Finite Group Databasebeta

← #61707

Subgroups — #61707

Order 256.

1043 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3496C211yesmaximal, characteristic, complemented
2C2×#553C211yesmaximal, characteristic, complemented
3#3345C218yesmaximal, complemented
4#3318C218yesmaximal, complemented
5#3234C212yesmaximal, complemented
6#3192C212yesmaximal, complemented
7#3186C212yesmaximal, complemented
8#3109C212yesmaximal, complemented
9#3094C212yesmaximal, complemented
10C2×#481C212yesmaximal, complemented
11#2973C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13#553—24no—
14#553C2218yescomplemented
15#547C2214yescomplemented
16#546—24no—
17D42C2214yescomplemented
18#537—21no—
19#534—21no—
20#534—21no—
21#532C2212yes—
22C2×#126—21no—
23C2×#125C2211yescharacteristic, complemented
24C2×#123—21no—
25C2×#121—21no—
26C2×#121C2212yes—
27C2×#121C2212yes—
28C2×#121C2211yescharacteristic, complemented
29C2×#121C2211yescharacteristic, complemented
30C2×C4×D4—21no—
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic, complemented
33C2×#117C2211yescharacteristic, complemented
34C22×C4⋊C4C2211yescharacteristic, complemented
35#501C2214yescomplemented
36#490C2214yescomplemented
37#482C2218yescomplemented
38#481C2218yescomplemented
39#481C2214yescomplemented
40#481C2214yescomplemented
41#481C2214yescomplemented
42#480C2214yescomplemented
43#480C2214yescomplemented
44#478C2218yescomplemented
45#476C2214yescomplemented
46#469C2218yescomplemented
47#466C2218yescomplemented
48#459C2214yescomplemented
49#447C2214yescomplemented
50#447C2214yescomplemented
51#447C2214yescomplemented
52#442C2218yescomplemented
53#435C2218yescomplemented
54#432C2212yescomplemented
55#428C2214yescomplemented
56C2×#106C2214yescomplemented
57#418C2214yescomplemented
58C2×#102C2214yescomplemented
59#407C2212yescomplemented
60C2×C22⋊C8C2212yescomplemented
61H(5,𝔽2)—48no—
62C2×#41—24no—
63C2×#41—22no—
64C22×D4—21no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—24no—
69C22×D4D411yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2312yes—
72C22×D4C2312yes—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×D8C2318yes—
78C2×M16C2314yes—
79C22×C8C2314yes—
80C4⋊Q8C2312yes—
81#127C2312yes—
82#126—28no—
83#125—22no—
84#125C2314yes—
85#124—22no—
86#123—28no—
87#123—24no—
88#123—24no—
89C22⋊Q8—24no—
90#121—28no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—22no—
98#121—24no—
99#121C2314yes—
100#121C2314yes—
101#121C2314yes—
102#121C2314yes—
103#121C2314yes—
104#120—24no—
105#120—24no—
106C4×D4—28no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4D414yes—
110C4×D4C2314yes—
111C4×D4C2314yes—
112C4×D4C2314yes—
113#117—22no—
114#117C2314yes—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4D411yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2314yes—
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42—21no—
135C2×C42C2311yescharacteristic
136#107C2318yes—
137#106C2318yes—
138#106C2318yes—
139C4⋊C8C2314yes—
140#102C2318yes—
141#102C2318yes—
142#102C2318yes—
143#102C2318yes—
144C22⋊C8C2314yes—
145C22⋊C8C2314yes—
146C22⋊C8C2314yes—
147#97C2314yes—
148C24—21no—
149C24—22no—
150C24—22no—
151C24C2×D411yescharacteristic
152C24C2×D411yescharacteristic
153#41—48no—
154#41—416no—
155C2×Q8—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—416no—
171C2×D4—48no—
172C2×D4—44no—
173C2×D4—48no—
174C2×D4—82no—
175C2×D4C2×D414yes—
176C2×D4C2414yes—
177C2×D4C2412yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D412yes—
195C22×C4C2×D412yes—
196C22×C4C2411yescharacteristic
197C22×C4C2412yes—
198C22×C4C2412yes—
199D8—48no—
200M16—44no—
201C2×C8—44no—
202C2×C8C2418yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—21no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4C2×D414yes—
213C4⋊C4C2412yes—
214C4⋊C4C2412yes—
215C4⋊C4C2412yes—
216C4⋊C4C2414yes—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4C2×D414yes—
226C22⋊C4C2×D414yes—
227C42—24no—
228C42C2×D412yes—
229C42C2412yes—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—42no—
241C23—44no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23AGL(1,ℤ/8)14yes—
247C23C22×D412yes—
248C23C22×D412yes—
249C23C22×D411yescharacteristic
250Q8—44no—
251D4—28no—
252D4—24no—
253D4—28no—
254D4—42no—
255D4—44no—
256D4—48no—
257D4—48no—
258D4—48no—
259D4—48no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4—84no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—48no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4C22×D412yes—
286C2×C4C22×D411yescharacteristic
287C2×C4C22×D412yes—
288C2×C4C22×D412yes—
289C2×C4C22×D412yes—
290C2×C4C2511yescharacteristic
291C8—28no—
292C22—21no—
293C22—22no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—44no—
300C22—42no—
301C22—44no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—82no—
307C22—84no—
308C22—164no—
309C22—164no—
310C22C2×AGL(1,ℤ/8)12yes—
311C22C2×AGL(1,ℤ/8)14yes—
312C22C23×D411yescharacteristic
313C4—22no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—42no—
320C4—44no—
321C4—42no—
322C4#58512yes—
323C2—22no—
324C2—24no—
325C2—42no—
326C2—82no—
327C2—84no—
328C2#348511yescharacteristic
329C2C22×AGL(1,ℤ/8)11yescharacteristic
330C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3496C2semidirect
2. C2×#553C2semidirect
3. #3345C2semidirect
4. #3318C2semidirect
5. #3234C2semidirect
6. #3192C2semidirect
7. #3186C2semidirect
8. #3109C2semidirect
9. #3094C2semidirect
10. C2×#481C2semidirect
11. #2973C2semidirect
14. #553C22semidirect
15. #547C22semidirect
17. D42C22semidirect
23. C2×#125C22semidirect
28. C2×#121C22semidirect
29. C2×#121C22semidirect
32. C2×C4×D4C22semidirect
33. C2×#117C22semidirect
34. C22×C4⋊C4C22semidirect
35. #501C22semidirect
36. #490C22semidirect
37. #482C22semidirect
38. #481C22semidirect
39. #481C22semidirect
40. #481C22semidirect
41. #481C22semidirect
42. #480C22semidirect
43. #480C22semidirect
44. #478C22semidirect
45. #476C22semidirect
46. #469C22semidirect
47. #466C22semidirect
48. #459C22semidirect
49. #447C22semidirect
50. #447C22semidirect
51. #447C22semidirect
52. #442C22semidirect
53. #435C22semidirect
54. #432C22semidirect
55. #428C22semidirect
56. C2×#106C22semidirect
57. #418C22semidirect
58. C2×#102C22semidirect
59. #407C22semidirect
60. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34962
C2×#5532
#33452
#33182
#32342
#31922
#31862
#31092
#30942
C2×#4812
#29732