Finite Group Databasebeta

← #62888

Subgroups — #62888

Order 256.

3453 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3562C218yesmaximal, complemented
2#3560C218yesmaximal, complemented
3#3559C218yesmaximal, complemented
4#3552C218yesmaximal, complemented
5#3541C218yesmaximal, complemented
6#3539C214yesmaximal, complemented
7#3538C214yesmaximal, complemented
8#3512C212yesmaximal, complemented
9#3507C212yesmaximal, complemented
10#3499C212yesmaximal, complemented
11#3497C212yesmaximal, complemented
12#3489C214yesmaximal, complemented
13#3455C211yesmaximal, characteristic, complemented
14C2×#541C211yesmaximal, characteristic, complemented
15#3449C211yesmaximal, characteristic, complemented
16C2×2−1+4C2214yescomplemented
17C22×#41C2212yes—
18#563C22116yescomplemented
19#563C22116yescomplemented
20#562C22116yescomplemented
21#560C22116yescomplemented
22#559C2214yescomplemented
23#557C2214yescomplemented
24#556C2214yescomplemented
25#556C2218yescomplemented
26#554C2214yescomplemented
27#554C22116yescomplemented
28#553C2218yescomplemented
29#553C22116yescomplemented
30#553C22116yescomplemented
31#553C22116yescomplemented
32#552C2218yescomplemented
33#552C2218yescomplemented
34#551C22116yescomplemented
35#550C2214yescomplemented
36#550C2214yescomplemented
37D4×Q8C2218yescomplemented
38D4×Q8C2214yescomplemented
39D4×Q8C2214yescomplemented
40D4×Q8C2218yescomplemented
41#548C2218yes—
42#548C2218yescomplemented
43#548C2218yescomplemented
44#547C2218yes—
45#547C22116yescomplemented
46#547C2218yescomplemented
47#547C22116yescomplemented
48#546C22116yes—
49#546C2218yescomplemented
50#544C22116yescomplemented
51#542C2214yescomplemented
52#542C22116yescomplemented
53#541C2214yescomplemented
54#541C2214yescomplemented
55#541C2218yescomplemented
56#541C2212yescomplemented
57#541C2214yescomplemented
58#540C2212yescomplemented
59#540C2214yescomplemented
60#540C22116yescomplemented
61#539C2214yescomplemented
62#539C2218yescomplemented
63#539C22116yescomplemented
64#539C22116yescomplemented
65#539C22116yescomplemented
66#538C2214yes—
67#538C22116yescomplemented
68#538C22116yescomplemented
69#537C2214yes—
70#537C2218yes—
71#537C2214yescomplemented
72#537C2218yescomplemented
73#536C2214yes—
74#536C2218yes—
75#536C2214yescomplemented
76#536C2218yescomplemented
77#535C2214yes—
78#534C2214yes—
79#533C2211yescharacteristic, complemented
80#532C2212yes—
81#532C2212yes—
82#532C2211yescharacteristic, complemented
83#532C2218yescomplemented
84#532C2214yescomplemented
85C2×C4⋊Q8C2211yescharacteristic, complemented
86C2×C4⋊Q8C2214yescomplemented
87#529C2218yescomplemented
88#529C2218yescomplemented
89#529C2218yescomplemented
90C2×#126C2214yes—
91C2×#125C2211yescharacteristic, complemented
92C2×#125C2214yescomplemented
93#525C2214yes—
94#525C2214yes—
95#525C2212yes—
96#525C2218yescomplemented
97C2×C22⋊Q8C2212yes—
98C2×C22⋊Q8C2218yescomplemented
99#520C2214yescomplemented
100C4×#41C2212yescomplemented
101C2×C4×Q8C2212yescomplemented
102C2×#117C2211yescharacteristic
1032−1+4—416no—
104C2×#41—48no—
105C2×#41C2312yes—
106C2×#41C2314yes—
107C2×#41C2314yes—
108C2×#41C2318yes—
109C2×#41C2318yes—
110C2×#41C23116yes—
111C2×#41C2314yes—
112C2×#41C2314yes—
113C22×Q8C2312yes—
114C22×Q8C2318yes—
115C22×Q8C2318yes—
116C22×Q8C2312yes—
117C22×D4C2314yes—
118C22×D4C2312yes—
119C23×C4C2311yescharacteristic
120C23×C4C2314yes—
121C4⋊Q8C2312yes—
122C4⋊Q8C2314yes—
123C4⋊Q8C2318yes—
124C4⋊Q8C2314yes—
125C4⋊Q8C23116yes—
126C4⋊Q8C2312yes—
127C4⋊Q8C2311yescharacteristic
128C4⋊Q8C2318yes—
129C4⋊Q8C2312yes—
130C4⋊Q8C23116yes—
131#127C2312yes—
132#127C2312yes—
133#127C2311yescharacteristic
134#127C2318yes—
135#126C23116yes—
136#126C23116yes—
137#126C2318yes—
138#126C23116yes—
139#126C23116yes—
140#126C2318yes—
141#126C23116yes—
142#126C23116yes—
143#125C2314yes—
144#125C23116yes—
145#125C2314yes—
146#125C23116yes—
147#125C2318yes—
148#125C2318yes—
149#125C2312yes—
150#125C23116yes—
151#125C23116yes—
152#124C2314yes—
153#124C2318yes—
154#124C2318yes—
155#124C2314yes—
156#124C2318yes—
157#124C23116yes—
158#124C2312yes—
159#124C23116yes—
160#123C2318yes—
161#123C2318yes—
162#123C2314yes—
163#123C23116yes—
164#123C23116yes—
165#123C2318yes—
166#123C2318yes—
167#123C23116yes—
168#123C23116yes—
169#123C23116yes—
170#123C23116yes—
171#123C23116yes—
172#123C23116yes—
173#123C23116yes—
174C22⋊Q8C2318yes—
175C22⋊Q8C2318yes—
176C22⋊Q8C2314yes—
177C22⋊Q8C2318yes—
178C22⋊Q8C2318yes—
179C22⋊Q8C23116yes—
180C22⋊Q8C23116yes—
181C22⋊Q8C2318yes—
182C22⋊Q8C2318yes—
183C22⋊Q8C2318yes—
184C22⋊Q8C2314yes—
185C22⋊Q8C2318yes—
186C22⋊Q8C2318yes—
187C22⋊Q8C23116yes—
188C22⋊Q8C23116yes—
189C22⋊Q8C23116yes—
190C22⋊Q8C23116yes—
191C22⋊Q8C23116yes—
192C22⋊Q8C23116yes—
193#121C2314yes—
194#121C2318yes—
195#121C2318yes—
196#121C2318yes—
197#121C2314yes—
198#121C2318yes—
199#121C2318yes—
200#121C2314yes—
201#121C2318yes—
202#121C2314yes—
203#121C23116yes—
204#121C23116yes—
205#121C23116yes—
206#121C23116yes—
207#121C23116yes—
208#121C23116yes—
209#121C23116yes—
210#121C23116yes—
211#120C2318yes—
212#120C2318yes—
213#120C2314yes—
214#120C23116yes—
215C4×Q8C2314yes—
216C4×Q8C2314yes—
217C4×Q8C2314yes—
218C4×Q8C2314yes—
219C4×Q8C2314yes—
220C4×Q8C2318yes—
221C4×Q8C2314yes—
222C4×Q8C2318yes—
223C4×D4C2318yes—
224C4×D4C2314yes—
225C4×D4C23116yes—
226C4×D4C2318yes—
227C4×D4C2318yes—
228C4×D4C2314yes—
229C4×D4C2314yes—
230C4×D4C2318yes—
231C4×D4C23116yes—
232C4×D4C23116yes—
233C4×D4C2314yes—
234C4×D4C2318yes—
235C4×D4C23116yes—
236C4×D4C23116yes—
237C4×D4C23116yes—
238C4×D4C2318yes—
239C4×D4C2318yes—
240#117C2312yes—
241#117C2312yes—
242#117C2314yes—
243#117C2312yes—
244#117C2314yes—
245#117C2318yes—
246#117C23116yes—
247#117C23116yes—
248C2×C4⋊C4C2312yes—
249C2×C4⋊C4C2312yes—
250C2×C4⋊C4C2314yes—
251C2×C4⋊C4C2318yes—
252C2×C4⋊C4C2314yes—
253C2×C4⋊C4C23116yes—
254C2×C4⋊C4C2314yes—
255C2×C4⋊C4C2314yes—
256C2×C4⋊C4C23116yes—
257C2×C22⋊C4C2312yes—
258C2×C22⋊C4C2314yes—
259C2×C22⋊C4C23116yes—
260C2×C42C2311yescharacteristic
261C2×C42C2311yescharacteristic
262C2×C42C2314yes—
263C2×C42C2314yes—
264C24C2411yescharacteristic
265C24C2414yes—
266#41—216no—
267#41—28no—
268#41—48no—
269#41—48no—
270#41—432no—
271#41—48no—
272#41—416no—
273#41—416no—
274C2×Q8—28no—
275C2×Q8—44no—
276C2×Q8—416no—
277C2×Q8—416no—
278C2×Q8C2412yes—
279C2×Q8C2414yes—
280C2×Q8C2414yes—
281C2×Q8C2414yes—
282C2×Q8C2414yes—
283C2×Q8C2414yes—
284C2×Q8C24116yes—
285C2×D4—44no—
286C2×D4—48no—
287C2×D4C2414yes—
288C2×D4C2418yes—
289C2×D4C2412yes—
290C2×D4C2414yes—
291C2×D4C2418yes—
292C2×D4C2418yes—
293C2×D4C2414yes—
294C2×D4C2414yes—
295C2×D4C24116yes—
296C2×D4C2414yes—
297C2×D4C2418yes—
298C2×D4C24116yes—
299C2×D4C2414yes—
300C22×C4—24no—
301C22×C4—48no—
302C22×C4C2414yes—
303C22×C4C2414yes—
304C22×C4C2411yescharacteristic
305C22×C4C2418yes—
306C22×C4C2412yes—
307C22×C4C2412yes—
308C22×C4C2412yes—
309C22×C4C2412yes—
310C22×C4C2414yes—
311C22×C4C2418yes—
312C22×C4C24116yes—
313C22×C4C24116yes—
314C22×C4C24116yes—
315C22×C4C2418yes—
316C22×C4C2414yes—
317C22×C4C2411yescharacteristic
318C4⋊C4C2414yes—
319C4⋊C4C2414yes—
320C4⋊C4C24116yes—
321C4⋊C4C24116yes—
322C4⋊C4C2418yes—
323C4⋊C4C2418yes—
324C4⋊C4C24116yes—
325C4⋊C4C2414yes—
326C4⋊C4C2414yes—
327C4⋊C4C24116yes—
328C4⋊C4C24116yes—
329C4⋊C4C24116yes—
330C4⋊C4C2414yes—
331C4⋊C4C24116yes—
332C4⋊C4C2418yes—
333C4⋊C4C2418yes—
334C4⋊C4C2418yes—
335C4⋊C4C2418yes—
336C4⋊C4C2414yes—
337C4⋊C4C2414yes—
338C4⋊C4C2414yes—
339C4⋊C4C2414yes—
340C22⋊C4C24116yes—
341C22⋊C4C2418yes—
342C22⋊C4C2414yes—
343C22⋊C4C2414yes—
344C22⋊C4C24116yes—
345C22⋊C4C24116yes—
346C22⋊C4C2418yes—
347C22⋊C4C2418yes—
348C22⋊C4C2414yes—
349C22⋊C4C24116yes—
350C22⋊C4C2418yes—
351C22⋊C4C2418yes—
352C22⋊C4C2418yes—
353C22⋊C4C24116yes—
354C22⋊C4C24116yes—
355C22⋊C4C2418yes—
356C22⋊C4C2418yes—
357C42C2412yes—
358C42C2412yes—
359C42C2411yescharacteristic
360C42C2414yes—
361C42C2411yescharacteristic
362C42C2418yes—
363C42C2412yes—
364C42C2418yes—
365C42C2418yes—
366C42C24116yes—
367C23—22no—
368C23—44no—
369C23—42no—
370C23—48no—
371C23C2512yes—
372C23C2518yes—
373C23C2514yes—
374C23C2511yescharacteristic
375Q8—28no—
376Q8—28no—
377Q8—24no—
378Q8—28no—
379Q8—416no—
380Q8—44no—
381Q8—44no—
382D4—216no—
383D4—28no—
384D4—28no—
385D4—24no—
386D4—44no—
387D4—48no—
388D4—48no—
389D4—416no—
390D4—48no—
391D4—44no—
392D4—44no—
393D4—44no—
394D4—416no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—216no—
400C2×C4—28no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—416no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—416no—
409C2×C4—416no—
410C2×C4H(5,𝔽2)14yes—
411C2×C4C2512yes—
412C2×C4C2512yes—
413C2×C4C2518yes—
414C2×C4C2514yes—
415C2×C4C2518yes—
416C2×C4C2514yes—
417C2×C4C2518yes—
418C2×C4C2518yes—
419C2×C4C2512yes—
420C2×C4C2511yescharacteristic
421C2×C4C2511yescharacteristic
422C22—22no—
423C22—22no—
424C22—24no—
425C22—28no—
426C22—28no—
427C22—28no—
428C22—28no—
429C22—24no—
430C22—82no—
431C22—88no—
432C22C2×H(5,𝔽2)12yes—
433C22C2611yescharacteristic
434C4—22no—
435C4—28no—
436C4—24no—
437C4—28no—
438C4—28no—
439C4—24no—
440C4—28no—
441C4—22no—
442C4—22no—
443C4C2×H(5,𝔽2)12yes—
444C4C2×H(5,𝔽2)12yes—
445C2—22no—
446C2—44no—
447C2—48no—
448C2—42no—
449C2C22×H(5,𝔽2)11yescharacteristic
450C22−1+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. #3560C2semidirect
3. #3559C2semidirect
4. #3552C2semidirect
5. #3541C2semidirect
6. #3539C2semidirect
7. #3538C2semidirect
8. #3512C2semidirect
9. #3507C2semidirect
10. #3499C2semidirect
11. #3497C2semidirect
12. #3489C2semidirect
13. #3455C2semidirect
14. C2×#541C2semidirect
15. #3449C2semidirect
16. C2×2−1+4C22semidirect
18. #563C22semidirect
19. #563C22semidirect
20. #562C22semidirect
21. #560C22semidirect
22. #559C22semidirect
23. #557C22semidirect
24. #556C22semidirect
25. #556C22semidirect
26. #554C22semidirect
27. #554C22semidirect
28. #553C22semidirect
29. #553C22semidirect
30. #553C22semidirect
31. #553C22semidirect
32. #552C22semidirect
33. #552C22semidirect
34. #551C22semidirect
35. #550C22semidirect
36. #550C22semidirect
37. D4×Q8C22semidirect
38. D4×Q8C22semidirect
39. D4×Q8C22semidirect
40. D4×Q8C22semidirect
42. #548C22semidirect
43. #548C22semidirect
45. #547C22semidirect
46. #547C22semidirect
47. #547C22semidirect
49. #546C22semidirect
50. #544C22semidirect
51. #542C22semidirect
52. #542C22semidirect
53. #541C22semidirect
54. #541C22semidirect
55. #541C22semidirect
56. #541C22semidirect
57. #541C22semidirect
58. #540C22semidirect
59. #540C22semidirect
60. #540C22semidirect
61. #539C22semidirect
62. #539C22semidirect
63. #539C22semidirect
64. #539C22semidirect
65. #539C22semidirect
67. #538C22semidirect
68. #538C22semidirect
71. #537C22semidirect
72. #537C22semidirect
75. #536C22semidirect
76. #536C22semidirect
79. #533C22semidirect
82. #532C22semidirect
83. #532C22semidirect
84. #532C22semidirect
85. C2×C4⋊Q8C22semidirect
86. C2×C4⋊Q8C22semidirect
87. #529C22semidirect
88. #529C22semidirect
89. #529C22semidirect
91. C2×#125C22semidirect
92. C2×#125C22semidirect
96. #525C22semidirect
98. C2×C22⋊Q8C22semidirect
99. #520C22semidirect
100. C4×#41C22semidirect
101. C2×C4×Q8C22semidirect

Maximal subgroups

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

subgroupindex
#35622
#35602
#35592
#35522
#35412
#35392
#35382
#35122
#35072
#34992
#34972
#34892
#34552
C2×#5412
#34492