Finite Group Databasebeta

← #24285

Subgroups — #24285

Order 256.

668 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 133 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#3584C211yesmaximal, characteristic, complemented
2#3050C212yesmaximal, complemented
3#3025C212yesmaximal, complemented
4#2929C211yesmaximal, characteristic, complemented
5#2053C212yesmaximal, complemented
6#2010C212yesmaximal, complemented
7#1976C212yesmaximal, complemented
8#1812C211yesmaximal, characteristic, complemented
9#1794C212yesmaximal, complemented
10#585C2212yes—
11#578—22no—
12#577—24no—
13#576—22no—
14#575—22no—
15#575C2212yes—
16C2×#135—22no—
17C2×#135C2212yes—
18C2×#131C2211yescharacteristic
19C23×C8C2211yescharacteristic, complemented
20#525C2211yescharacteristic
21#525C2214yescomplemented
22#467—22no—
23#466—22no—
24#465—24no—
25#458—24no—
26B(4,𝔽2)—24no—
27#435—22no—
28C8×D4—22no—
29#427C2212yescomplemented
30#424C2211yescharacteristic, complemented
31#421C2214yescomplemented
32#416C2214yescomplemented
33#410C2212yes—
34#410C2214yescomplemented
35#408C2211yescharacteristic
36#407C2211yescharacteristic, complemented
37#405C2211yescharacteristic, complemented
38H(3,ℤ/4)C2212yescomplemented
392−1+4—24no—
402−1+4—22no—
41H(5,𝔽2)—24no—
42H(5,𝔽2)—24no—
43H(5,𝔽2)—22no—
44C2×#41—24no—
45C2×#41—44no—
46C2×#41C2314yes—
47C2×#41C2311yescharacteristic
48C23×C4C2311yescharacteristic
49#137—28no—
50#137—24no—
51AGL(1,ℤ/8)—28no—
52AGL(1,ℤ/8)—24no—
53#135—24no—
54#135—24no—
55#135—22no—
56#135—24no—
57#135—22no—
58#135—22no—
59#135—22no—
60C2×Q16—22no—
61C2×Q16D412yescomplemented
62C2×SD16—24no—
63C2×SD16D414yescomplemented
64C2×D8—22no—
65C2×D8D412yescomplemented
66#131—22no—
67#131—22no—
68C2×M16D412yescomplemented
69C2×M16C2311yescharacteristic
70C2×M16C2311yescharacteristic
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8D412yescomplemented
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#123—24no—
79#123D412yes—
80C22⋊Q8—21no—
81C22⋊Q8—24no—
82#121—21no—
83#121—24no—
84#120—24no—
85#120D412yes—
86C4×D4—22no—
87C4×D4—44no—
88#117C2314yes—
89#117C2311yescharacteristic
90#107—22no—
91#106—22no—
92C4⋊C8—22no—
93#104—44no—
94#103—24no—
95#102—24no—
96#99—24no—
97#99—28no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100#97—21no—
101C4×C8—21no—
102C24C2×D411yescharacteristic
103#41—22no—
104#41—22no—
105#41—28no—
106#41—24no—
107#41—24no—
108#41—24no—
109#41—24no—
110#41—48no—
111#41—44no—
112#41—44no—
113#41—44no—
114#41—44no—
115C2×Q8—24no—
116C2×Q8—44no—
117C2×Q8C2×D414yes—
118C2×Q8#4111yescharacteristic
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—24no—
122C2×D4—44no—
123C2×D4—48no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4C2×D412yes—
127C2×D4C2×D414yes—
128C2×D4#4111yescharacteristic
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—44no—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138Q16—24no—
139Q16—22no—
140Q16—22no—
141SD16—28no—
142SD16—24no—
143SD16—24no—
144D8—24no—
145D8—22no—
146D8—22no—
147M16—22no—
148M16—24no—
149M16—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—42no—
158C2×C8C2×D412yes—
159C2×C8C2×D411yescharacteristic
160C2×C8C2×D411yescharacteristic
161C2×C8C2×D412yes—
162C2×C8C2×D412yes—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—24no—
166C4⋊C4—44no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—24no—
170C22⋊C4—44no—
171C22⋊C4—44no—
172C22⋊C4C2×D412yes—
173C42—21no—
174C42—24no—
175C23—21no—
176C23—22no—
177C23—24no—
178C23—42no—
179C23—44no—
180C23#12112yes—
181C23C22×D411yescharacteristic
182Q8—22no—
183Q8—24no—
184Q8—24no—
185Q8—44no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—24no—
190D4—44no—
191D4—44no—
192D4—48no—
193D4—44no—
194D4—44no—
195D4—88no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—48no—
203C2×C4—44no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4#12112yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D412yes—
213C2×C4C2×#4111yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C8—22no—
216C8—41no—
217C8—41no—
218C8#12011yescharacteristic
219C8#12011yescharacteristic
220C8#12012yes—
221C22—22no—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—44no—
230C22—84no—
231C22—164no—
232C22C2×#12111yescharacteristic
233C22D4212yes—
234C4—22no—
235C4—42no—
236C4—41no—
237C4—44no—
238C4—84no—
239C4C2×#12011yescharacteristic
240C4#53511yescharacteristic
241C4#54612yes—
242C2—21no—
243C2—22no—
244C2—42no—
245C2—42no—
246C2—84no—
247C2#240811yescharacteristic
subgroupG/H
centerC4#535
commutator, Frattini, cosocleC22×C4C24
socleC2#2408
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3584C2semidirect
2. #3050C2semidirect
3. #3025C2semidirect
4. #2929C2semidirect
5. #2053C2semidirect
6. #2010C2semidirect
7. #1976C2semidirect
8. #1812C2semidirect
9. #1794C2semidirect
19. C23×C8C22semidirect
21. #525C22semidirect
29. #427C22semidirect
30. #424C22semidirect
31. #421C22semidirect
32. #416C22semidirect
34. #410C22semidirect
36. #407C22semidirect
37. #405C22semidirect
38. H(3,ℤ/4)C22semidirect
61. C2×Q16D4semidirect
63. C2×SD16D4semidirect
65. C2×D8D4semidirect
68. C2×M16D4semidirect
75. C22×C8D4semidirect

Maximal subgroups

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

subgroupindex
#35842
#30502
#30252
#29292
#20532
#20102
#19762
#18122
#17942