Finite Group Databasebeta

← #23906

Subgroups — #23906

Order 256.

1251 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 147 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#3484C211yesmaximal, characteristic, complemented
2C22×#127C211yesmaximal, characteristic, complemented
3C2×#453C212yesmaximal, complemented
4#3003C212yesmaximal, complemented
5#2002C214yesmaximal, complemented
6#1896C212yesmaximal, complemented
7#1887C212yesmaximal, complemented
8#1783C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic, complemented
10C23×D4—22no—
11C23×D4C2211yescharacteristic
12C23×D4C2211yescharacteristic, complemented
13C2×AGL(1,ℤ/8)C2214yescomplemented
14#546—24no—
15D42—24no—
16#535C2212yescomplemented
17#534—22no—
18C2×#127—22no—
19C2×#127—28no—
20C2×#127C2212yes—
21C2×#127C2212yescomplemented
22#525C2212yes—
23C2×#120—22no—
24#518C2211yescharacteristic, complemented
25C22×C42C2211yescharacteristic, complemented
26#453—28no—
27#450—24no—
28#447—24no—
29C2×#104C2214yescomplemented
30#418C2214yescomplemented
31C2×#100C2212yescomplemented
32C2×#100C2212yescomplemented
33#407C2212yescomplemented
34#339C2214yescomplemented
35C25—21no—
36C25D412yes—
37H(5,𝔽2)—24no—
38H(5,𝔽2)—42no—
39C2×#41—22no—
40C2×#41C2312yes—
41C22×D4—216no—
42C22×D4—22no—
43C22×D4—24no—
44C22×D4—24no—
45C22×D4—24no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—42no—
53C22×D4—42no—
54C22×D4—416no—
55C22×D4D412yes—
56C22×D4C2311yescharacteristic
57C22×D4C2312yes—
58C22×D4C2311yescharacteristic
59C23×C4—21no—
60C23×C4C2311yescharacteristic
61AGL(1,ℤ/8)—28no—
62AGL(1,ℤ/8)—44no—
63C2×SD16—24no—
64C2×D8—24no—
65C2×M16C2314yes—
66#127—216no—
67#127—24no—
68#127—24no—
69#127—22no—
70#127—24no—
71#127D418yescomplemented
72#124—22no—
73#123—24no—
74#123—22no—
75C22⋊Q8D414yescomplemented
76#121—28no—
77#121—24no—
78#121D414yescomplemented
79#120—28no—
80#120—24no—
81#120—22no—
82#120—24no—
83C4×D4—24no—
84C4×D4—24no—
85#117C2312yes—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C42—21no—
89C2×C42—24no—
90C2×C42C2312yes—
91#104—28no—
92#102—28no—
93#100—24no—
94#100—42no—
95C22⋊C8—24no—
96C24—24no—
97C24—22no—
98C24—24no—
99C24—22no—
100C24—24no—
101C24—42no—
102C24—48no—
103C24—42no—
104C24—48no—
105C24—42no—
106C24C2×D411yescharacteristic
107C24C2×D412yes—
108C24C2×D412yes—
109#41—24no—
110#41—24no—
111#41—48no—
112C2×Q8C2×D412yescomplemented
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—216no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—216no—
119C2×D4—22no—
120C2×D4—28no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—24no—
129C2×D4—22no—
130C2×D4—416no—
131C2×D4—416no—
132C2×D4—42no—
133C2×D4—416no—
134C2×D4—416no—
135C2×D4—464no—
136C2×D4—42no—
137C2×D4—48no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—42no—
141C2×D4—88no—
142C2×D4C2×D414yes—
143C2×D4C2×D412yescomplemented
144C2×D4C2×D414yescomplemented
145C22×C4—24no—
146C22×C4—28no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D414yes—
155C22×C4C2411yescharacteristic
156SD16—48no—
157D8—48no—
158M16—24no—
159M16—24no—
160C2×C8—24no—
161C4⋊C4—24no—
162C4⋊C4C2×D414yescomplemented
163C22⋊C4—24no—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C22⋊C4—22no—
167C22⋊C4—24no—
168C42—24no—
169C42—22no—
170C42—22no—
171C42C2×D414yescomplemented
172C23—21no—
173C23—21no—
174C23—24no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—48no—
183C23—44no—
184C23—44no—
185C23—416no—
186C23—416no—
187C23—48no—
188C23—48no—
189C23—44no—
190C23—42no—
191C23—42no—
192C23—48no—
193C23—84no—
194C23—88no—
195C23—88no—
196C23—84no—
197C23—84no—
198C23—84no—
199C23#12012yes—
200C23#12014yes—
201C23C22×D411yescharacteristic
202Q8—24no—
203D4—28no—
204D4—24no—
205D4—28no—
206D4—48no—
207D4—432no—
208D4—432no—
209D4—48no—
210D4—48no—
211D4—42no—
212D4—42no—
213D4—48no—
214D4—44no—
215D4—44no—
216D4—88no—
217C2×C4—216no—
218C2×C4—28no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—28no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—84no—
228C2×C4#12012yes—
229C2×C4C22×D412yes—
230C2×C4C22×D412yes—
231C2×C4H(5,𝔽2)12yes—
232C8—44no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—24no—
239C22—44no—
240C22—44no—
241C22—48no—
242C22—44no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—44no—
247C22—44no—
248C22—82no—
249C22—816no—
250C22—82no—
251C22—84no—
252C22—816no—
253C22—84no—
254C22—164no—
255C22#45314yescomplemented
256C22C2×#12011yescharacteristic
257C22C2×#12011yescharacteristic
258C22#53411yescharacteristic
259C4—28no—
260C4—22no—
261C4—42no—
262C4—44no—
263C4D4214yes—
264C2—21no—
265C2—21no—
266C2—24no—
267C2—44no—
268C2—82no—
269C2—84no—
270C2—82no—
271C2#240311yescharacteristic
272C2C2×#45312yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3484C2semidirect
2. C22×#127C2semidirect
3. C2×#453C2semidirect
4. #3003C2semidirect
5. #2002C2semidirect
6. #1896C2semidirect
7. #1887C2semidirect
8. #1783C2semidirect
9. C2×H(5,𝔽2)C22semidirect
12. C23×D4C22semidirect
13. C2×AGL(1,ℤ/8)C22semidirect
16. #535C22semidirect
21. C2×#127C22semidirect
24. #518C22semidirect
25. C22×C42C22semidirect
29. C2×#104C22semidirect
30. #418C22semidirect
31. C2×#100C22semidirect
32. C2×#100C22semidirect
33. #407C22semidirect
34. #339C22semidirect
71. #127D4semidirect
75. C22⋊Q8D4semidirect
78. #121D4semidirect
112. C2×Q8C2×D4semidirect
143. C2×D4C2×D4semidirect
144. C2×D4C2×D4semidirect
162. C4⋊C4C2×D4semidirect
171. C42C2×D4semidirect
255. C22#453semidirect

Maximal subgroups

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

subgroupindex
#34842
C22×#1272
C2×#4532
#30032
#20022
#18962
#18872
#17832