Finite Group Databasebeta

← #33526

Subgroups — #33526

Order 256.

575 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#576C211yesmaximal, characteristic, complemented
2C2×#509C211yesmaximal, characteristic, complemented
3C2×#508C211yesmaximal, characteristic, complemented
4#3061C211yesmaximal, characteristic, complemented
5#2189C212yesmaximal, complemented
6#2187C212yesmaximal, complemented
7#2186C212yesmaximal, complemented
8#2184C212yesmaximal, complemented
9#2139C211yesmaximal, characteristic, complemented
10C2×#357C211yesmaximal, characteristic, complemented
11#2113C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13#576—24no—
14#576—24no—
15C2×AGL(1,ℤ/8)—21no—
16C2×AGL(1,ℤ/8)—22no—
17C2×#135C2212yes—
18C2×#135C2211yescharacteristic
19C2×#135C2211yescharacteristic, complemented
20C22×D8C2212yes—
21C22×D8C2211yescharacteristic
22C2×#131C2211yescharacteristic
23#537—21no—
24#509—24no—
25#508—24no—
26C2×Q32C2211yescharacteristic, complemented
27C2×SD32C2211yescharacteristic, complemented
28C2×SD32C2211yescharacteristic, complemented
29C2×SD32C2212yescomplemented
30C2×D16C2211yescharacteristic, complemented
31C2×D16C2212yescomplemented
32C2×M32C2211yescharacteristic, complemented
33C22×C16C2211yescharacteristic, complemented
34#468—22no—
35#467C2212yescomplemented
36#466C2212yescomplemented
37C2×#107C2211yescharacteristic, complemented
38#417—21no—
39#358C2212yescomplemented
40#357C2212yescomplemented
41#357C2212yescomplemented
42#357C2212yescomplemented
43C22⋊C16C2212yescomplemented
44C22⋊C16C2212yescomplemented
45H(5,𝔽2)—28no—
46H(5,𝔽2)—44no—
47C2×#41—22no—
48C2×#41—21no—
49C2×#41—21no—
50C2×#41—41no—
51C2×#41D411yescharacteristic
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—42no—
55C22×D4—41no—
56AGL(1,ℤ/8)—28no—
57AGL(1,ℤ/8)—216no—
58#135—24no—
59#135—24no—
60#135—22no—
61#135—22no—
62#135—22no—
63#135D414yescomplemented
64C2×Q16C2311yescharacteristic
65C2×Q16C2311yescharacteristic
66C2×SD16—22no—
67C2×SD16—21no—
68C2×SD16—21no—
69C2×D8—24no—
70C2×D8—24no—
71C2×D8—22no—
72C2×D8—21no—
73C2×D8—22no—
74C2×D8D414yescomplemented
75C2×D8C2311yescharacteristic
76C2×D8C2311yescharacteristic
77C2×D8C2312yes—
78#131—22no—
79#131—22no—
80C2×M16D411yescharacteristic
81C2×M16D412yes—
82C22×C8C2312yes—
83C22×C8C2311yescharacteristic
84C22⋊Q8—22no—
85C22⋊Q8—41no—
86#121—22no—
87#121—41no—
88#121—42no—
89C2×C4⋊C4—21no—
90Q32—22no—
91SD32—22no—
92SD32—22no—
93SD32—24no—
94D16—22no—
95D16—24no—
96M32—22no—
97C2×C16—22no—
98C2×C16C2311yescharacteristic
99C2×C16C2311yescharacteristic
100C2×C16C2312yes—
101#107—21no—
102#107C2312yes—
103#103—22no—
104#102—22no—
105C24—42no—
106C24—41no—
107#41—22no—
108#41—22no—
109#41—28no—
110#41—24no—
111#41—44no—
112#41—48no—
113#41—42no—
114#41—42no—
115#41—42no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8D811yescharacteristic
119C2×D4—28no—
120C2×D4—24no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—44no—
125C2×D4—416no—
126C2×D4—48no—
127C2×D4—42no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—41no—
133C2×D4—42no—
134C2×D4D812yes—
135C2×D4D811yescharacteristic
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142Q16—21no—
143Q16—21no—
144Q16—21no—
145Q16C2×D412yes—
146SD16—28no—
147SD16—24no—
148SD16—42no—
149D8—28no—
150D8—24no—
151D8—22no—
152D8—21no—
153D8—21no—
154D8—21no—
155D8C2×D412yes—
156D8C2×D414yes—
157M16—22no—
158M16—24no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8C2×D412yes—
162C2×C8C2×D411yescharacteristic
163C2×C8C2×D412yes—
164C2×C8C2×D412yes—
165C2×C8C2411yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C16—21no—
172C16—21no—
173C16—22no—
174C23—48no—
175C23—42no—
176C23—41no—
177C23—44no—
178C23—41no—
179C23—161no—
180C23—161no—
181C23—162no—
182C23—162no—
183C23C2×D812yes—
184C23C2×D811yescharacteristic
185Q8—22no—
186Q8—22no—
187Q8—41no—
188Q8—41no—
189Q8—41no—
190D4—24no—
191D4—22no—
192D4—24no—
193D4—22no—
194D4—48no—
195D4—48no—
196D4—44no—
197D4—41no—
198D4—41no—
199D4—42no—
200D4—41no—
201D4—44no—
202D4—82no—
203D4—162no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—48no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—161no—
213C2×C4#12012yes—
214C2×C4C2×D812yes—
215C2×C4C2×D811yescharacteristic
216C2×C4AGL(1,ℤ/8)12yes—
217C2×C4C22×D411yescharacteristic
218C8—22no—
219C8#12012yes—
220C8C22×D411yescharacteristic
221C8C22×D411yescharacteristic
222C22—21no—
223C22—21no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—44no—
228C22—42no—
229C22—82no—
230C22—81no—
231C22—82no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—164no—
237C22—168no—
238C22#44712yes—
239C22C22×D811yescharacteristic
240C4—22no—
241C4—42no—
242C4—82no—
243C4—81no—
244C4#44712yes—
245C4C2×#12011yescharacteristic
246C4C2×AGL(1,ℤ/8)11yescharacteristic
247C2—22no—
248C2—42no—
249C2—84no—
250C2—82no—
251C2—161no—
252C2C2×#44711yescharacteristic
253C2#341511yescharacteristic
254C2#341611yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#576C2semidirect
2. C2×#509C2semidirect
3. C2×#508C2semidirect
4. #3061C2semidirect
5. #2189C2semidirect
6. #2187C2semidirect
7. #2186C2semidirect
8. #2184C2semidirect
9. #2139C2semidirect
10. C2×#357C2semidirect
11. #2113C2semidirect
19. C2×#135C22semidirect
26. C2×Q32C22semidirect
27. C2×SD32C22semidirect
28. C2×SD32C22semidirect
29. C2×SD32C22semidirect
30. C2×D16C22semidirect
31. C2×D16C22semidirect
32. C2×M32C22semidirect
33. C22×C16C22semidirect
35. #467C22semidirect
36. #466C22semidirect
37. C2×#107C22semidirect
39. #358C22semidirect
40. #357C22semidirect
41. #357C22semidirect
42. #357C22semidirect
43. C22⋊C16C22semidirect
44. C22⋊C16C22semidirect
63. #135D4semidirect
74. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5762
C2×#5092
C2×#5082
#30612
#21892
#21872
#21862
#21842
#21392
C2×#3572
#21132