Finite Group Databasebeta

← #61651

Subgroups — #61651

Order 256.

1079 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3537C211yesmaximal, characteristic, complemented
2#3487C211yesmaximal, characteristic, complemented
3#3326C214yesmaximal, complemented
4#3309C214yesmaximal, complemented
5#3274C214yesmaximal, complemented
6#3265C214yesmaximal, complemented
7#3220C212yesmaximal, complemented
8#3199C212yesmaximal, complemented
9#3194C212yesmaximal, complemented
10#3126C212yesmaximal, complemented
11#3010C212yesmaximal, complemented
12#3006C212yesmaximal, complemented
13#2978C211yesmaximal, characteristic, complemented
14C2×2−1+4C2211yescharacteristic, complemented
15C22×#41C2211yescharacteristic, complemented
16C2×AGL(1,ℤ/8)C2214yescomplemented
17C2×AGL(1,ℤ/8)C2212yescomplemented
18C22×SD16C2212yescomplemented
19#560—22no—
20#553C2214yescomplemented
21#550—22no—
22#550C2214yescomplemented
23#548—22no—
24#548C2214yescomplemented
25D42—22no—
26D42C2214yescomplemented
27#540C2212yes—
28#538—22no—
29#535—22no—
30#535C2212yes—
31#535C2211yescharacteristic, complemented
32#535C2211yescharacteristic, complemented
33#534—22no—
34#525—22no—
35#525C2211yescharacteristic
36#525C2211yescharacteristic, complemented
37C2×#121C2212yes—
38#519C2211yescharacteristic, complemented
39#496C2212yescomplemented
40#493C2212yescomplemented
41#490C2212yescomplemented
42#488C2212yescomplemented
43#483C2212yescomplemented
44#482C2214yescomplemented
45#482C2214yescomplemented
46#480C2212yescomplemented
47#479C2214yescomplemented
48#469C2214yescomplemented
49#469C2214yescomplemented
50#468C2214yescomplemented
51#466C2214yescomplemented
52#463C2214yescomplemented
53#460C2214yescomplemented
54#459C2214yescomplemented
55#450C2212yescomplemented
56#449C2214yescomplemented
57#449C2214yescomplemented
58#449C2214yescomplemented
59#449C2214yescomplemented
60#448C2214yescomplemented
61#448C2214yescomplemented
62#447C2212yescomplemented
63#447C2214yescomplemented
64#447C2214yescomplemented
65#440C2214yescomplemented
66C4×SD16C2214yescomplemented
67#436C2214yescomplemented
68#435C2214yescomplemented
69#433C2212yescomplemented
70#419C2212yescomplemented
71#417C2214yescomplemented
72#416C2212yescomplemented
73#408C2212yescomplemented
74#407C2212yescomplemented
752−1+4—44no—
76C2×#41—22no—
77C2×#41—21no—
78C2×#41—22no—
79C2×#41—44no—
80C2×#41C2314yes—
81C2×#41C2312yes—
82C2×#41C2311yescharacteristic
83C2×#41C2311yescharacteristic
84C22×Q8—21no—
85C22×Q8C2311yescharacteristic
86C22×Q8C2312yes—
87C22×D4—21no—
88C22×D4—22no—
89C22×D4—24no—
90C22×D4C2312yes—
91C22×D4C2312yes—
92C22×D4C2311yescharacteristic
93C23×C4—21no—
94C23×C4C2311yescharacteristic
95AGL(1,ℤ/8)—48no—
96AGL(1,ℤ/8)—44no—
97C2×SD16—44no—
98C2×SD16C2314yes—
99C2×SD16C2314yes—
100C2×SD16C2314yes—
101C2×D8C2314yes—
102C2×D8C2314yes—
103C2×D8C2314yes—
104C2×M16C2314yes—
105C2×M16C2312yes—
106C22×C8C2312yes—
107#127—24no—
108#127—22no—
109#127—21no—
110#127C2312yes—
111#127C2312yes—
112#126—24no—
113#125C2312yes—
114#124—22no—
115#124—21no—
116#124—24no—
117#124C2312yes—
118#123—21no—
119#123—24no—
120#123—24no—
121#123D412yes—
122C22⋊Q8—24no—
123C22⋊Q8D412yes—
124C22⋊Q8C2312yes—
125#121—24no—
126#121—24no—
127#121—22no—
128#121—22no—
129#121—22no—
130#121—24no—
131#121—24no—
132#121—24no—
133#121—24no—
134#121D412yes—
135#121C2314yes—
136#121C2314yes—
137#121C2312yes—
138#121C2314yes—
139#121C2314yes—
140#120—21no—
141#120—22no—
142#120—22no—
143#120—22no—
144#120—22no—
145#120—24no—
146#120—24no—
147#120D412yes—
148C4×Q8—22no—
149C4×Q8C2314yes—
150C4×D4—22no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4C2314yes—
155C4×D4C2314yes—
156#117—22no—
157#117C2311yescharacteristic
158#117C2312yes—
159#117C2311yescharacteristic
160C2×C4⋊C4C2312yes—
161C2×C22⋊C4—22no—
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—22no—
180C24—21no—
181C24—22no—
182C24C2×D411yescharacteristic
183#41—28no—
184#41—24no—
185#41—42no—
186#41—44no—
187#41—42no—
188#41—42no—
189#41—48no—
190#41—48no—
191#41—44no—
192C2×Q8—24no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—42no—
197C2×Q8—44no—
198C2×Q8C2×D411yescharacteristic
199C2×Q8C2411yescharacteristic
200C2×Q8C2412yes—
201C2×D4—24no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—21no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—88no—
220C2×D4—84no—
221C2×D4C2×D414yes—
222C2×D4C2×D412yes—
223C2×D4C2×D411yescharacteristic
224C2×D4C2412yes—
225C2×D4C2411yescharacteristic
226C2×D4C2414yes—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4C2×D412yes—
237C22×C4C2×D411yescharacteristic
238C22×C4C2411yescharacteristic
239C22×C4C2412yes—
240C22×C4C2412yes—
241SD16—28no—
242SD16—28no—
243SD16—44no—
244D8—44no—
245D8—44no—
246D8—44no—
247M16—44no—
248M16—42no—
249C2×C8—42no—
250C2×C8C2414yes—
251C2×C8C2414yes—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4C2×D412yes—
257C4⋊C4C2×D414yes—
258C4⋊C4C2412yes—
259C4⋊C4C2414yes—
260C22⋊C4—21no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4C2×D412yes—
271C22⋊C4C2×D414yes—
272C22⋊C4C2×D414yes—
273C42—24no—
274C42—21no—
275C42—22no—
276C42C2412yes—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—44no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23—84no—
289C23C22×D412yes—
290C23C22×D411yescharacteristic
291C23C22×D412yes—
292Q8—24no—
293Q8—22no—
294Q8—22no—
295Q8—41no—
296Q8—44no—
297Q8H(5,𝔽2)14yes—
298D4—24no—
299D4—24no—
300D4—22no—
301D4—24no—
302D4—44no—
303D4—41no—
304D4—42no—
305D4—48no—
306D4—44no—
307D4—42no—
308D4—48no—
309D4—42no—
310D4—84no—
311D4—82no—
312D4—82no—
313D4—84no—
314D4—82no—
315D4—84no—
316D4—84no—
317D4—84no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4C22×D414yes—
336C2×C4C22×D412yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C22×D412yes—
339C2×C4C2511yescharacteristic
340C8—24no—
341C8—24no—
342C22—22no—
343C22—22no—
344C22—24no—
345C22—22no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—44no—
353C22—41no—
354C22—42no—
355C22—44no—
356C22—82no—
357C22—84no—
358C22—164no—
359C22—164no—
360C22C23×D411yescharacteristic
361C4—24no—
362C4—24no—
363C4—22no—
364C4—21no—
365C4—22no—
366C4—44no—
367C4—42no—
368C4—41no—
369C4C2×H(5,𝔽2)12yes—
370C2—41no—
371C2—42no—
372C2—42no—
373C2—81no—
374C2—82no—
375C2—84no—
376C2#348411yescharacteristic
377C2#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. #3487C2semidirect
3. #3326C2semidirect
4. #3309C2semidirect
5. #3274C2semidirect
6. #3265C2semidirect
7. #3220C2semidirect
8. #3199C2semidirect
9. #3194C2semidirect
10. #3126C2semidirect
11. #3010C2semidirect
12. #3006C2semidirect
13. #2978C2semidirect
14. C2×2−1+4C22semidirect
15. C22×#41C22semidirect
16. C2×AGL(1,ℤ/8)C22semidirect
17. C2×AGL(1,ℤ/8)C22semidirect
18. C22×SD16C22semidirect
20. #553C22semidirect
22. #550C22semidirect
24. #548C22semidirect
26. D42C22semidirect
31. #535C22semidirect
32. #535C22semidirect
36. #525C22semidirect
38. #519C22semidirect
39. #496C22semidirect
40. #493C22semidirect
41. #490C22semidirect
42. #488C22semidirect
43. #483C22semidirect
44. #482C22semidirect
45. #482C22semidirect
46. #480C22semidirect
47. #479C22semidirect
48. #469C22semidirect
49. #469C22semidirect
50. #468C22semidirect
51. #466C22semidirect
52. #463C22semidirect
53. #460C22semidirect
54. #459C22semidirect
55. #450C22semidirect
56. #449C22semidirect
57. #449C22semidirect
58. #449C22semidirect
59. #449C22semidirect
60. #448C22semidirect
61. #448C22semidirect
62. #447C22semidirect
63. #447C22semidirect
64. #447C22semidirect
65. #440C22semidirect
66. C4×SD16C22semidirect
67. #436C22semidirect
68. #435C22semidirect
69. #433C22semidirect
70. #419C22semidirect
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
#34872
#33262
#33092
#32742
#32652
#32202
#31992
#31942
#31262
#30102
#30062
#29782