Finite Group Databasebeta

← #33516

Subgroups — #33516

Order 256.

575 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×#135C211yesmaximal, characteristic, complemented
2C2×#508C212yesmaximal, complemented
3#3050C211yesmaximal, characteristic, complemented
4#2189C212yesmaximal, complemented
5#2188C211yesmaximal, characteristic, complemented
6#2187C211yesmaximal, characteristic, complemented
7#2186C212yesmaximal, complemented
8#2185C211yesmaximal, characteristic, complemented
9#2184C211yesmaximal, characteristic, complemented
10#2138C212yesmaximal, complemented
11C2×C22⋊C16C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C2×#135—28no—
14C2×#135—21no—
15C2×#135—21no—
16C2×#135C2212yes—
17C2×#135C2212yes—
18C2×#135C2211yescharacteristic, complemented
19C22×Q16C2211yescharacteristic
20C22×SD16—21no—
21C22×D8C2211yescharacteristic
22C23×C8C2211yescharacteristic
23#525—21no—
24#508—28no—
25C2×Q32C2212yescomplemented
26C2×SD32C2212yescomplemented
27C2×SD32C2212yescomplemented
28C2×D16C2212yescomplemented
29C22×C16C2212yescomplemented
30#467C2211yescharacteristic
31#467C2211yescharacteristic, complemented
32#466C2211yescharacteristic
33#466C2211yescharacteristic, complemented
34#465—21no—
35#465—21no—
36#427C2211yescharacteristic, complemented
37#416—21no—
38#358C2212yescomplemented
39#358C2212yescomplemented
40#357C2212yescomplemented
41#357C2212yescomplemented
42C22⋊C16C2211yescharacteristic
43C22⋊C16C2211yescharacteristic
44C22⋊C16C2212yescomplemented
45C2×#41—28no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41—21no—
49C2×#41—44no—
50C2×#41—41no—
51C2×#41—41no—
52C22×Q8—21no—
53C22×D4—21no—
54C22×D4—41no—
55C23×C4—41no—
56C23×C4D411yescharacteristic
57#135—28no—
58#135—28no—
59#135—24no—
60#135—24no—
61#135—24no—
62#135—24no—
63#135D418yescomplemented
64C2×Q16—24no—
65C2×Q16—21no—
66C2×Q16C2311yescharacteristic
67C2×Q16C2312yes—
68C2×SD16—28no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—21no—
72C2×D8—24no—
73C2×D8—21no—
74C2×D8C2311yescharacteristic
75C2×D8C2312yes—
76C22×C8—24no—
77C22×C8D411yescharacteristic
78C22×C8D411yescharacteristic
79C22×C8D411yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#123—41no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86#121—21no—
87#121—21no—
88#120—41no—
89C4×D4—41no—
90C4×D4—41no—
91#117—21no—
92Q32—24no—
93SD32—24no—
94SD32—24no—
95D16—24no—
96C2×C16—24no—
97C2×C16C2312yes—
98C2×C16C2312yes—
99#107C2311yescharacteristic
100#107C2311yescharacteristic
101#106—21no—
102#103—21no—
103#103—21no—
104#102—21no—
105#102—21no—
106C24—41no—
107C24D811yescharacteristic
108#41—28no—
109#41—28no—
110#41—48no—
111#41—44no—
112#41—48no—
113#41—44no—
114#41—42no—
115#41—42no—
116C2×Q8—24no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—42no—
120C2×Q8—41no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—48no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—48no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4D811yescharacteristic
140C22×C4D811yescharacteristic
141C22×C4D811yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145Q16—24no—
146Q16—22no—
147Q16—22no—
148Q16C2×D414yes—
149SD16—28no—
150SD16—28no—
151SD16—42no—
152D8—24no—
153D8—22no—
154D8—22no—
155D8C2×D414yes—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8C2×D411yescharacteristic
162C2×C8C2×D411yescharacteristic
163C2×C8C2×D411yescharacteristic
164C2×C8C2×D414yes—
165C2×C8C2411yescharacteristic
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C42—41no—
176C16—22no—
177C16—22no—
178C23—22no—
179C23—22no—
180C23—41no—
181C23—44no—
182C23—42no—
183C23—41no—
184C23—82no—
185C23—82no—
186C23C2×D811yescharacteristic
187C23C2×D811yescharacteristic
188C23C2×D811yescharacteristic
189Q8—24no—
190Q8—44no—
191Q8—42no—
192Q8—41no—
193Q8—41no—
194D4—24no—
195D4—44no—
196D4—48no—
197D4—48no—
198D4—41no—
199D4—42no—
200D4—41no—
201D4—82no—
202D4—82no—
203D4—82no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—48no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—82no—
219C2×C4#12012yes—
220C2×C4C2×D811yescharacteristic
221C2×C4C2×D811yescharacteristic
222C2×C4C2×D811yescharacteristic
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4C22×D411yescharacteristic
225C8—21no—
226C8—21no—
227C8#12012yes—
228C8C22×D412yes—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—44no—
237C22—44no—
238C22—88no—
239C22—82no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22#44712yes—
245C22#50812yescomplemented
246C22C22×D811yescharacteristic
247C4—21no—
248C4—21no—
249C4—44no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4#44712yes—
254C4C2×#12011yescharacteristic
255C4C2×AGL(1,ℤ/8)11yescharacteristic
256C2—22no—
257C2—22no—
258C2—41no—
259C2—84no—
260C2—161no—
261C2C2×#44711yescharacteristic
262C2C2×#50811yescharacteristic
263C2#341411yescharacteristic
subgroupG/H
centerC2×C4C2×D8
commutator, Frattini, cosocleC2×C8C24
socleC22C22×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#135C2semidirect
2. C2×#508C2semidirect
3. #3050C2semidirect
4. #2189C2semidirect
5. #2188C2semidirect
6. #2187C2semidirect
7. #2186C2semidirect
8. #2185C2semidirect
9. #2184C2semidirect
10. #2138C2semidirect
11. C2×C22⋊C16C2semidirect
18. C2×#135C22semidirect
25. C2×Q32C22semidirect
26. C2×SD32C22semidirect
27. C2×SD32C22semidirect
28. C2×D16C22semidirect
29. C22×C16C22semidirect
31. #467C22semidirect
33. #466C22semidirect
36. #427C22semidirect
38. #358C22semidirect
39. #358C22semidirect
40. #357C22semidirect
41. #357C22semidirect
44. C22⋊C16C22semidirect
63. #135D4semidirect
245. C22#508semidirect

Maximal subgroups

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

subgroupindex
C22×#1352
C2×#5082
#30502
#21892
#21882
#21872
#21862
#21852
#21842
#21382
C2×C22⋊C162