Finite Group Databasebeta

← #18620

Subgroups — #18620

Order 256.

705 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3498C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3C2×#493C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic, complemented
5#1621C214yesmaximal, complemented
6#1619C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11C2×AGL(1,ℤ/8)—21no—
12C22×D8—22no—
13#550—21no—
14#550C2212yescomplemented
15D42—22no—
16D42—22no—
17D42—21no—
18D42C2212yescomplemented
19#535—22no—
20#535—21no—
21#532—21no—
22#532C2211yescharacteristic
23C2×#127—21no—
24C2×#127C2211yescharacteristic
25C2×#127C2211yescharacteristic, complemented
26C4×#41C2211yescharacteristic
27#493—22no—
28#493—21no—
29#493C2214yescomplemented
30#486—21no—
31#460—21no—
32#460C2212yescomplemented
33#459—21no—
34#459C2212yescomplemented
35#459C2214yescomplemented
36#447—22no—
37#431C2211yescharacteristic, complemented
38#423C2211yescharacteristic, complemented
39#418—21no—
40C2×C4×C8C2211yescharacteristic, complemented
41#331C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43D4⋊C8C2212yescomplemented
44D4⋊C8C2214yescomplemented
45H(5,𝔽2)—44no—
46H(5,𝔽2)—44no—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—21no—
50C2×#41D411yescharacteristic
51C22×D4—22no—
52C22×D4—24no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—22no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4—21no—
61C22×D4D412yes—
62AGL(1,ℤ/8)—44no—
63C2×SD16—22no—
64C2×D8—28no—
65C2×D8—22no—
66C2×D8—24no—
67C2×D8—48no—
68C2×M16—21no—
69C22×C8—21no—
70C4⋊Q8D411yescharacteristic
71#127—22no—
72#127—22no—
73#127—22no—
74#127—21no—
75#127—21no—
76#127—21no—
77#127—21no—
78#127—21no—
79#127D411yescharacteristic
80#127D412yes—
81#127C2312yes—
82#127C2312yes—
83#124—22no—
84#124—21no—
85#121—22no—
86#121—24no—
87#121—24no—
88#121—21no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121D412yes—
93#120—24no—
94#120—24no—
95#120—24no—
96C4×Q8D411yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×D4—22no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4D411yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2312yes—
106#117—21no—
107#117D411yescharacteristic
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C4⋊C8—21no—
111C4⋊C8C2312yes—
112C4⋊C8C2312yes—
113#102—22no—
114#102—22no—
115#102—24no—
116C22⋊C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8C2312yes—
120C24—24no—
121C24—22no—
122C24—41no—
123#41—22no—
124#41—48no—
125#41—44no—
126#41—44no—
127#41—42no—
128#41—42no—
129#41D814yescomplemented
130C2×Q8—21no—
131C2×Q8C2×D411yescharacteristic
132C2×D4—21no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—48no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—416no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4—84no—
164C2×D4—84no—
165C2×D4C2×D412yes—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D414yes—
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176SD16—44no—
177D8—416no—
178D8—44no—
179D8—84no—
180M16—42no—
181C2×C8—24no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4C2×D411yescharacteristic
189C4⋊C4C2×D411yescharacteristic
190C4⋊C4C2×D412yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C22⋊C4—22no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2411yescharacteristic
205C23—21no—
206C23—22no—
207C23—24no—
208C23—48no—
209C23—44no—
210C23—42no—
211C23—82no—
212C23—82no—
213C23—82no—
214C23—82no—
215C23—84no—
216C23—84no—
217C23—82no—
218C23#12011yescharacteristic
219Q8—21no—
220Q8—42no—
221Q8C2×D812yescomplemented
222D4—24no—
223D4—21no—
224D4—24no—
225D4—44no—
226D4—48no—
227D4—44no—
228D4—48no—
229D4—44no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—44no—
235D4—44no—
236D4—42no—
237D4—44no—
238D4—44no—
239D4—42no—
240D4—82no—
241D4—82no—
242D4—84no—
243D4—84no—
244D4—82no—
245D4—82no—
246D4C2×D812yescomplemented
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—48no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#12011yescharacteristic
260C2×C4#12011yescharacteristic
261C2×C4#12011yescharacteristic
262C2×C4C2×D812yes—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C8—24no—
268C8—42no—
269C8—42no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—44no—
276C22—42no—
277C22—42no—
278C22—44no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—164no—
285C22—164no—
286C22—168no—
287C22#44712yes—
288C22C2×#12011yescharacteristic
289C4—22no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4#44712yes—
296C4#45312yes—
297C4C22×D811yescharacteristic
298C4C2×AGL(1,ℤ/8)11yescharacteristic
299C4#57612yes—
300C2—22no—
301C2—42no—
302C2—82no—
303C2—84no—
304C2—162no—
305C2C2×#44711yescharacteristic
306C2#300811yescharacteristic
307C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3498C2semidirect
2. #3205C2semidirect
3. C2×#493C2semidirect
4. #3038C2semidirect
5. #1621C2semidirect
6. #1619C2semidirect
7. #1545C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
14. #550C22semidirect
18. D42C22semidirect
25. C2×#127C22semidirect
29. #493C22semidirect
32. #460C22semidirect
34. #459C22semidirect
35. #459C22semidirect
37. #431C22semidirect
38. #423C22semidirect
40. C2×C4×C8C22semidirect
41. #331C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect
129. #41D8semidirect
221. Q8C2×D8semidirect
246. D4C2×D8semidirect

Maximal subgroups

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

subgroupindex
#34982
#32052
C2×#4932
#30382
#16212
#16192
#15452
#14922
#14762