Finite Group Databasebeta

← #18317

Subgroups — #18317

Order 256.

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

normal subgroups only

This group has 701 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
2#3193C212yesmaximal, complemented
3#2561C211yesmaximal, characteristic, complemented
4#1607C214yesmaximal, complemented
5#1601C214yesmaximal, complemented
6#1507C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic, complemented
8C23×D4—21no—
9#534—24no—
10#534—22no—
11#534C2214yes—
12C2×#123—22no—
13C2×C22⋊Q8C2211yescharacteristic, complemented
14C2×#121—21no—
15C2×#121C2212yes—
16C2×#120—22no—
17C22×C22⋊C4C2211yescharacteristic
18#484—22no—
19#480C2214yescomplemented
20#468—24no—
21#450—22no—
22#448C2214yescomplemented
23#407C2212yescomplemented
24#397C2212yescomplemented
25#393C2212yescomplemented
26#388—22no—
27#381—21no—
28#379—21no—
29C4×C22⋊C4C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32C25—21no—
33C25D411yescharacteristic
34C22×Q8C2311yescharacteristic, complemented
35C22×D4—22no—
36C22×D4—24no—
37C22×D4—22no—
38C22×D4—24no—
39C22×D4—21no—
40C22×D4—48no—
41C22×D4D412yes—
42C23×C4—21no—
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C2×SD16—24no—
46C2×M16—22no—
47#123—28no—
48#123—28no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51#121—24no—
52#121—24no—
53#121—22no—
54#121—22no—
55#121D414yes—
56#121C2314yes—
57#120—28no—
58#120—28no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D412yes—
62C2×C4⋊C4C2311yescharacteristic, complemented
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4D412yes—
72C2×C22⋊C4C2312yes—
73C2×C42—21no—
74#107—24no—
75#103—24no—
76#102—24no—
77C22⋊C8—22no—
78C22⋊C8C2314yescomplemented
79#95—22no—
80#95—22no—
81#95C2312yescomplemented
82C24—21no—
83C24—21no—
84C24—24no—
85C24—24no—
86C24—22no—
87C24—44no—
88C24—42no—
89C24—81no—
90C24—82no—
91C24—82no—
92C24C2×D411yescharacteristic
93C24C2×D412yes—
94C2×Q8—22no—
95C2×Q8—81no—
96C2×D4—28no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—48no—
102C2×D4—416no—
103C2×D4—48no—
104C2×D4—48no—
105C2×D4—48no—
106C2×D4—82no—
107C2×D4—82no—
108C2×D4C2×D414yes—
109C22×C4—24no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—81no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2411yescharacteristic
124SD16—84no—
125M16—82no—
126C2×C8—24no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4C2×D414yes—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—42no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—44no—
141C22⋊C4—82no—
142C42—81no—
143C23—24no—
144C23—24no—
145C23—22no—
146C23—22no—
147C23—22no—
148C23—22no—
149C23—24no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—44no—
155C23—48no—
156C23—44no—
157C23—42no—
158C23—44no—
159C23—81no—
160C23—81no—
161C23—88no—
162C23—84no—
163C23—84no—
164C23—816no—
165C23#12011yescharacteristic
166C23#12011yescharacteristic
167C23AGL(1,ℤ/8)14yes—
168C23C22×D411yescharacteristic
169Q8—44no—
170D4—44no—
171D4—416no—
172D4—48no—
173D4—48no—
174D4—82no—
175D4—84no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—44no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4#12012yes—
195C2×C4C22×D412yes—
196C8—44no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—44no—
206C22—44no—
207C22—48no—
208C22—42no—
209C22—42no—
210C22—44no—
211C22—44no—
212C22—44no—
213C22—82no—
214C22—88no—
215C22—84no—
216C22—816no—
217C22#45512yes—
218C22C2×#12011yescharacteristic
219C22C2×AGL(1,ℤ/8)12yes—
220C4—22no—
221C4—22no—
222C4—44no—
223C4—42no—
224C4—82no—
225C4—81no—
226C4—81no—
227C2—22no—
228C2—22no—
229C2—41no—
230C2—44no—
231C2—84no—
232C2#300311yescharacteristic
233C2C2×#45511yescharacteristic
234C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#534C2
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
2. #3193C2semidirect
3. #2561C2semidirect
4. #1607C2semidirect
5. #1601C2semidirect
6. #1507C2semidirect
7. #1462C2semidirect
13. C2×C22⋊Q8C22semidirect
19. #480C22semidirect
22. #448C22semidirect
23. #407C22semidirect
24. #397C22semidirect
25. #393C22semidirect
29. C4×C22⋊C4C22semidirect
30. #327C22semidirect
31. #323C22semidirect
34. C22×Q8C23semidirect
62. C2×C4⋊C4C23semidirect
78. C22⋊C8C23semidirect
81. #95C23semidirect

Maximal subgroups

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

subgroupindex
C2×#5342
#31932
#25612
#16072
#16012
#15072
#14622