Finite Group Databasebeta

← #18316

Subgroups — #18316

Order 256.

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

normal subgroups only

This group has 758 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×#534C211yesmaximal, characteristic, complemented
2#3192C212yesmaximal, complemented
3#2539C211yesmaximal, characteristic, complemented
4#1613C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1507C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic, complemented
8C23×D4—21no—
9#534—24no—
10#534—22no—
11#534C2214yes—
12C2×#123—21no—
13C2×#123—22no—
14C2×#121—21no—
15C2×#121C2212yes—
16C2×#121C2211yescharacteristic, complemented
17C2×#120—22no—
18C2×#120—21no—
19C22×C22⋊C4C2211yescharacteristic
20#481—22no—
21#481C2214yescomplemented
22#469—24no—
23#447—22no—
24#447C2214yescomplemented
25#407C2212yescomplemented
26#399C2212yescomplemented
27#392C2212yescomplemented
28#390—22no—
29C4×C22⋊C4C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32C25—21no—
33C25D411yescharacteristic
34C22×D4—22no—
35C22×D4—24no—
36C22×D4—22no—
37C22×D4—24no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4—22no—
41C22×D4—48no—
42C22×D4D412yescomplemented
43C22×D4C2311yescharacteristic, complemented
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C2×D8—24no—
48C2×M16—22no—
49#123—28no—
50#123—28no—
51#123—44no—
52#121—24no—
53#121—24no—
54#121—22no—
55#121—22no—
56#121—22no—
57#121—22no—
58#121D414yescomplemented
59#121C2314yes—
60#120—28no—
61#120—28no—
62#120—44no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D412yescomplemented
66C2×C4⋊C4C2311yescharacteristic, complemented
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4D412yes—
76C2×C22⋊C4C2312yes—
77C2×C42—21no—
78#106—24no—
79#102—24no—
80#102—24no—
81C22⋊C8—22no—
82C22⋊C8C2314yescomplemented
83#95C2312yescomplemented
84C24—21no—
85C24—21no—
86C24—24no—
87C24—24no—
88C24—22no—
89C24—21no—
90C24—44no—
91C24—42no—
92C24—81no—
93C24—82no—
94C24—82no—
95C24C2×D411yescharacteristic
96C24C2×D412yes—
97C2×D4—28no—
98C2×D4—24no—
99C2×D4—24no—
100C2×D4—24no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—44no—
104C2×D4—48no—
105C2×D4—416no—
106C2×D4—48no—
107C2×D4—48no—
108C2×D4—48no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—44no—
114C2×D4—81no—
115C2×D4—82no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4C2×D414yes—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—44no—
126C22×C4—42no—
127C22×C4—81no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2411yescharacteristic
133D8—84no—
134M16—82no—
135C2×C8—24no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—44no—
140C4⋊C4C2×D414yes—
141C22⋊C4—28no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—42no—
150C42—81no—
151C23—24no—
152C23—24no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—24no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—44no—
163C23—48no—
164C23—44no—
165C23—41no—
166C23—41no—
167C23—42no—
168C23—42no—
169C23—44no—
170C23—41no—
171C23—42no—
172C23—81no—
173C23—81no—
174C23—88no—
175C23—84no—
176C23—84no—
177C23—816no—
178C23#12011yescharacteristic
179C23#12011yescharacteristic
180C23AGL(1,ℤ/8)14yes—
181C23C22×D411yescharacteristic
182D4—44no—
183D4—44no—
184D4—416no—
185D4—48no—
186D4—48no—
187D4—82no—
188D4—84no—
189D4—88no—
190D4—82no—
191D4—82no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—48no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4#12012yes—
210C2×C4C22×D412yes—
211C8—44no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—24no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—44no—
221C22—44no—
222C22—48no—
223C22—42no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22—44no—
228C22—82no—
229C22—88no—
230C22—82no—
231C22—82no—
232C22—84no—
233C22—816no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22#45312yes—
238C22C2×#12011yescharacteristic
239C22C2×AGL(1,ℤ/8)12yes—
240C4—22no—
241C4—22no—
242C4—44no—
243C4—44no—
244C4—42no—
245C4—81no—
246C2—22no—
247C2—22no—
248C2—41no—
249C2—44no—
250C2—84no—
251C2—161no—
252C2#300311yescharacteristic
253C2C2×#45311yescharacteristic
254C2#302611yescharacteristic
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×#534C2semidirect
2. #3192C2semidirect
3. #2539C2semidirect
4. #1613C2semidirect
5. #1595C2semidirect
6. #1507C2semidirect
7. #1462C2semidirect
16. C2×#121C22semidirect
21. #481C22semidirect
24. #447C22semidirect
25. #407C22semidirect
26. #399C22semidirect
27. #392C22semidirect
29. C4×C22⋊C4C22semidirect
30. #327C22semidirect
31. #323C22semidirect
42. C22×D4D4semidirect
43. C22×D4C23semidirect
58. #121D4semidirect
65. C2×C4⋊C4D4semidirect
66. C2×C4⋊C4C23semidirect
82. C22⋊C8C23semidirect
83. #95C23semidirect

Maximal subgroups

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

subgroupindex
C2×#5342
#31922
#25392
#16132
#15952
#15072
#14622