Finite Group Databasebeta

← #60810

Subgroups — #60810

Order 256.

1049 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2#3449C211yesmaximal, characteristic, complemented
3#3225C218yesmaximal, complemented
4#3224C218yesmaximal, complemented
5#3153C212yesmaximal, complemented
6#3141C212yesmaximal, complemented
7#3037C212yesmaximal, complemented
8#3036C212yesmaximal, complemented
9#3007C212yesmaximal, complemented
10#3006C212yesmaximal, complemented
11#2921C211yesmaximal, characteristic, complemented
12C22×#41C2212yes—
13C2×#137C2214yescomplemented
14C2×AGL(1,ℤ/8)C2214yescomplemented
15#542—24no—
16#542C2218yescomplemented
17#537—21no—
18#537—21no—
19#536C2214yescomplemented
20#535C2214yescomplemented
21#534—21no—
22#534—21no—
23#532C2212yes—
24#532C2212yes—
25C2×C4⋊Q8C2211yescharacteristic, complemented
26C2×#126—21no—
27C2×#124—21no—
28C2×#124C2211yescharacteristic, complemented
29#525—24no—
30C2×#123—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8C2211yescharacteristic
33C2×#121C2211yescharacteristic
34C2×C4×Q8C2211yescharacteristic, complemented
35C2×C4×D4C2211yescharacteristic, complemented
36C2×#117C2211yescharacteristic
37#497C2214yescomplemented
38#496C2214yescomplemented
39#495C2218yescomplemented
40#489C2218yescomplemented
41#487C2214yescomplemented
42#486C2214yescomplemented
43#464C2218yescomplemented
44#463C2218yescomplemented
45#462C2214yescomplemented
46#461C2214yescomplemented
47#460C2214yescomplemented
48#459C2214yescomplemented
49#452C2218yescomplemented
50#451C2214yescomplemented
51#450C2214yescomplemented
52#449C2218yes—
53#448C2214yescomplemented
54#447C2214yescomplemented
55#433C2218yescomplemented
56#423C2212yescomplemented
57#417C2214yescomplemented
58#417C2214yescomplemented
59#407C2212yes—
60C4×M16C2212yescomplemented
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—48no—
64C2×#41C2314yes—
65C2×#41C2314yes—
66C22×Q8—21no—
67C22×Q8C2311yescharacteristic
68C22×Q8C2312yes—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C22×D4C2312yes—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76#137—48no—
77AGL(1,ℤ/8)—48no—
78C2×Q16C2318yes—
79C2×SD16C2318yes—
80C2×SD16C2318yes—
81C2×D8C2318yes—
82C2×M16C2314yes—
83C4⋊Q8—22no—
84C4⋊Q8C2312yes—
85C4⋊Q8C2312yes—
86C4⋊Q8C2314yes—
87#127C2312yes—
88#127C2312yes—
89#126—28no—
90#124—28no—
91#124—22no—
92#124C2314yes—
93#124C2314yes—
94#124C2314yes—
95#123—28no—
96#123—24no—
97#123—24no—
98C22⋊Q8—22no—
99C22⋊Q8—28no—
100C22⋊Q8—24no—
101C22⋊Q8—24no—
102C22⋊Q8C2314yes—
103#121—22no—
104#121—22no—
105#121—22no—
106#121—22no—
107#121—22no—
108#121—24no—
109#121—24no—
110#121C2314yes—
111#120—24no—
112#120—24no—
113C4×Q8—22no—
114C4×Q8C2314yes—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4C2314yes—
121#117D414yes—
122#117C2314yes—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4D411yescharacteristic
127C2×C4⋊C4D411yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D411yescharacteristic
136C2×C22⋊C4D411yescharacteristic
137C2×C42—21no—
138C2×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140C4⋊C8C2314yes—
141C4⋊C8C2314yes—
142#103C2318yes—
143#103C2318yes—
144#102C2318yes—
145#102C2318yes—
146C22⋊C8C2314yes—
147C22⋊C8C2314yes—
148#97C2314yes—
149C4×C8C2314yes—
150C24—21no—
151C24—22no—
152C24C2×D411yescharacteristic
153#41—28no—
154#41—28no—
155#41—44no—
156#41—44no—
157#41—48no—
158#41—48no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—24no—
162C2×Q8—24no—
163C2×Q8—44no—
164C2×Q8—82no—
165C2×Q8C2414yes—
166C2×Q8C2412yes—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—24no—
177C2×D4—44no—
178C2×D4—48no—
179C2×D4—82no—
180C2×D4C2414yes—
181C2×D4C2412yes—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—48no—
191C22×C4—82no—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2412yes—
199C22×C4C2411yescharacteristic
200Q16—48no—
201SD16—48no—
202SD16—48no—
203D8—48no—
204M16—28no—
205C2×C8C2418yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4C2×D414yes—
216C4⋊C4C2×D414yes—
217C4⋊C4C2412yes—
218C4⋊C4C2412yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4C2×D414yes—
229C22⋊C4C2×D414yes—
230C42—24no—
231C42C2×D412yes—
232C42C2×D412yes—
233C42C2412yes—
234C42C2412yes—
235C23—22no—
236C23—22no—
237C23—24no—
238C23—41no—
239C23—41no—
240C23—44no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—82no—
245C23—82no—
246C23C22×D412yes—
247C23C22×D411yescharacteristic
248Q8—24no—
249Q8—24no—
250Q8—24no—
251Q8—44no—
252Q8—42no—
253Q8—84no—
254D4—24no—
255D4—24no—
256D4—24no—
257D4—44no—
258D4—48no—
259D4—48no—
260D4—42no—
261D4—82no—
262D4—82no—
263D4—82no—
264D4—82no—
265D4—84no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—48no—
283C2×C4—84no—
284C2×C4C22×D412yes—
285C2×C4C22×D412yes—
286C2×C4C22×D412yes—
287C2×C4C22×D412yes—
288C2×C4C22×D412yes—
289C2×C4C22×D411yescharacteristic
290C2×C4H(5,𝔽2)14yes—
291C2×C4C2511yescharacteristic
292C8—28no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—42no—
299C22—44no—
300C22—44no—
301C22—42no—
302C22—44no—
303C22—42no—
304C22—82no—
305C22—84no—
306C22—88no—
307C22#53512yes—
308C22C23×D411yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—44no—
317C4—42no—
318C4#53512yes—
319C4C2×H(5,𝔽2)12yes—
320C2—22no—
321C2—42no—
322C2—82no—
323C2—84no—
324C2C2×#53511yescharacteristic
325C2#358511yescharacteristic
326C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. #3449C2semidirect
3. #3225C2semidirect
4. #3224C2semidirect
5. #3153C2semidirect
6. #3141C2semidirect
7. #3037C2semidirect
8. #3036C2semidirect
9. #3007C2semidirect
10. #3006C2semidirect
11. #2921C2semidirect
13. C2×#137C22semidirect
14. C2×AGL(1,ℤ/8)C22semidirect
16. #542C22semidirect
19. #536C22semidirect
20. #535C22semidirect
25. C2×C4⋊Q8C22semidirect
28. C2×#124C22semidirect
34. C2×C4×Q8C22semidirect
35. C2×C4×D4C22semidirect
37. #497C22semidirect
38. #496C22semidirect
39. #495C22semidirect
40. #489C22semidirect
41. #487C22semidirect
42. #486C22semidirect
43. #464C22semidirect
44. #463C22semidirect
45. #462C22semidirect
46. #461C22semidirect
47. #460C22semidirect
48. #459C22semidirect
49. #452C22semidirect
50. #451C22semidirect
51. #450C22semidirect
53. #448C22semidirect
54. #447C22semidirect
55. #433C22semidirect
56. #423C22semidirect
57. #417C22semidirect
58. #417C22semidirect
60. C4×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
#34492
#32252
#32242
#31532
#31412
#30372
#30362
#30072
#30062
#29212