Finite Group Databasebeta

← #62342

Subgroups — #62342

Order 256.

974 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3548C211yesmaximal, characteristic, complemented
3#3357C212yesmaximal, complemented
4#3354C212yesmaximal, complemented
5#3305C212yesmaximal, complemented
6#3303C212yesmaximal, complemented
7#3295C212yesmaximal, complemented
8#3294C212yesmaximal, complemented
9#3274C212yesmaximal, complemented
10#3272C212yesmaximal, complemented
11#3254C211yesmaximal, characteristic, complemented
12#3253C211yesmaximal, characteristic, complemented
13#3198C214yesmaximal, complemented
14#3169C211yesmaximal, characteristic, complemented
15#3167C211yesmaximal, characteristic, complemented
16#3062C214yesmaximal, complemented
17#2931C211yesmaximal, characteristic, complemented
18C2×AGL(1,ℤ/8)C2214yescomplemented
19C2×#131C2212yescomplemented
20#562—22no—
21H(3,𝔽4)—22no—
22#555—21no—
23#555C2212yes—
24#555C2212yes—
25#554—21no—
26#554C2212yescomplemented
27#550—21no—
28#550C2212yescomplemented
29#547C2212yescomplemented
30#547C2212yescomplemented
31D42—21no—
32D42C2212yes—
33D42C2212yes—
34#540—22no—
35#537—22no—
36#537C2214yescomplemented
37#535—22no—
38#535—22no—
39#535C2214yes—
40#533C2211yescharacteristic, complemented
41#532C2211yescharacteristic
42#532C2211yescharacteristic
43#532C2211yescharacteristic
44#532C2211yescharacteristic, complemented
45#500C2211yescharacteristic, complemented
46#498C2211yescharacteristic, complemented
47#493C2211yescharacteristic
48#492C2211yescharacteristic
49#487C2211yescharacteristic, complemented
50#486C2211yescharacteristic
51#486C2211yescharacteristic
52#486C2211yescharacteristic, complemented
53#485C2214yescomplemented
54#482C2214yescomplemented
55#481C2214yescomplemented
56#480C2214yescomplemented
57#477C2212yescomplemented
58#476C2212yescomplemented
59#476C2212yescomplemented
60#474C2212yescomplemented
61#469C2214yescomplemented
62#469C2214yescomplemented
63#468C2214yescomplemented
64#468C2214yescomplemented
65#466C2212yescomplemented
66#466C2212yescomplemented
67#466C2214yescomplemented
68#465C2212yescomplemented
69#465C2212yescomplemented
70#465C2214yescomplemented
71#461C2212yescomplemented
72#460C2212yes—
73#459C2212yes—
74#459C2212yescomplemented
75#450C2214yescomplemented
76#449C2214yescomplemented
77#449C2214yescomplemented
78#447C2214yescomplemented
79C4×SD16C2212yescomplemented
80C4×SD16C2212yescomplemented
81C4×D8C2212yescomplemented
82C4×D8C2212yescomplemented
83#436C2212yescomplemented
84#436C2212yescomplemented
85C8×D4C2212yescomplemented
86C8×D4C2212yescomplemented
87#428C2214yescomplemented
88#423C2211yescharacteristic, complemented
89#416C2214yescomplemented
90C2×#102C2214yescomplemented
91#408C2212yescomplemented
92C2×C4×C8C2211yescharacteristic, complemented
93C2×#41—22no—
94C2×#41—21no—
95C2×#41—21no—
96C2×#41—21no—
97C2×#41C2314yes—
98C2×#41C2312yes—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4C2314yes—
102AGL(1,ℤ/8)—48no—
103C2×SD16C2312yes—
104C2×SD16C2314yes—
105C2×SD16C2312yes—
106C2×D8C2312yes—
107C2×D8C2314yes—
108C2×D8C2312yes—
109#131—44no—
110C2×M16C2314yes—
111C2×M16C2312yes—
112C22×C8C2311yescharacteristic
113C22×C8C2314yes—
114C22×C8C2311yescharacteristic
115C4⋊Q8D411yescharacteristic
116C4⋊Q8C2311yescharacteristic
117C4⋊Q8C2311yescharacteristic
118C4⋊Q8C2311yescharacteristic
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2311yescharacteristic
121#127—24no—
122#127—22no—
123#127D411yescharacteristic
124#127C2311yescharacteristic
125#127C2311yescharacteristic
126#127C2311yescharacteristic
127#125—22no—
128#125—21no—
129#124—24no—
130#124—24no—
131#124—24no—
132#124—21no—
133#124—21no—
134#124—21no—
135#124D412yes—
136#123—24no—
137#123—24no—
138C22⋊Q8—24no—
139C22⋊Q8—21no—
140C22⋊Q8—22no—
141C22⋊Q8—24no—
142C22⋊Q8C2312yes—
143C22⋊Q8C2314yes—
144#121—22no—
145#121—22no—
146#121—24no—
147#121—24no—
148#121—22no—
149#121—24no—
150#121—21no—
151#121—21no—
152#121—21no—
153#121—22no—
154#121—22no—
155#121—24no—
156#121D412yes—
157#121D412yes—
158#121C2312yes—
159#121C2312yes—
160#121C2312yes—
161#121C2314yes—
162#121C2314yes—
163#121C2314yes—
164#120—24no—
165#120—24no—
166#120—24no—
167#120—24no—
168C4×Q8—21no—
169C4×Q8C2312yes—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4C2312yes—
176C4×D4C2312yes—
177C4×D4C2312yes—
178C4×D4C2312yes—
179C4×D4C2312yes—
180#117—21no—
181#117—22no—
182#117—22no—
183#117C2314yes—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C2314yes—
186C2×C42C2311yescharacteristic
187#107C2312yes—
188#107C2314yes—
189#107C2312yes—
190#106C2312yes—
191#106C2314yes—
192#106C2312yes—
193C4⋊C8C2312yes—
194C4⋊C8C2312yes—
195#103C2312yes—
196#103C2314yes—
197#103C2312yes—
198#102C2312yes—
199#102C2312yes—
200#102C2314yes—
201#102C2314yes—
202#102C2312yes—
203#102C2314yes—
204#102C2312yes—
205#102C2312yes—
206#102C2312yes—
207C22⋊C8C2314yes—
208C22⋊C8C2314yes—
209C4×C8C2311yescharacteristic
210C4×C8C2311yescharacteristic
211C4×C8C2311yescharacteristic
212C4×C8C2311yescharacteristic
213C24—24no—
214#41—42no—
215#41—42no—
216#41—44no—
217#41—44no—
218#41—82no—
219#41—82no—
220#41—82no—
221#41—84no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8C2×D412yes—
227C2×Q8C2412yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—24no—
241C2×D4—21no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—84no—
245C2×D4—84no—
246C2×D4C2×D412yes—
247C2×D4C2×D412yes—
248C2×D4C2×D412yes—
249C2×D4C2×D414yes—
250C2×D4C2412yes—
251C2×D4C2412yes—
252C2×D4C2412yes—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D411yescharacteristic
260C22×C4C2414yes—
261C22×C4C2411yescharacteristic
262SD16—24no—
263SD16—42no—
264SD16—44no—
265D8—24no—
266D8—42no—
267D8—44no—
268M16—28no—
269M16—42no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—44no—
273C2×C8C2411yescharacteristic
274C2×C8C2411yescharacteristic
275C2×C8C2411yescharacteristic
276C2×C8C2414yes—
277C2×C8C2411yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4C2×D412yes—
289C4⋊C4C2×D412yes—
290C4⋊C4C2412yes—
291C4⋊C4C2412yes—
292C4⋊C4C2412yes—
293C4⋊C4C2412yes—
294C22⋊C4—22no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—24no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4C2×D414yes—
307C22⋊C4C2×D414yes—
308C42—24no—
309C42—22no—
310C42C2×D411yescharacteristic
311C42C2×D411yescharacteristic
312C42C2411yescharacteristic
313C42C2411yescharacteristic
314C23—22no—
315C23—22no—
316C23—22no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23C22×D414yes—
321C23C22×D411yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328Q8—81no—
329D4—24no—
330D4—24no—
331D4—22no—
332D4—22no—
333D4—22no—
334D4—22no—
335D4—22no—
336D4—22no—
337D4—44no—
338D4—42no—
339D4—42no—
340D4—42no—
341D4—42no—
342D4—84no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—84no—
347D4—84no—
348D4—84no—
349D4—82no—
350D4—82no—
351D4—84no—
352D4—82no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4C22×D414yes—
370C2×C4C22×D411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D411yescharacteristic
375C2×C4C2511yescharacteristic
376C8—21no—
377C8—24no—
378C8—21no—
379C8—21no—
380C8H(5,𝔽2)12yes—
381C22—21no—
382C22—24no—
383C22—21no—
384C22—24no—
385C22—21no—
386C22—24no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—168no—
397C22C23×D411yescharacteristic
398C4—24no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4#57612yes—
408C4#57712yes—
409C4C2×H(5,𝔽2)11yescharacteristic
410C4#58511yescharacteristic
411C2—41no—
412C2—44no—
413C2—82no—
414C2—82no—
415C2—82no—
416C2#349611yescharacteristic
417C2C2×#57611yescharacteristic
418C2C2×#57711yescharacteristic
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. #3548C2semidirect
3. #3357C2semidirect
4. #3354C2semidirect
5. #3305C2semidirect
6. #3303C2semidirect
7. #3295C2semidirect
8. #3294C2semidirect
9. #3274C2semidirect
10. #3272C2semidirect
11. #3254C2semidirect
12. #3253C2semidirect
13. #3198C2semidirect
14. #3169C2semidirect
15. #3167C2semidirect
16. #3062C2semidirect
17. #2931C2semidirect
18. C2×AGL(1,ℤ/8)C22semidirect
19. C2×#131C22semidirect
26. #554C22semidirect
28. #550C22semidirect
29. #547C22semidirect
30. #547C22semidirect
36. #537C22semidirect
40. #533C22semidirect
44. #532C22semidirect
45. #500C22semidirect
46. #498C22semidirect
49. #487C22semidirect
52. #486C22semidirect
53. #485C22semidirect
54. #482C22semidirect
55. #481C22semidirect
56. #480C22semidirect
57. #477C22semidirect
58. #476C22semidirect
59. #476C22semidirect
60. #474C22semidirect
61. #469C22semidirect
62. #469C22semidirect
63. #468C22semidirect
64. #468C22semidirect
65. #466C22semidirect
66. #466C22semidirect
67. #466C22semidirect
68. #465C22semidirect
69. #465C22semidirect
70. #465C22semidirect
71. #461C22semidirect
74. #459C22semidirect
75. #450C22semidirect
76. #449C22semidirect
77. #449C22semidirect
78. #447C22semidirect
79. C4×SD16C22semidirect
80. C4×SD16C22semidirect
81. C4×D8C22semidirect
82. C4×D8C22semidirect
83. #436C22semidirect
84. #436C22semidirect
85. C8×D4C22semidirect
86. C8×D4C22semidirect
87. #428C22semidirect
88. #423C22semidirect
89. #416C22semidirect
90. C2×#102C22semidirect
91. #408C22semidirect
92. C2×C4×C8C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#35482
#33572
#33542
#33052
#33032
#32952
#32942
#32742
#32722
#32542
#32532
#31982
#31692
#31672
#30622
#29312