Finite Group Databasebeta

← #17620

Subgroups — #17620

Order 256.

635 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 183 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#3496C211yesmaximal, characteristic
2#2973C211yesmaximal, characteristic, complemented
3#2353C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1514C212yesmaximal, complemented
6#1510C212yesmaximal, complemented
7#1507C212yesmaximal, complemented
8#1505C212yesmaximal, complemented
9C2×#327C212yesmaximal, complemented
10C2×H(5,𝔽2)—21no—
11#547C414yescomplemented
12#546—24no—
13D42C414yescomplemented
14#537—21no—
15#534—21no—
16#534—21no—
17#532C2212yes—
18C2×#121C2212yes—
19C2×#121C2212yes—
20C2×C4×D4C2211yescharacteristic
21#435—24no—
22#432C2212yescomplemented
23#407C2212yescomplemented
24C2×C22⋊C8C2212yescomplemented
25#388—21no—
26#387—21no—
27#386—21no—
28#383C2211yescharacteristic, complemented
29#381C2211yescharacteristic, complemented
30#380C2211yescharacteristic, complemented
31C4×C22⋊C4—21no—
32C4×C22⋊C4C2211yescharacteristic, complemented
33#376C2211yescharacteristic, complemented
34C2×#95C2211yescharacteristic, complemented
35#327C2214yescomplemented
36#327C2214yescomplemented
37#327C2214yescomplemented
38#327C2214yescomplemented
39H(5,𝔽2)—44no—
40H(5,𝔽2)—44no—
41C2×#41—24no—
42C2×#41—22no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—24no—
46C22×D4C2×C412yes—
47C22×D4C2×C412yescomplemented
48C22×D4D411yescharacteristic
49C23×C4—21no—
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C2×M16—22no—
53C22×C8—22no—
54C4⋊Q8C2×C412yescomplemented
55#127C2×C412yescomplemented
56#124—22no—
57#123—24no—
58#123—24no—
59C22⋊Q8—24no—
60#121—22no—
61#121—22no—
62#121—22no—
63#121—22no—
64#121—22no—
65#121—24no—
66#121C2×C414yes—
67#121C2×C414yescomplemented
68#121C2314yes—
69#120—24no—
70#120—24no—
71C4×D4—22no—
72C4×D4C2×C414yes—
73C4×D4C2×C414yescomplemented
74C4×D4D414yes—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2×C412yes—
77C2×C4⋊C4C2×C412yescomplemented
78C2×C4⋊C4D411yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4D411yescharacteristic
85C2×C22⋊C4D411yescharacteristic
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89C4⋊C8—22no—
90C22⋊C8—22no—
91C22⋊C8—22no—
92C22⋊C8C2314yes—
93#97—22no—
94#95—21no—
95#95—21no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95C2311yescharacteristic
100#95C2311yescharacteristic
101#95C2311yescharacteristic
102#95C2311yescharacteristic
103C24—22no—
104C24—22no—
105C24C2×D411yescharacteristic
106C24C2×D411yescharacteristic
107#41—44no—
108#41—44no—
109#41—416no—
110C2×Q8—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—416no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4C22⋊C414yes—
128C2×D4C22×C414yes—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—24no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4C22×C412yes—
142C22×C4C22×C412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D412yes—
148C22×C4C2×D412yes—
149C22×C4C2411yescharacteristic
150M16—82no—
151C2×C8—24no—
152C2×C8—82no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—81no—
157C4⋊C4C22⋊C414yes—
158C4⋊C4C22×C414yes—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—81no—
166C22⋊C4C22⋊C414yes—
167C22⋊C4C22⋊C414yes—
168C42—42no—
169C42—81no—
170C42C22×C412yes—
171C42C2×D412yes—
172C23—21no—
173C23—21no—
174C23—24no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—44no—
180C23—44no—
181C23—82no—
182C23—82no—
183C23—82no—
184C23—82no—
185C23C2×C22⋊C412yes—
186C23C2×C22⋊C412yes—
187C23AGL(1,ℤ/8)12yes—
188C23AGL(1,ℤ/8)12yes—
189C23C22×D411yescharacteristic
190Q8—44no—
191D4—28no—
192D4—44no—
193D4—44no—
194D4—48no—
195D4—44no—
196D4—44no—
197D4—48no—
198D4—82no—
199D4—82no—
200D4—82no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—48no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—41no—
217C2×C4—44no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—84no—
228C2×C4C2×C22⋊C412yes—
229C2×C4C2×C22⋊C412yes—
230C2×C4C2×C22⋊C412yes—
231C2×C4C2×C22⋊C412yes—
232C2×C4C23×C411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C8—44no—
235C22—21no—
236C22—22no—
237C22—21no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22—44no—
245C22—44no—
246C22—44no—
247C22—81no—
248C22—81no—
249C22—164no—
250C22—164no—
251C22#41814yes—
252C22C22×C22⋊C411yescharacteristic
253C22C2×AGL(1,ℤ/8)11yescharacteristic
254C22C2×AGL(1,ℤ/8)11yescharacteristic
255C4—21no—
256C4—21no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—44no—
263C4—81no—
264C4—81no—
265C4—81no—
266C2—21no—
267C2—21no—
268C2—24no—
269C2—42no—
270C2—84no—
271C2#288311yescharacteristic
272C2C2×#41811yescharacteristic
273C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3496C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2973C2semidirect
3. #2353C2semidirect
4. #1519C2semidirect
5. #1514C2semidirect
6. #1510C2semidirect
7. #1507C2semidirect
8. #1505C2semidirect
9. C2×#327C2semidirect
11. #547C4semidirect
13. D42C4semidirect
22. #432C22semidirect
23. #407C22semidirect
24. C2×C22⋊C8C22semidirect
28. #383C22semidirect
29. #381C22semidirect
30. #380C22semidirect
32. C4×C22⋊C4C22semidirect
33. #376C22semidirect
34. C2×#95C22semidirect
35. #327C22semidirect
36. #327C22semidirect
37. #327C22semidirect
38. #327C22semidirect
47. C22×D4C2×C4semidirect
54. C4⋊Q8C2×C4semidirect
55. #127C2×C4semidirect
67. #121C2×C4semidirect
73. C4×D4C2×C4semidirect
77. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34962
#29732
#23532
#15192
#15142
#15102
#15072
#15052
C2×#3272