Finite Group Databasebeta

← #61679

Subgroups — #61679

Order 256.

1112 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3496C211yesmaximal, characteristic, complemented
3#3325C212yesmaximal, complemented
4#3319C214yesmaximal, complemented
5#3303C212yesmaximal, complemented
6D4×D8C212yesmaximal, complemented
7#3272C212yesmaximal, complemented
8#3264C214yesmaximal, complemented
9#3214C211yesmaximal, characteristic, complemented
10#3207C211yesmaximal, characteristic, complemented
11#3198C214yesmaximal, complemented
12#3109C211yesmaximal, characteristic, complemented
13#3100C211yesmaximal, characteristic, complemented
14#3008C214yesmaximal, complemented
15#2981C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2212yescomplemented
17C2×AGL(1,ℤ/8)C2214yescomplemented
18C22×D8C2214yescomplemented
19H(3,𝔽4)—22no—
20#555—21no—
21#555C2212yescomplemented
22#555C2212yescomplemented
23#547—21no—
24#547C2212yescomplemented
25#546—22no—
26#546C2214yescomplemented
27D42—21no—
28D42—21no—
29D42C2212yescomplemented
30D42C2212yescomplemented
31D42C2212yescomplemented
32#537C2212yescomplemented
33#535—22no—
34#535—22no—
35#535C2214yescomplemented
36#534—22no—
37#532C2211yescharacteristic
38#532C2211yescharacteristic, complemented
39#532C2211yescharacteristic, complemented
40#532C2211yescharacteristic, complemented
41C2×#121—22no—
42C2×C4×D4C2211yescharacteristic, complemented
43#496C2212yescomplemented
44#493C2211yescharacteristic, complemented
45#492C2211yescharacteristic, complemented
46#489C2212yescomplemented
47#487C2211yescharacteristic, complemented
48#486C2211yescharacteristic, complemented
49#485C2214yescomplemented
50#482C2214yescomplemented
51#481C2212yescomplemented
52#480C2212yescomplemented
53#476C2212yescomplemented
54#474C2212yescomplemented
55#469C2214yescomplemented
56#468C2214yescomplemented
57#466C2214yescomplemented
58#465C2214yescomplemented
59#463C2214yescomplemented
60#461C2212yescomplemented
61#459C2212yescomplemented
62#459C2212yescomplemented
63#459C2212yescomplemented
64#450C2214yescomplemented
65#449C2214yescomplemented
66#449C2214yescomplemented
67#449C2214yescomplemented
68#449C2214yescomplemented
69#447C2212yescomplemented
70#447C2212yescomplemented
71#447C2214yescomplemented
72#447C2214yescomplemented
73#447C2214yescomplemented
74#442C2214yescomplemented
75C4×D8C2212yescomplemented
76C4×D8C2212yescomplemented
77#436C2212yescomplemented
78#435C2214yescomplemented
79C8×D4C2212yescomplemented
80#432C2211yescharacteristic, complemented
81#431C2211yescharacteristic, complemented
82#418C2214yescomplemented
83#416C2214yescomplemented
84#408C2214yescomplemented
85H(5,𝔽2)—48no—
86C2×#41—22no—
87C2×#41—21no—
88C2×#41—21no—
89C2×#41C2314yes—
90C2×#41C2312yes—
91C2×#41C2312yes—
92C22×D4—24no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4C2311yescharacteristic
98C22×D4C2314yes—
99C22×D4C2314yes—
100C23×C4—21no—
101AGL(1,ℤ/8)—48no—
102C2×SD16C2314yes—
103C2×SD16C2314yes—
104C2×D8—48no—
105C2×D8C2312yes—
106C2×D8C2314yes—
107C2×D8C2312yes—
108C2×D8C2314yes—
109C2×D8C2314yes—
110C2×M16C2314yes—
111C22×C8C2314yes—
112C4⋊Q8D411yescharacteristic
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115C4⋊Q8C2311yescharacteristic
116#127—24no—
117#127D411yescharacteristic
118#127C2311yescharacteristic
119#127C2311yescharacteristic
120#127C2311yescharacteristic
121#124—24no—
122#124—24no—
123#124—24no—
124#124—21no—
125#124—21no—
126#124D412yes—
127#124C2312yes—
128#123—24no—
129#123—24no—
130C22⋊Q8—22no—
131C22⋊Q8C2314yes—
132#121—24no—
133#121—22no—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—21no—
138#121—21no—
139#121—21no—
140#121—21no—
141#121—22no—
142#121—22no—
143#121—22no—
144#121—24no—
145#121D412yes—
146#121D412yes—
147#121C2312yes—
148#121C2312yes—
149#121C2314yes—
150#121C2314yes—
151#121C2314yes—
152#120—24no—
153#120—24no—
154#120—24no—
155#120—24no—
156#120—24no—
157#120—24no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4C2312yes—
165C4×D4C2312yes—
166C4×D4C2312yes—
167C4×D4C2312yes—
168C4×D4C2312yes—
169C4×D4C2312yes—
170#117—22no—
171#117—22no—
172#117C2314yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C2311yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—24no—
177C2×C42C2311yescharacteristic
178#107C2312yes—
179#107C2312yes—
180#106C2314yes—
181C4⋊C8C2312yes—
182C4⋊C8C2312yes—
183#103C2314yes—
184#103C2314yes—
185#102C2312yes—
186#102C2314yes—
187#102C2312yes—
188#102C2314yes—
189#102C2314yes—
190#102C2312yes—
191#102C2312yes—
192#102C2314yes—
193C22⋊C8C2312yes—
194C22⋊C8C2312yes—
195C22⋊C8C2314yes—
196C22⋊C8C2314yes—
197#97C2312yes—
198C4×C8C2311yescharacteristic
199C4×C8C2311yescharacteristic
200C24—21no—
201C24—24no—
202C24—24no—
203#41—28no—
204#41—44no—
205#41—44no—
206#41—44no—
207#41—48no—
208#41—82no—
209#41—82no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8C2×D412yes—
213C2×Q8C2412yes—
214C2×D4—24no—
215C2×D4—28no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—24no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—24no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—48no—
236C2×D4—416no—
237C2×D4—84no—
238C2×D4—84no—
239C2×D4—84no—
240C2×D4C2×D412yes—
241C2×D4C2×D412yes—
242C2×D4C2×D412yes—
243C2×D4C2×D414yes—
244C2×D4C2412yes—
245C2×D4C2412yes—
246C2×D4C2412yes—
247C2×D4C2412yes—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—82no—
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2411yescharacteristic
259C22×C4C2414yes—
260SD16—44no—
261SD16—44no—
262D8—24no—
263D8—28no—
264D8—24no—
265D8—44no—
266D8—44no—
267M16—44no—
268C2×C8—44no—
269C2×C8C2412yes—
270C2×C8C2412yes—
271C2×C8C2414yes—
272C4⋊C4—21no—
273C4⋊C4—24no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4C2×D412yes—
279C4⋊C4C2×D412yes—
280C4⋊C4C2412yes—
281C4⋊C4C2412yes—
282C4⋊C4C2412yes—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4C2×D414yes—
297C22⋊C4C2×D414yes—
298C42—24no—
299C42C2×D411yescharacteristic
300C42C2×D411yescharacteristic
301C42C2411yescharacteristic
302C42C2411yescharacteristic
303C23—24no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—48no—
309C23—81no—
310C23—81no—
311C23—84no—
312C23—84no—
313C23—84no—
314C23—84no—
315C23—84no—
316C23C22×D414yes—
317C23C22×D411yescharacteristic
318Q8—24no—
319Q8—42no—
320Q8—81no—
321Q8—81no—
322D4—24no—
323D4—24no—
324D4—22no—
325D4—22no—
326D4—24no—
327D4—22no—
328D4—24no—
329D4—42no—
330D4—44no—
331D4—42no—
332D4—48no—
333D4—42no—
334D4—48no—
335D4—44no—
336D4—42no—
337D4—42no—
338D4—81no—
339D4—84no—
340D4—81no—
341D4—81no—
342D4—84no—
343D4—84no—
344D4—84no—
345D4—82no—
346D4—82no—
347D4H(5,𝔽2)14yes—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4C22×D414yes—
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C2511yescharacteristic
371C8—22no—
372C8—24no—
373C8—22no—
374C22—21no—
375C22—24no—
376C22—21no—
377C22—24no—
378C22—24no—
379C22—21no—
380C22—24no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—84no—
393C22—168no—
394C22—168no—
395C22C23×D411yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—24no—
400C4—21no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4#57612yes—
406C4C2×H(5,𝔽2)12yes—
407C2—41no—
408C2—44no—
409C2—44no—
410C2—82no—
411C2—82no—
412C2—82no—
413C2#348411yescharacteristic
414C2C2×#57611yescharacteristic
415C2#358511yescharacteristic
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. #3496C2semidirect
3. #3325C2semidirect
4. #3319C2semidirect
5. #3303C2semidirect
6. D4×D8C2semidirect
7. #3272C2semidirect
8. #3264C2semidirect
9. #3214C2semidirect
10. #3207C2semidirect
11. #3198C2semidirect
12. #3109C2semidirect
13. #3100C2semidirect
14. #3008C2semidirect
15. #2981C2semidirect
16. C2×H(5,𝔽2)C22semidirect
17. C2×AGL(1,ℤ/8)C22semidirect
18. C22×D8C22semidirect
21. #555C22semidirect
22. #555C22semidirect
24. #547C22semidirect
26. #546C22semidirect
29. D42C22semidirect
30. D42C22semidirect
31. D42C22semidirect
32. #537C22semidirect
35. #535C22semidirect
38. #532C22semidirect
39. #532C22semidirect
40. #532C22semidirect
42. C2×C4×D4C22semidirect
43. #496C22semidirect
44. #493C22semidirect
45. #492C22semidirect
46. #489C22semidirect
47. #487C22semidirect
48. #486C22semidirect
49. #485C22semidirect
50. #482C22semidirect
51. #481C22semidirect
52. #480C22semidirect
53. #476C22semidirect
54. #474C22semidirect
55. #469C22semidirect
56. #468C22semidirect
57. #466C22semidirect
58. #465C22semidirect
59. #463C22semidirect
60. #461C22semidirect
61. #459C22semidirect
62. #459C22semidirect
63. #459C22semidirect
64. #450C22semidirect
65. #449C22semidirect
66. #449C22semidirect
67. #449C22semidirect
68. #449C22semidirect
69. #447C22semidirect
70. #447C22semidirect
71. #447C22semidirect
72. #447C22semidirect
73. #447C22semidirect
74. #442C22semidirect
75. C4×D8C22semidirect
76. C4×D8C22semidirect
77. #436C22semidirect
78. #435C22semidirect
79. C8×D4C22semidirect
80. #432C22semidirect
81. #431C22semidirect
82. #418C22semidirect
83. #416C22semidirect
84. #408C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#34962
#33252
#33192
#33032
D4×D82
#32722
#32642
#32142
#32072
#31982
#31092
#31002
#30082
#29812