Finite Group Databasebeta

← #60768

Subgroups — #60768

Order 256.

1134 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3488C212yesmaximal, complemented
2#3203C212yesmaximal, complemented
3#3199C212yesmaximal, complemented
4#3198C214yesmaximal, complemented
5#3191C212yesmaximal, complemented
6#3188C214yesmaximal, complemented
7#3186C212yesmaximal, complemented
8#3098C212yesmaximal, complemented
9#3096C212yesmaximal, complemented
10#3055C212yesmaximal, complemented
11#3050C211yesmaximal, characteristic, complemented
12C2×#466C211yesmaximal, characteristic, complemented
13#3012C212yesmaximal, complemented
14#3008C212yesmaximal, complemented
15C2×#408C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2212yescomplemented
17C22×#41C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2212yescomplemented
19C2×#135C2211yescharacteristic, complemented
20C22×D8C2211yescharacteristic, complemented
21C22×M16C2211yescharacteristic, complemented
22C23×C8C2211yescharacteristic, complemented
23#550—22no—
24#548—22no—
25#546—24no—
26#546—22no—
27D42—24no—
28D42—22no—
29#537C2212yes—
30#537C2214yescomplemented
31#535C2212yes—
32#535C2214yescomplemented
33#525C2214yescomplemented
34#525C2212yescomplemented
35C2×#121C2214yescomplemented
36C2×#121C2212yescomplemented
37#520C2212yescomplemented
38#485C2214yescomplemented
39#484C2214yescomplemented
40#482C2214yescomplemented
41#482C2214yescomplemented
42#482C2214yescomplemented
43#481C2214yescomplemented
44#480C2214yescomplemented
45#480C2214yescomplemented
46#469C2214yescomplemented
47#468C2214yescomplemented
48#467C2211yescharacteristic, complemented
49#467C2211yescharacteristic, complemented
50#466C2211yescharacteristic, complemented
51#466C2214yescomplemented
52#466C2212yescomplemented
53#466C2212yescomplemented
54#466C2212yescomplemented
55#466C2211yescharacteristic, complemented
56#466C2212yescomplemented
57#465C2212yescomplemented
58#465C2214yescomplemented
59#465C2212yescomplemented
60#452C2214yescomplemented
61#450C2214yescomplemented
62#449C2214yescomplemented
63#449C2214yescomplemented
64#449C2214yescomplemented
65#447C2214yescomplemented
66#447C2214yescomplemented
67#447C2214yescomplemented
68#428C2212yescomplemented
69#427C2211yescharacteristic, complemented
70C2×#107C2211yescharacteristic, complemented
71#418C2212yescomplemented
72#417C2212yescomplemented
73#416C2212yescomplemented
74C2×#102C2212yescomplemented
75#408C2212yescomplemented
76#408C2212yescomplemented
77#408C2212yescomplemented
78#408C2212yescomplemented
79C2×C22⋊C8C2212yescomplemented
80C2×C22⋊C8C2212yescomplemented
81H(5,𝔽2)—44no—
82H(5,𝔽2)—44no—
83C2×#41—22no—
84C2×#41—42no—
85C2×#41—42no—
86C2×#41D412yes—
87C2×#41D412yes—
88C2×#41C2312yes—
89C2×#41C2314yes—
90C2×#41C2312yes—
91C2×#41C2312yes—
92C22×Q8D411yescharacteristic
93C22×D4—22no—
94C22×D4—24no—
95C22×D4—24no—
96C22×D4—22no—
97C22×D4D411yescharacteristic
98C22×D4D412yes—
99C22×D4C2312yes—
100C22×D4C2314yes—
101C23×C4C2311yescharacteristic
102C23×C4C2312yes—
103AGL(1,ℤ/8)—44no—
104#135—42no—
105C2×Q16C2311yescharacteristic
106C2×SD16C2312yes—
107C2×SD16C2314yes—
108C2×D8—42no—
109C2×D8C2312yes—
110C2×D8C2312yes—
111C2×D8C2311yescharacteristic
112C2×D8C2314yes—
113C2×M16—42no—
114C2×M16C2312yes—
115C2×M16C2312yes—
116C22×C8—42no—
117C22×C8C2311yescharacteristic
118C22×C8C2311yescharacteristic
119C22×C8C2311yescharacteristic
120C22×C8C2311yescharacteristic
121C22×C8C2311yescharacteristic
122C22×C8C2311yescharacteristic
123C22×C8C2312yes—
124#127—22no—
125#127—24no—
126#124—22no—
127#124—24no—
128#123—22no—
129#123—24no—
130C22⋊Q8—22no—
131C22⋊Q8—24no—
132C22⋊Q8C2312yes—
133C22⋊Q8C2314yes—
134C22⋊Q8C2312yes—
135#121—24no—
136#121—22no—
137#121—24no—
138#121—24no—
139#121—24no—
140#121—22no—
141#121—24no—
142#121—22no—
143#121—24no—
144#121C2312yes—
145#121C2314yes—
146#121C2314yes—
147#121C2314yes—
148#121C2314yes—
149#121C2314yes—
150#121C2312yes—
151#120—22no—
152#120—24no—
153#120—24no—
154#120—24no—
155#120—24no—
156C4×Q8—22no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—24no—
163#117C2312yes—
164#117C2314yes—
165C2×C4⋊C4C2312yes—
166C2×C4⋊C4C2314yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169#107C2312yes—
170#107C2312yes—
171#107C2311yescharacteristic
172#107C2311yescharacteristic
173#107C2314yes—
174#106C2312yes—
175#106C2314yes—
176#103C2312yes—
177#103C2312yes—
178#103C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2312yes—
182#102C2312yes—
183#102C2314yes—
184#102C2314yes—
185#102C2314yes—
186C22⋊C8C2314yes—
187C22⋊C8C2314yes—
188C22⋊C8C2314yes—
189C22⋊C8C2314yes—
190C24—24no—
191C24—22no—
192C24C2×D411yescharacteristic
193C24C2×D412yes—
194#41—24no—
195#41—24no—
196#41—24no—
197#41—24no—
198#41—42no—
199#41—42no—
200#41—44no—
201#41—44no—
202#41—44no—
203#41—44no—
204#41—48no—
205C2×Q8—42no—
206C2×Q8C2×D412yes—
207C2×Q8C2×D412yes—
208C2×Q8C2412yes—
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—416no—
227C2×D4—48no—
228C2×D4—84no—
229C2×D4—88no—
230C2×D4C2×D414yes—
231C2×D4C2×D412yes—
232C2×D4C2×D412yes—
233C2×D4C2×D414yes—
234C2×D4C2414yes—
235C2×D4C2412yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D414yes—
248C22×C4C2×D412yes—
249C22×C4C2414yes—
250C22×C4C2411yescharacteristic
251C22×C4C2411yescharacteristic
252C22×C4C2411yescharacteristic
253Q16—41no—
254SD16—42no—
255SD16—44no—
256D8—42no—
257D8—42no—
258D8—41no—
259D8—44no—
260M16—24no—
261M16—42no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8C2411yescharacteristic
270C2×C8C2411yescharacteristic
271C2×C8C2414yes—
272C2×C8C2411yescharacteristic
273C2×C8C2411yescharacteristic
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4C2412yes—
280C4⋊C4C2414yes—
281C4⋊C4C2412yes—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—22no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—22no—
294C42—24no—
295C42—22no—
296C23—22no—
297C23—22no—
298C23—24no—
299C23—24no—
300C23—22no—
301C23—41no—
302C23—42no—
303C23—42no—
304C23—48no—
305C23—44no—
306C23—82no—
307C23—84no—
308C23—82no—
309C23—84no—
310C23C22×D414yes—
311C23C22×D411yescharacteristic
312C23C22×D411yescharacteristic
313C23C22×D411yescharacteristic
314Q8—24no—
315Q8—22no—
316Q8—22no—
317Q8—42no—
318D4—28no—
319D4—24no—
320D4—24no—
321D4—22no—
322D4—24no—
323D4—22no—
324D4—42no—
325D4—44no—
326D4—48no—
327D4—48no—
328D4—44no—
329D4—48no—
330D4—44no—
331D4—44no—
332D4—82no—
333D4—84no—
334D4—84no—
335D4—84no—
336D4—84no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4C22×D414yes—
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4H(5,𝔽2)12yes—
361C2×C4H(5,𝔽2)12yes—
362C2×C4C2511yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—21no—
367C8—24no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—21no—
378C22—44no—
379C22—44no—
380C22—42no—
381C22—44no—
382C22—42no—
383C22—44no—
384C22—42no—
385C22—84no—
386C22—164no—
387C22—168no—
388C22#53412yes—
389C22#57612yes—
390C22C23×D411yescharacteristic
391C4—21no—
392C4—21no—
393C4—24no—
394C4—42no—
395C4—42no—
396C4—44no—
397C4—42no—
398C4#53412yes—
399C4C2×H(5,𝔽2)11yescharacteristic
400C4C2×H(5,𝔽2)11yescharacteristic
401C2—22no—
402C2—22no—
403C2—41no—
404C2—44no—
405C2—84no—
406C2—82no—
407C2C2×#53411yescharacteristic
408C2C2×#57611yescharacteristic
409C2#358411yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3488C2semidirect
2. #3203C2semidirect
3. #3199C2semidirect
4. #3198C2semidirect
5. #3191C2semidirect
6. #3188C2semidirect
7. #3186C2semidirect
8. #3098C2semidirect
9. #3096C2semidirect
10. #3055C2semidirect
11. #3050C2semidirect
12. C2×#466C2semidirect
13. #3012C2semidirect
14. #3008C2semidirect
15. C2×#408C2semidirect
16. C2×H(5,𝔽2)C22semidirect
18. C2×AGL(1,ℤ/8)C22semidirect
19. C2×#135C22semidirect
20. C22×D8C22semidirect
21. C22×M16C22semidirect
22. C23×C8C22semidirect
30. #537C22semidirect
32. #535C22semidirect
33. #525C22semidirect
34. #525C22semidirect
35. C2×#121C22semidirect
36. C2×#121C22semidirect
37. #520C22semidirect
38. #485C22semidirect
39. #484C22semidirect
40. #482C22semidirect
41. #482C22semidirect
42. #482C22semidirect
43. #481C22semidirect
44. #480C22semidirect
45. #480C22semidirect
46. #469C22semidirect
47. #468C22semidirect
48. #467C22semidirect
49. #467C22semidirect
50. #466C22semidirect
51. #466C22semidirect
52. #466C22semidirect
53. #466C22semidirect
54. #466C22semidirect
55. #466C22semidirect
56. #466C22semidirect
57. #465C22semidirect
58. #465C22semidirect
59. #465C22semidirect
60. #452C22semidirect
61. #450C22semidirect
62. #449C22semidirect
63. #449C22semidirect
64. #449C22semidirect
65. #447C22semidirect
66. #447C22semidirect
67. #447C22semidirect
68. #428C22semidirect
69. #427C22semidirect
70. C2×#107C22semidirect
71. #418C22semidirect
72. #417C22semidirect
73. #416C22semidirect
74. C2×#102C22semidirect
75. #408C22semidirect
76. #408C22semidirect
77. #408C22semidirect
78. #408C22semidirect
79. C2×C22⋊C8C22semidirect
80. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34882
#32032
#31992
#31982
#31912
#31882
#31862
#30982
#30962
#30552
#30502
C2×#4662
#30122
#30082
C2×#4082