Finite Group Databasebeta

← #61673

Subgroups — #61673

Order 256.

1080 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3500C211yesmaximal, characteristic, complemented
3#3321C214yesmaximal, complemented
4#3319C214yesmaximal, complemented
5#3265C214yesmaximal, complemented
6#3264C214yesmaximal, complemented
7#3213C212yesmaximal, complemented
8#3200C212yesmaximal, complemented
9#3199C212yesmaximal, complemented
10#3115C212yesmaximal, complemented
11#3010C212yesmaximal, complemented
12#3008C212yesmaximal, complemented
13#2980C211yesmaximal, characteristic, complemented
14C2×2−1+4C2211yescharacteristic, complemented
15C2×H(5,𝔽2)C2211yescharacteristic, complemented
16C2×AGL(1,ℤ/8)C2212yescomplemented
17C2×AGL(1,ℤ/8)C2212yescomplemented
18C22×SD16C2212yescomplemented
19C22×D8C2212yescomplemented
20H(3,𝔽4)—21no—
21H(3,𝔽4)—21no—
22#555—21no—
23#555C2214yescomplemented
24#548C2214yescomplemented
25#546—24no—
26#546C2214yescomplemented
27D42—21no—
28D42C2214yescomplemented
29#536—21no—
30#536C2211yescharacteristic, complemented
31#535—21no—
32#535—24no—
33#535C2212yes—
34#535C2211yescharacteristic, complemented
35#535C2212yescomplemented
36#532—21no—
37#532C2211yescharacteristic
38#532C2212yescomplemented
39C2×#124—21no—
40C2×#124C2211yescharacteristic, complemented
41C4×#41C2211yescharacteristic, complemented
42#496C2214yescomplemented
43#489C2214yescomplemented
44#485C2214yescomplemented
45#482C2214yescomplemented
46#482C2214yescomplemented
47#475C2212yescomplemented
48#474C2212yescomplemented
49#469C2214yescomplemented
50#468C2214yescomplemented
51#466C2214yescomplemented
52#465C2214yescomplemented
53#463C2214yescomplemented
54#463C2214yescomplemented
55#460C2212yescomplemented
56#459C2212yescomplemented
57#450C2214yescomplemented
58#449C2214yescomplemented
59#449C2214yescomplemented
60#449C2214yescomplemented
61#448C2214yescomplemented
62#448C2214yescomplemented
63#447C2214yescomplemented
64#447C2214yescomplemented
65#447C2214yescomplemented
66#442C2214yescomplemented
67#440C2214yescomplemented
68#435C2214yescomplemented
69#435C2214yescomplemented
70#432C2212yescomplemented
71#419C2212yescomplemented
72#418C2212yescomplemented
73#416C2212yescomplemented
74#416C2212yescomplemented
75#408C2212yescomplemented
76#408C2212yescomplemented
772−1+4—42no—
782−1+4—42no—
79H(5,𝔽2)—42no—
80H(5,𝔽2)—42no—
81C2×#41—21no—
82C2×#41—21no—
83C2×#41—22no—
84C2×#41—22no—
85C2×#41C2311yescharacteristic
86C2×#41C2312yes—
87C2×#41C2312yes—
88C2×#41C2311yescharacteristic
89C2×#41C2311yescharacteristic
90C22×Q8—21no—
91C22×Q8C2312yes—
92C22×Q8C2311yescharacteristic
93C22×D4—21no—
94C22×D4—22no—
95C22×D4—24no—
96C22×D4C2312yes—
97C22×D4C2311yescharacteristic
98C22×D4C2312yes—
99C22×D4C2312yes—
100AGL(1,ℤ/8)—44no—
101AGL(1,ℤ/8)—44no—
102C2×SD16—44no—
103C2×SD16C2314yes—
104C2×SD16C2314yes—
105C2×SD16C2314yes—
106C2×D8—44no—
107C2×D8C2314yes—
108C2×D8C2314yes—
109C2×D8C2314yes—
110C2×M16C2312yes—
111C2×M16C2312yes—
112C22×C8C2312yes—
113C22×C8C2312yes—
114C4⋊Q8—21no—
115C4⋊Q8D411yescharacteristic
116C4⋊Q8C2312yes—
117#127—21no—
118#127—22no—
119#127—22no—
120#127D411yescharacteristic
121#127C2312yes—
122#124—21no—
123#124—22no—
124#124—22no—
125#124—21no—
126#124—24no—
127#124—24no—
128#124—24no—
129#124—21no—
130#124—21no—
131#124—22no—
132#124D411yescharacteristic
133#124D411yescharacteristic
134#124C2314yes—
135#123—22no—
136#123—24no—
137C22⋊Q8—24no—
138C22⋊Q8C2314yes—
139#121—24no—
140#121—24no—
141#121—22no—
142#121—24no—
143#121—24no—
144#121D414yes—
145#121C2314yes—
146#121C2314yes—
147#121C2314yes—
148#121C2314yes—
149#120—22no—
150#120—22no—
151#120—24no—
152#120—22no—
153#120—22no—
154#120—22no—
155#120—24no—
156#120—24no—
157C4×Q8C2312yes—
158C4×D4—22no—
159C4×D4—24no—
160C4×D4—22no—
161C4×D4C2312yes—
162C4×D4C2314yes—
163C4×D4C2314yes—
164#117—24no—
165#117—21no—
166#117C2311yescharacteristic
167#117C2312yes—
168#117C2312yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C42—21no—
172C2×C42C2311yescharacteristic
173#107C2314yes—
174#106C2314yes—
175C4⋊C8C2312yes—
176C4⋊C8C2312yes—
177#103C2314yes—
178#103C2314yes—
179#103C2314yes—
180#102C2314yes—
181#102C2314yes—
182#102C2314yes—
183#102C2314yes—
184#102C2314yes—
185C22⋊C8C2314yes—
186C22⋊C8C2314yes—
187C22⋊C8C2314yes—
188#97C2314yes—
189C24—22no—
190C24—21no—
191C24—22no—
192C24—22no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—24no—
197#41—41no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—42no—
202#41—42no—
203#41—44no—
204#41—44no—
205#41—48no—
206#41—84no—
207C2×Q8—22no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—44no—
215C2×Q8C2×D411yescharacteristic
216C2×Q8C2×D411yescharacteristic
217C2×Q8C2412yes—
218C2×Q8C2411yescharacteristic
219C2×D4—21no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—44no—
241C2×D4—48no—
242C2×D4—88no—
243C2×D4C2×D414yes—
244C2×D4C2×D412yes—
245C2×D4C2×D411yescharacteristic
246C2×D4C2×D411yescharacteristic
247C2×D4C2×D412yes—
248C2×D4C2412yes—
249C2×D4C2414yes—
250C2×D4C2411yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4C2×D412yes—
259C22×C4C2411yescharacteristic
260C22×C4C2412yes—
261C22×C4C2412yes—
262SD16—28no—
263SD16—44no—
264SD16—44no—
265D8—28no—
266D8—44no—
267D8—44no—
268M16—42no—
269M16—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8C2414yes—
273C2×C8C2414yes—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4C2×D414yes—
278C4⋊C4C2412yes—
279C4⋊C4C2414yes—
280C22⋊C4—24no—
281C22⋊C4—21no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4C2×D414yes—
293C22⋊C4C2×D414yes—
294C42—21no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42C2×D411yescharacteristic
299C42C2×D411yescharacteristic
300C42C2412yes—
301C23—21no—
302C23—22no—
303C23—24no—
304C23—44no—
305C23—42no—
306C23—82no—
307C23—82no—
308C23—84no—
309C23—84no—
310C23—82no—
311C23—84no—
312C23C22×D412yes—
313C23C22×D412yes—
314C23C22×D411yescharacteristic
315Q8—21no—
316Q8—22no—
317Q8—22no—
318Q8—41no—
319Q8—41no—
320Q8—44no—
321Q8—42no—
322Q8—82no—
323Q8H(5,𝔽2)12yes—
324D4—24no—
325D4—21no—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—24no—
330D4—41no—
331D4—42no—
332D4—44no—
333D4—41no—
334D4—42no—
335D4—44no—
336D4—44no—
337D4—42no—
338D4—44no—
339D4—44no—
340D4—82no—
341D4—82no—
342D4—82no—
343D4—84no—
344D4—84no—
345D4—84no—
346D4—84no—
347D4H(5,𝔽2)12yes—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4C22×D411yescharacteristic
367C2×C4C22×D412yes—
368C2×C4C22×D412yes—
369C2×C4C22×D412yes—
370C2×C4C22×D412yes—
371C2×C4C2511yescharacteristic
372C8—24no—
373C8—24no—
374C22—22no—
375C22—24no—
376C22—24no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—44no—
383C22—44no—
384C22—44no—
385C22—42no—
386C22—44no—
387C22—164no—
388C22—162no—
389C22—164no—
390C22—164no—
391C22C23×D411yescharacteristic
392C4—22no—
393C4—22no—
394C4—21no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—44no—
399C4—42no—
400C4—42no—
401C4C2×H(5,𝔽2)11yescharacteristic
402C4C2×H(5,𝔽2)11yescharacteristic
403C2—41no—
404C2—42no—
405C2—42no—
406C2—42no—
407C2—84no—
408C2—82no—
409C2#348411yescharacteristic
410C2#358512yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #532 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3549C2semidirect
2. #3500C2semidirect
3. #3321C2semidirect
4. #3319C2semidirect
5. #3265C2semidirect
6. #3264C2semidirect
7. #3213C2semidirect
8. #3200C2semidirect
9. #3199C2semidirect
10. #3115C2semidirect
11. #3010C2semidirect
12. #3008C2semidirect
13. #2980C2semidirect
14. C2×2−1+4C22semidirect
15. C2×H(5,𝔽2)C22semidirect
16. C2×AGL(1,ℤ/8)C22semidirect
17. C2×AGL(1,ℤ/8)C22semidirect
18. C22×SD16C22semidirect
19. C22×D8C22semidirect
23. #555C22semidirect
24. #548C22semidirect
26. #546C22semidirect
28. D42C22semidirect
30. #536C22semidirect
34. #535C22semidirect
35. #535C22semidirect
38. #532C22semidirect
40. C2×#124C22semidirect
41. C4×#41C22semidirect
42. #496C22semidirect
43. #489C22semidirect
44. #485C22semidirect
45. #482C22semidirect
46. #482C22semidirect
47. #475C22semidirect
48. #474C22semidirect
49. #469C22semidirect
50. #468C22semidirect
51. #466C22semidirect
52. #465C22semidirect
53. #463C22semidirect
54. #463C22semidirect
55. #460C22semidirect
56. #459C22semidirect
57. #450C22semidirect
58. #449C22semidirect
59. #449C22semidirect
60. #449C22semidirect
61. #448C22semidirect
62. #448C22semidirect
63. #447C22semidirect
64. #447C22semidirect
65. #447C22semidirect
66. #442C22semidirect
67. #440C22semidirect
68. #435C22semidirect
69. #435C22semidirect
70. #432C22semidirect
71. #419C22semidirect
72. #418C22semidirect
73. #416C22semidirect
74. #416C22semidirect
75. #408C22semidirect
76. #408C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#35002
#33212
#33192
#32652
#32642
#32132
#32002
#31992
#31152
#30102
#30082
#29802