Finite Group Databasebeta

← #18142

Subgroups — #18142

Order 256.

725 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 725 conjugacy classes of subgroups — too many to draw. Only its 117 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×#537C211yesmaximal, characteristic, complemented
2#3006C212yesmaximal, complemented
3#2406C211yesmaximal, characteristic, complemented
4#1604C214yesmaximal, complemented
5#1598C214yesmaximal, complemented
6#1510C212yesmaximal, complemented
7#1463C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9C22×#41C2211yescharacteristic
10C2×AGL(1,ℤ/8)—22no—
11#537—24no—
12#537—22no—
13#537C2214yes—
14C2×#123—21no—
15C2×C22⋊Q8—22no—
16C2×#121—22no—
17C2×#121—21no—
18C2×#121C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20#449—24no—
21#448C2214yescomplemented
22#447C2214yescomplemented
23#417—22no—
24#407C2212yes—
25#395C2212yescomplemented
26#394C2212yescomplemented
27#386—21no—
28#385—22no—
29#381C2211yescharacteristic
30#327C2218yescomplemented
31#324C2214yescomplemented
32C2×#41—24no—
33C2×#41—24no—
34C2×#41—22no—
35C2×#41—22no—
36C2×#41—48no—
37C2×#41—44no—
38C22×Q8—21no—
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4—22no—
42C22×D4—21no—
43C22×D4—21no—
44C22×D4D412yes—
45C22×D4C2311yescharacteristic
46C23×C4—22no—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51AGL(1,ℤ/8)—48no—
52C2×SD16—24no—
53C2×D8—24no—
54C2×M16—22no—
55#123—44no—
56C22⋊Q8—28no—
57C22⋊Q8—28no—
58#121—24no—
59#121—28no—
60#121—28no—
61#121—24no—
62#121—22no—
63#121—22no—
64#121D414yes—
65#121C2314yes—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4D412yes—
71C2×C4⋊C4D412yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78#103—24no—
79#102—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yes—
82#95—21no—
83#95—22no—
84#95C2312yes—
85C24—22no—
86C24—21no—
87C24C2×D411yescharacteristic
88#41—28no—
89#41—416no—
90#41—48no—
91#41—44no—
92#41—44no—
93#41—44no—
94#41—48no—
95C2×Q8—24no—
96C2×Q8—22no—
97C2×Q8—42no—
98C2×Q8—44no—
99C2×D4—24no—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—42no—
109C2×D4—48no—
110C2×D4—42no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—82no—
114C2×D4C2×D414yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—28no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—48no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2411yescharacteristic
136SD16—48no—
137D8—48no—
138M16—44no—
139C2×C8—24no—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—44no—
145C4⋊C4C2×D414yes—
146C22⋊C4—28no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—24no—
156C23—41no—
157C23—44no—
158C23—41no—
159C23—42no—
160C23—44no—
161C23—41no—
162C23—42no—
163C23—44no—
164C23#12011yescharacteristic
165C23#12011yescharacteristic
166C23C22×D411yescharacteristic
167Q8—24no—
168Q8—48no—
169Q8—42no—
170D4—24no—
171D4—44no—
172D4—48no—
173D4—42no—
174D4—48no—
175D4—44no—
176D4—48no—
177D4—44no—
178D4—82no—
179D4—84no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—416no—
192C2×C4—48no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—81no—
202C2×C4#12012yes—
203C2×C4AGL(1,ℤ/8)14yescomplemented
204C2×C4C22×D412yes—
205C8—44no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—44no—
214C22—44no—
215C22—44no—
216C22—44no—
217C22—42no—
218C22—44no—
219C22—82no—
220C22—88no—
221C22—81no—
222C22—81no—
223C22B(4,𝔽2)12yes—
224C22C2×#12011yescharacteristic
225C22C2×AGL(1,ℤ/8)12yes—
226C4—24no—
227C4—22no—
228C4—22no—
229C4—44no—
230C4—42no—
231C4—42no—
232C4—82no—
233C2—22no—
234C2—22no—
235C2—41no—
236C2—82no—
237C2—84no—
238C2#300311yescharacteristic
239C2#302011yescharacteristic
240C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#537C2semidirect
2. #3006C2semidirect
3. #2406C2semidirect
4. #1604C2semidirect
5. #1598C2semidirect
6. #1510C2semidirect
7. #1463C2semidirect
21. #448C22semidirect
22. #447C22semidirect
25. #395C22semidirect
26. #394C22semidirect
30. #327C22semidirect
31. #324C22semidirect
203. C2×C4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
#30062
#24062
#16042
#15982
#15102
#14632