplemented
100C2×C22⋊Q8C2214yescomplemented
101#519C2212yescomplemented
102#519C2211yescharacteristic, complemented
103#518C2211yescharacteristic
104C4×#41C2212yes—
105C4×#41C2212yescomplemented
106C2×#117C2211yescharacteristic, complemented
1072−1+4—48no—
1082−1+4—44no—
109H(5,𝔽2)—44no—
110C2×#41—44no—
111C2×#41C2312yes—
112C2×#41C2312yes—
113C2×#41C2314yes—
114C2×#41C2318yes—
115C2×#41C2314yes—
116C2×#41C2314yes—
117C2×#41C2314yes—
118C2×#41C2314yes—
119C2×#41C2318yes—
120C22×Q8—44no—
121C22×Q8C2312yes—
122C22×Q8C2311yescharacteristic
123C22×Q8C2312yes—
124C22×Q8C2314yes—
125C22×Q8C2314yes—
126C22×Q8C2314yes—
127C22×Q8C2318yes—
128C22×D4C2311yescharacteristic
129C22×D4C2314yes—
130C22×D4C2312yes—
131C22×D4C2314yes—
132C23×C4C2311yescharacteristic
133C23×C4C2312yes—
134C23×C4C2312yes—
135C4⋊Q8C2312yes—
136C4⋊Q8C2314yes—
137C4⋊Q8C2314yes—
138C4⋊Q8C2314yes—
139C4⋊Q8C2318yes—
140C4⋊Q8C2314yes—
141C4⋊Q8C23116yes—
142C4⋊Q8C2314yes—
143C4⋊Q8C2314yes—
144C4⋊Q8C23116yes—
145#127C2312yes—
146#127C2314yes—
147#127C2318yes—
148#126C2318yes—
149#126C23116yes—
150#126C2318yes—
151#126C2318yes—
152#126C2318yes—
153#126C23116yes—
154#126C23116yes—
155#126C23116yes—
156#125C2314yes—
157#125C2318yes—
158#125C2318yes—
159#125C23116yes—
160#125C23116yes—
161#125C23116yes—
162#124C2314yes—
163#124C2314yes—
164#124C2318yes—
165#124C2312yes—
166#124C2314yes—
167#124C2314yes—
168#124C23116yes—
169#124C2318yes—
170#124C2318yes—
171#124C2312yes—
172#124C2314yes—
173#124C2314yes—
174#124C2318yes—
175#124C23116yes—
176#124C23116yes—
177#123C2314yes—
178#123C2318yes—
179#123C23116yes—
180#123C23116yes—
181#123C2318yes—
182#123C2318yes—
183#123C2314yes—
184#123C2318yes—
185#123C2318yes—
186#123C2318yes—
187#123C2318yes—
188#123C2318yes—
189#123C23116yes—
190#123C2318yes—
191#123C2318yes—
192#123C23116yes—
193#123C23116yes—
194#123C23116yes—
195#123C23116yes—
196C22⋊Q8C2318yes—
197C22⋊Q8C2318yes—
198C22⋊Q8C23116yes—
199C22⋊Q8C2314yes—
200C22⋊Q8C23116yes—
201C22⋊Q8C2314yes—
202C22⋊Q8C2318yes—
203C22⋊Q8C2318yes—
204C22⋊Q8C2318yes—
205C22⋊Q8C2318yes—
206C22⋊Q8C2318yes—
207C22⋊Q8C2318yes—
208C22⋊Q8C2318yes—
209C22⋊Q8C23116yes—
210C22⋊Q8C23116yes—
211C22⋊Q8C23116yes—
212C22⋊Q8C23116yes—
213C22⋊Q8C23116yes—
214C22⋊Q8C23116yes—
215C22⋊Q8C23116yes—
216#121C2314yes—
217#121C2318yes—
218#121C23116yes—
219#121C2314yes—
220#121C2318yes—
221#121C2318yes—
222#121C2318yes—
223#121C2318yes—
224#121C23116yes—
225#121C23116yes—
226#121C23116yes—
227#121C23116yes—
228#121C23116yes—
229#120C2314yes—
230#120C2314yes—
231#120C2318yes—
232#120C2318yes—
233#120C23116yes—
234C4×Q8C2314yes—
235C4×Q8C2314yes—
236C4×Q8C2318yes—
237C4×Q8C2318yes—
238C4×Q8C2318yes—
239C4×Q8C2318yes—
240C4×Q8C23116yes—
241C4×D4C2318yes—
242C4×D4C2314yes—
243C4×D4C2318yes—
244C4×D4C2314yes—
245C4×D4C2318yes—
246C4×D4C2318yes—
247C4×D4C2318yes—
248C4×D4C23116yes—
249C4×D4C23116yes—
250C4×D4C23116yes—
251C4×D4C2318yes—
252C4×D4C23116yes—
253C4×D4C23116yes—
254C4×D4C23116yes—
255#117C2314yes—
256#117C2314yes—
257#117C2312yes—
258#117C2314yes—
259#117C2312yes—
260#117C2314yes—
261#117C2314yes—
262#117C2318yes—
263#117C23116yes—
264#117C23116yes—
265C2×C4⋊C4C2312yes—
266C2×C4⋊C4C2318yes—
267C2×C4⋊C4C2318yes—
268C2×C4⋊C4C23116yes—
269C2×C22⋊C4C2312yes—
270C2×C22⋊C4C2314yes—
271C2×C22⋊C4C2318yes—
272C2×C22⋊C4C2318yes—
273C2×C22⋊C4C23116yes—
274C2×C42C2312yes—
275C2×C42C2314yes—
276C2×C42C2314yes—
277C24C2412yes—
278C24C2411yescharacteristic
279C24C2414yes—
280#41—28no—
281#41—28no—
282#41—48no—
283#41—416no—
284#41—416no—
285#41—48no—
286#41—48no—
287#41—48no—
288#41—48no—
289C2×Q8—28no—
290C2×Q8—24no—
291C2×Q8—44no—
292C2×Q8—416no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—48no—
296C2×Q8C2414yes—
297C2×Q8C2414yes—
298C2×Q8C2412yes—
299C2×Q8C2414yes—
300C2×Q8C2412yes—
301C2×Q8C2418yes—
302C2×Q8C2418yes—
303C2×Q8C2418yes—
304C2×Q8C2414yes—
305C2×D4—24no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4C2414yes—
310C2×D4C2414yes—
311C2×D4C2414yes—
312C2×D4C2412yes—
313C2×D4C2412yes—
314C2×D4C2418yes—
315C2×D4C2418yes—
316C2×D4C24116yes—
317C2×D4C2418yes—
318C2×D4C2418yes—
319C2×D4C2418yes—
320C2×D4C2418yes—
321C2×D4C2414yes—
322C22×C4—24no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4C2414yes—
326C22×C4C2414yes—
327C22×C4C2414yes—
328C22×C4C2412yes—
329C22×C4C2411yescharacteristic
330C22×C4C2414yes—
331C22×C4C2414yes—
332C22×C4C2418yes—
333C22×C4C2418yes—
334C22×C4C2418yes—
335C22×C4C2418yes—
336C22×C4C2418yes—
337C22×C4C2418yes—
338C22×C4C2418yes—
339C22×C4C2414yes—
340C22×C4C2418yes—
341C22×C4C2411yescharacteristic
342C4⋊C4C24116yes—
343C4⋊C4C2418yes—
344C4⋊C4C2418yes—
345C4⋊C4C24116yes—
346C4⋊C4C2412yes—
347C4⋊C4C2414yes—
348C4⋊C4C2412yes—
349C4⋊C4C24116yes—
350C4⋊C4C24116yes—
351C4⋊C4C2418yes—
352C4⋊C4C2418yes—
353C4⋊C4C2414yes—
354C4⋊C4C2414yes—
355C4⋊C4C2418yes—
356C4⋊C4C2418yes—
357C4⋊C4C2418yes—
358C4⋊C4C24116yes—
359C4⋊C4C2418yes—
360C4⋊C4C24116yes—
361C4⋊C4C2418yes—
362C22⋊C4C2414yes—
363C22⋊C4C2418yes—
364C22⋊C4C24116yes—
365C22⋊C4C2418yes—
366C22⋊C4C2412yes—
367C22⋊C4C2414yes—
368C22⋊C4C2412yes—
369C22⋊C4C24116yes—
370C22⋊C4C24116yes—
371C22⋊C4C24116yes—
372C22⋊C4C24116yes—
373C22⋊C4C2418yes—
374C22⋊C4C2414yes—
375C22⋊C4C2418yes—
376C22⋊C4C2418yes—
377C22⋊C4C2418yes—
378C22⋊C4C2418yes—
379C22⋊C4C2418yes—
380C22⋊C4C2418yes—
381C22⋊C4C24116yes—
382C42C2412yes—
383C42C2412yes—
384C42C2414yes—
385C42C2414yes—
386C42C2418yes—
387C42C2414yes—
388C42C2418yes—
389C42C24116yes—
390C42C2418yes—
391C23—22no—
392C23—24no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—48no—
397C23C2514yes—
398C23C2514yes—
399C23C2512yes—
400C23C2514yes—
401C23C2511yescharacteristic
402Q8—28no—
403Q8—28no—
404Q8—28no—
405Q8—24no—
406Q8—24no—
407Q8—48no—
408Q8—48no—
409Q8—44no—
410Q8—48no—
411D4—28no—
412D4—28no—
413D4—28no—
414D4—24no—
415D4—24no—
416D4—48no—
417D4—48no—
418D4—48no—
419D4—48no—
420D4—416no—
421D4—48no—
422D4—44no—
423D4—48no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4H(5,𝔽2)14yes—
441C2×C4C2518yes—
442C2×C4C2514yes—
443C2×C4C2518yes—
444C2×C4C2514yes—
445C2×C4C2514yes—
446C2×C4C2518yes—
447C2×C4C2514yes—
448C2×C4C2514yes—
449C2×C4C2512yes—
450C2×C4C2512yes—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—28no—
457C22—28no—
458C22—28no—
459C22—24no—
460C22—44no—
461C22—84no—
462C22—164no—
463C22C2×H(5,𝔽2)12yes—
464C22C2611yescharacteristic
465C4—22no—
466C4—24no—
467C4—24no—
468C4—24no—
469C4—28no—
470C4—24no—
471C4—28no—
472C4—24no—
473C4—28no—
474C4C2×H(5,𝔽2)14yes—
475C2—22no—
476C2—44no—
477C2—42no—
478C2—44no—
479C2—44no—
480C2C22×H(5,𝔽2)11yescharacteristic
481C22−1+612yes—
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. #3554C2semidirect
4. #3552C2semidirect
5. #3535C2semidirect
6. #3533C2semidirect
7. #3510C2semidirect
8. #3509C2semidirect
9. #3500C2semidirect
10. #3499C2semidirect
11. #3487C2semidirect
12. #3486C2semidirect
13. #3485C2semidirect
14. #3455C2semidirect
15. C2×#536C2semidirect
17. C2×2−1+4C22semidirect
20. C23×Q8C22semidirect
21. #563C22semidirect
22. #563C22semidirect
23. #562C22semidirect
24. #562C22semidirect
25. #560C22semidirect
26. #559C22semidirect
27. #559C22semidirect
28. #557C22semidirect
29. #556C22semidirect
30. #556C22semidirect
31. #554C22semidirect
32. #553C22semidirect
33. #552C22semidirect
34. #551C22semidirect
35. #551C22semidirect
36. #551C22semidirect
37. #550C22semidirect
38. D4×Q8C22semidirect
39. D4×Q8C22semidirect
40. D4×Q8C22semidirect
41. #548C22semidirect
42. #548C22semidirect
43. #548C22semidirect
44. #548C22semidirect
46. #547C22semidirect
47. #547C22semidirect
50. #546C22semidirect
51. #544C22semidirect
52. #543C22semidirect
53. #542C22semidirect
54. #542C22semidirect
55. #542C22semidirect
56. #542C22semidirect
57. #542C22semidirect
58. #541C22semidirect
59. #541C22semidirect
60. #540C22semidirect
61. #540C22semidirect
62. #539C22semidirect
63. #539C22semidirect
64. #539C22semidirect
65. #539C22semidirect
66. #538C22semidirect
67. #538C22semidirect
68. #538C22semidirect
71. #536C22semidirect
72. #536C22semidirect
73. #536C22semidirect
74. #536C22semidirect
75. #536C22semidirect
76. #536C22semidirect
77. #536C22semidirect
79. #535C22semidirect
80. #535C22semidirect
82. #533C22semidirect
83. #533C22semidirect
85. #532C22semidirect
86. #532C22semidirect
87. C2×C4⋊Q8C22semidirect
88. C2×C4⋊Q8C22semidirect
89. #529C22semidirect
90. #529C22semidirect
91. #529C22semidirect
93. C2×#124C22semidirect
95. #525C22semidirect
96. #525C22semidirect
97. #525C22semidirect
99. C2×#123C22semidirect
100. C2×C22⋊Q8C22semidirect
101. #519C22semidirect
102. #519C22semidirect
105. C4×#41C22semidirect
106. C2×#117C22semidirect

Maximal subgroups

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

subgroupindex
#35622
#35612
#35542
#35522
#35352
#35332
#35102
#35092
#35002
#34992
#34872
#34862
#34852
#34552
C2×#5362