Finite Group Databasebeta

← #61797

Subgroups — #61797

Order 256.

901 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3512C211yesmaximal, characteristic, complemented
3#3348C214yesmaximal, complemented
4#3347C214yesmaximal, complemented
5#3318C214yesmaximal, complemented
6#3310C214yesmaximal, complemented
7#3234C212yesmaximal, complemented
8#3198C214yesmaximal, complemented
9#3115C212yesmaximal, complemented
10#3096C214yesmaximal, complemented
11#2980C211yesmaximal, characteristic, complemented
12H(3,𝔽4)—22no—
13#555—21no—
14#555C2214yes—
15#553—24no—
16#553C2214yescomplemented
17#552C2214yescomplemented
18D42—21no—
19D42C2214yes—
20#541—21no—
21#540—21no—
22#537C2212yescomplemented
23#535—24no—
24#535C2214yes—
25#532C2212yes—
26#532C2211yescharacteristic
27#529—22no—
28C2×#125C2211yescharacteristic, complemented
29#520C2212yescomplemented
30C4×#41C2211yescharacteristic, complemented
31#501C2214yescomplemented
32#490C2214yes—
33#485C2214yescomplemented
34#482C2214yescomplemented
35#482C2214yescomplemented
36#482C2214yescomplemented
37#482C2214yescomplemented
38#481C2214yescomplemented
39#481C2214yescomplemented
40#480C2214yescomplemented
41#480C2214yescomplemented
42#478C2214yescomplemented
43#478C2214yescomplemented
44#475C2212yescomplemented
45#474C2212yescomplemented
46#469C2214yescomplemented
47#468C2214yescomplemented
48#466C2214yescomplemented
49#465C2214yescomplemented
50#460C2212yes—
51#459C2212yes—
52#450C2214yescomplemented
53#449C2214yescomplemented
54#447C2214yescomplemented
55#442C2214yescomplemented
56#440C2214yescomplemented
57#435C2214yescomplemented
58#435C2214yescomplemented
59#432C2212yescomplemented
60#428C2214yescomplemented
61#427C2214yescomplemented
62#418C2214yescomplemented
63C2×#102C2214yescomplemented
64#408C2214yescomplemented
65C2×#41—22no—
66C2×#41C2311yescharacteristic
67C2×#41C2312yes—
68C22×D4—24no—
69C22×D4C2314yes—
70C2×SD16C2314yes—
71C2×D8C2314yes—
72C2×M16C2314yes—
73C22×C8C2314yes—
74C4⋊Q8D411yescharacteristic
75C4⋊Q8C2312yes—
76#127—24no—
77#127D411yescharacteristic
78#127C2312yes—
79#126—24no—
80#126—24no—
81#125—24no—
82#125—21no—
83#125—21no—
84#125C2314yes—
85#124—24no—
86#124—24no—
87#124—24no—
88#124—22no—
89#124D412yes—
90#123—24no—
91#123—24no—
92C22⋊Q8—22no—
93C22⋊Q8C2314yes—
94#121—22no—
95#121—24no—
96#121—24no—
97#121—24no—
98#121—22no—
99#121—24no—
100#121D414yes—
101#121C2314yes—
102#121C2314yes—
103#121C2314yes—
104#121C2314yes—
105#120—24no—
106#120—24no—
107#120—24no—
108#120—24no—
109C4×Q8—22no—
110C4×Q8C2312yes—
111C4×D4—22no—
112C4×D4—24no—
113C4×D4—22no—
114C4×D4—24no—
115C4×D4—22no—
116C4×D4C2312yes—
117C4×D4C2314yes—
118C4×D4C2314yes—
119#117—24no—
120#117—24no—
121#117—21no—
122#117C2311yescharacteristic
123#117C2314yes—
124#117C2314yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2314yes—
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130#107C2314yes—
131#107C2314yes—
132#107C2314yes—
133#106C2314yes—
134#106C2314yes—
135#106C2314yes—
136C4⋊C8C2312yes—
137C4⋊C8C2312yes—
138#103C2314yes—
139#102C2314yes—
140#102C2314yes—
141#102C2314yes—
142#102C2314yes—
143#102C2314yes—
144#102C2314yes—
145#102C2314yes—
146C22⋊C8C2314yes—
147C22⋊C8C2314yes—
148C22⋊C8C2314yes—
149#97C2314yes—
150C24—24no—
151#41—42no—
152#41—42no—
153#41—42no—
154#41—84no—
155C2×Q8—22no—
156C2×Q8C2×D412yes—
157C2×Q8C2411yescharacteristic
158C2×D4—21no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—24no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—88no—
169C2×D4C2×D412yes—
170C2×D4C2×D414yes—
171C2×D4C2×D414yes—
172C2×D4C2414yes—
173C2×D4C2411yescharacteristic
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4C2×D412yes—
180C22×C4C2411yescharacteristic
181C22×C4C2414yes—
182SD16—44no—
183D8—44no—
184M16—44no—
185C2×C8—44no—
186C2×C8C2414yes—
187C2×C8C2414yes—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4C2×D414yes—
196C4⋊C4C2412yes—
197C4⋊C4C2414yes—
198C4⋊C4C2414yes—
199C22⋊C4—24no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—24no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4C2×D414yes—
210C22⋊C4C2×D414yes—
211C42—22no—
212C42—21no—
213C42—24no—
214C42—24no—
215C42C2×D411yescharacteristic
216C42C2×D411yescharacteristic
217C42C2412yes—
218C23—21no—
219C23—24no—
220C23—84no—
221C23—84no—
222C23—84no—
223C23C22×D414yes—
224C23C22×D411yescharacteristic
225Q8—22no—
226Q8—42no—
227Q8—82no—
228D4—22no—
229D4—24no—
230D4—24no—
231D4—44no—
232D4—42no—
233D4—44no—
234D4—44no—
235D4—42no—
236D4—84no—
237D4—82no—
238D4—84no—
239D4—84no—
240D4—84no—
241D4—84no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—48no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4C22×D414yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D412yes—
258C2×C4C22×D412yes—
259C2×C4C2511yescharacteristic
260C8—24no—
261C8—24no—
262C22—22no—
263C22—24no—
264C22—24no—
265C22—21no—
266C22—24no—
267C22—44no—
268C22—42no—
269C22—44no—
270C22—41no—
271C22—44no—
272C22—168no—
273C22C23×D411yescharacteristic
274C4—22no—
275C4—22no—
276C4—21no—
277C4—24no—
278C4—44no—
279C4—42no—
280C4—44no—
281C4—41no—
282C4#58512yes—
283C2—41no—
284C2—44no—
285C2—81no—
286C2—84no—
287C2#348511yescharacteristic
288C2#358512yes—
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. #3512C2semidirect
3. #3348C2semidirect
4. #3347C2semidirect
5. #3318C2semidirect
6. #3310C2semidirect
7. #3234C2semidirect
8. #3198C2semidirect
9. #3115C2semidirect
10. #3096C2semidirect
11. #2980C2semidirect
16. #553C22semidirect
17. #552C22semidirect
22. #537C22semidirect
28. C2×#125C22semidirect
29. #520C22semidirect
30. C4×#41C22semidirect
31. #501C22semidirect
33. #485C22semidirect
34. #482C22semidirect
35. #482C22semidirect
36. #482C22semidirect
37. #482C22semidirect
38. #481C22semidirect
39. #481C22semidirect
40. #480C22semidirect
41. #480C22semidirect
42. #478C22semidirect
43. #478C22semidirect
44. #475C22semidirect
45. #474C22semidirect
46. #469C22semidirect
47. #468C22semidirect
48. #466C22semidirect
49. #465C22semidirect
52. #450C22semidirect
53. #449C22semidirect
54. #447C22semidirect
55. #442C22semidirect
56. #440C22semidirect
57. #435C22semidirect
58. #435C22semidirect
59. #432C22semidirect
60. #428C22semidirect
61. #427C22semidirect
62. #418C22semidirect
63. C2×#102C22semidirect
64. #408C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#35122
#33482
#33472
#33182
#33102
#32342
#31982
#31152
#30962
#29802