Finite Group Databasebeta

← #61648

Subgroups — #61648

Order 256.

1085 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3537C211yesmaximal, characteristic, complemented
2#3485C211yesmaximal, characteristic, complemented
3#3322C214yesmaximal, complemented
4#3307C214yesmaximal, complemented
5#3272C214yesmaximal, complemented
6#3264C214yesmaximal, complemented
7#3221C212yesmaximal, complemented
8#3200C212yesmaximal, complemented
9#3194C212yesmaximal, complemented
10#3122C212yesmaximal, complemented
11#3008C212yesmaximal, complemented
12#3006C212yesmaximal, complemented
13#2978C211yesmaximal, characteristic, complemented
14C2×H(5,𝔽2)C2211yescharacteristic, complemented
15C22×#41C2211yescharacteristic, complemented
16C2×AGL(1,ℤ/8)C2214yescomplemented
17C2×AGL(1,ℤ/8)C2212yescomplemented
18C22×D8C2212yescomplemented
19#560—22no—
20#553C2214yescomplemented
21#547—22no—
22#547C2214yescomplemented
23#546—22no—
24#546C2214yescomplemented
25D42—22no—
26D42C2214yescomplemented
27#540C2212yescomplemented
28#538—22no—
29#536C2211yescharacteristic, complemented
30#536C2211yescharacteristic, complemented
31#535C2212yes—
32#534—22no—
33#534—22no—
34#525C2211yescharacteristic
35#525C2211yescharacteristic, complemented
36C2×#123—22no—
37C2×#121C2212yescomplemented
38#518C2211yescharacteristic, complemented
39#496C2212yescomplemented
40#492C2212yescomplemented
41#491C2212yescomplemented
42#488C2212yescomplemented
43#485C2214yescomplemented
44#483C2212yescomplemented
45#482C2214yescomplemented
46#480C2212yescomplemented
47#478C2214yescomplemented
48#469C2214yescomplemented
49#468C2214yescomplemented
50#468C2214yescomplemented
51#465C2214yescomplemented
52#463C2214yescomplemented
53#461C2214yescomplemented
54#459C2214yescomplemented
55#450C2212yescomplemented
56#450C2214yescomplemented
57#449C2214yescomplemented
58#449C2214yescomplemented
59#449C2214yescomplemented
60#449C2214yescomplemented
61#448C2214yescomplemented
62#447C2214yescomplemented
63#447C2212yescomplemented
64#447C2214yescomplemented
65#442C2214yescomplemented
66C4×D8C2214yescomplemented
67#436C2214yescomplemented
68#435C2214yescomplemented
69#433C2212yescomplemented
70#418C2212yescomplemented
71#417C2214yescomplemented
72#416C2212yescomplemented
73#408C2212yescomplemented
74#407C2212yescomplemented
75H(5,𝔽2)—44no—
76C2×#41—21no—
77C2×#41—22no—
78C2×#41—44no—
79C2×#41C2314yes—
80C2×#41C2312yes—
81C2×#41C2311yescharacteristic
82C2×#41C2311yescharacteristic
83C22×Q8C2311yescharacteristic
84C22×D4—21no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—21no—
88C22×D4—24no—
89C22×D4C2311yescharacteristic
90C22×D4C2312yes—
91C22×D4C2312yes—
92C22×D4C2312yes—
93C23×C4—21no—
94C23×C4C2311yescharacteristic
95AGL(1,ℤ/8)—48no—
96AGL(1,ℤ/8)—44no—
97C2×SD16C2314yes—
98C2×SD16C2314yes—
99C2×SD16C2314yes—
100C2×D8—44no—
101C2×D8C2314yes—
102C2×D8C2314yes—
103C2×D8C2314yes—
104C2×M16C2314yes—
105C2×M16C2312yes—
106C22×C8C2312yes—
107C4⋊Q8—21no—
108C4⋊Q8C2312yes—
109#127—22no—
110#127C2312yes—
111#126—24no—
112#125C2312yes—
113#124—22no—
114#124—21no—
115#124C2312yes—
116#123—22no—
117#123—21no—
118#123—24no—
119#123—24no—
120#123—22no—
121#123—24no—
122#123—24no—
123#123—24no—
124#123D412yes—
125C22⋊Q8—22no—
126C22⋊Q8D412yes—
127C22⋊Q8C2312yes—
128C22⋊Q8C2314yes—
129#121—24no—
130#121—24no—
131#121—22no—
132#121—22no—
133#121—24no—
134#121—24no—
135#121—24no—
136#121D412yes—
137#121C2314yes—
138#121C2314yes—
139#121C2312yes—
140#121C2314yes—
141#120—21no—
142#120—22no—
143#120—24no—
144#120—22no—
145#120—24no—
146#120—24no—
147#120D412yes—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—24no—
151C4×D4C2314yes—
152C4×D4C2314yes—
153C4×D4C2314yes—
154#117C2311yescharacteristic
155#117C2312yes—
156#117C2311yescharacteristic
157C2×C4⋊C4—24no—
158C2×C4⋊C4C2312yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162#107C2314yes—
163#106C2314yes—
164C4⋊C8C2314yes—
165#103C2314yes—
166#103C2314yes—
167#103C2314yes—
168#102C2314yes—
169#102C2314yes—
170#102C2314yes—
171#102C2314yes—
172#102C2314yes—
173C22⋊C8C2314yes—
174C22⋊C8C2312yes—
175C22⋊C8C2312yes—
176C22⋊C8C2314yes—
177#97C2312yes—
178C4×C8C2312yes—
179C24—21no—
180C24—22no—
181C24—22no—
182C24—22no—
183C24C2×D411yescharacteristic
184#41—28no—
185#41—24no—
186#41—42no—
187#41—44no—
188#41—42no—
189#41—42no—
190#41—48no—
191#41—44no—
192C2×Q8—21no—
193C2×Q8—42no—
194C2×Q8C2×D411yescharacteristic
195C2×Q8C2412yes—
196C2×Q8C2411yescharacteristic
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—48no—
218C2×D4—88no—
219C2×D4—84no—
220C2×D4C2×D414yes—
221C2×D4C2×D411yescharacteristic
222C2×D4C2×D412yes—
223C2×D4C2411yescharacteristic
224C2×D4C2414yes—
225C2×D4C2412yes—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4C2×D412yes—
236C22×C4C2×D411yescharacteristic
237C22×C4C2411yescharacteristic
238C22×C4C2412yes—
239C22×C4C2412yes—
240SD16—44no—
241SD16—44no—
242SD16—44no—
243D8—28no—
244D8—28no—
245D8—44no—
246M16—44no—
247M16—42no—
248C2×C8—42no—
249C2×C8C2414yes—
250C2×C8C2414yes—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4C2×D414yes—
257C4⋊C4C2×D412yes—
258C4⋊C4C2412yes—
259C4⋊C4C2414yes—
260C22⋊C4—21no—
261C22⋊C4—24no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4C2×D412yes—
272C22⋊C4C2×D414yes—
273C22⋊C4C2×D414yes—
274C42—21no—
275C42—22no—
276C42C2412yes—
277C23—24no—
278C23—22no—
279C23—21no—
280C23—24no—
281C23—41no—
282C23—42no—
283C23—44no—
284C23—44no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23—84no—
290C23—84no—
291C23C22×D412yes—
292C23C22×D412yes—
293C23C22×D411yescharacteristic
294Q8—24no—
295Q8—22no—
296Q8—41no—
297Q8—42no—
298D4—24no—
299D4—22no—
300D4—24no—
301D4—24no—
302D4—22no—
303D4—41no—
304D4—42no—
305D4—44no—
306D4—44no—
307D4—44no—
308D4—48no—
309D4—42no—
310D4—48no—
311D4—82no—
312D4—82no—
313D4—84no—
314D4—82no—
315D4—84no—
316D4—84no—
317D4H(5,𝔽2)14yes—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4C22×D411yescharacteristic
336C2×C4C22×D412yes—
337C2×C4C22×D414yes—
338C2×C4C22×D412yes—
339C2×C4C2511yescharacteristic
340C8—24no—
341C8—24no—
342C22—22no—
343C22—22no—
344C22—24no—
345C22—22no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—42no—
354C22—41no—
355C22—44no—
356C22—84no—
357C22—82no—
358C22—164no—
359C22—164no—
360C22—164no—
361C22C23×D411yescharacteristic
362C4—24no—
363C4—22no—
364C4—21no—
365C4—22no—
366C4—44no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4C2×H(5,𝔽2)12yes—
371C2—41no—
372C2—42no—
373C2—42no—
374C2—44no—
375C2—84no—
376C2—81no—
377C2#348411yescharacteristic
378C2#358512yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #525 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3537C2semidirect
2. #3485C2semidirect
3. #3322C2semidirect
4. #3307C2semidirect
5. #3272C2semidirect
6. #3264C2semidirect
7. #3221C2semidirect
8. #3200C2semidirect
9. #3194C2semidirect
10. #3122C2semidirect
11. #3008C2semidirect
12. #3006C2semidirect
13. #2978C2semidirect
14. C2×H(5,𝔽2)C22semidirect
15. C22×#41C22semidirect
16. C2×AGL(1,ℤ/8)C22semidirect
17. C2×AGL(1,ℤ/8)C22semidirect
18. C22×D8C22semidirect
20. #553C22semidirect
22. #547C22semidirect
24. #546C22semidirect
26. D42C22semidirect
27. #540C22semidirect
29. #536C22semidirect
30. #536C22semidirect
35. #525C22semidirect
37. C2×#121C22semidirect
38. #518C22semidirect
39. #496C22semidirect
40. #492C22semidirect
41. #491C22semidirect
42. #488C22semidirect
43. #485C22semidirect
44. #483C22semidirect
45. #482C22semidirect
46. #480C22semidirect
47. #478C22semidirect
48. #469C22semidirect
49. #468C22semidirect
50. #468C22semidirect
51. #465C22semidirect
52. #463C22semidirect
53. #461C22semidirect
54. #459C22semidirect
55. #450C22semidirect
56. #450C22semidirect
57. #449C22semidirect
58. #449C22semidirect
59. #449C22semidirect
60. #449C22semidirect
61. #448C22semidirect
62. #447C22semidirect
63. #447C22semidirect
64. #447C22semidirect
65. #442C22semidirect
66. C4×D8C22semidirect
67. #436C22semidirect
68. #435C22semidirect
69. #433C22semidirect
70. #418C22semidirect
71. #417C22semidirect
72. #416C22semidirect
73. #408C22semidirect
74. #407C22semidirect

Maximal subgroups

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

subgroupindex
#35372
#34852
#33222
#33072
#32722
#32642
#32212
#32002
#31942
#31222
#30082
#30062
#29782