Finite Group Databasebeta

← #18285

Subgroups — #18285

Order 256.

704 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 127 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#3187C212yesmaximal, complemented
3#2626C211yesmaximal, characteristic, complemented
4#1613C214yesmaximal, complemented
5#1601C214yesmaximal, complemented
6#1519C212yesmaximal, complemented
7C2×#323C211yesmaximal, 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#483—22no—
19#481C2214yescomplemented
20#465—24no—
21#450—22no—
22#448C2214yescomplemented
23C2×C22⋊C8C2212yescomplemented
24#399C2212yescomplemented
25#396—22no—
26#393C2212yescomplemented
27#385—21no—
28#380C2211yescharacteristic, complemented
29#379—21no—
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—44no—
41C22×D4—44no—
42C22×D4D412yes—
43C23×C4—21no—
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C2×SD16—24no—
47C22×C8—22no—
48#123—28no—
49#123—28no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52#121—24no—
53#121—24no—
54#121—22no—
55#121—22no—
56#121D414yes—
57#121C2314yes—
58#120—28no—
59#120—28no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2311yescharacteristic, complemented
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C22⋊C4C2312yes—
76#106—24no—
77#103—24no—
78#102—24no—
79C22⋊C8—22no—
80C22⋊C8C2314yescomplemented
81#95—22no—
82#95C2312yescomplemented
83C24—21no—
84C24—21no—
85C24—22no—
86C24—24no—
87C24—22no—
88C24—44no—
89C24—81no—
90C24—81no—
91C24—82no—
92C24—82no—
93C24SD1614yescomplemented
94C24C2×D412yes—
95C24C2×D411yescharacteristic
96C2×Q8—22no—
97C2×Q8—81no—
98C2×D4—28no—
99C2×D4—24no—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—22no—
103C2×D4—48no—
104C2×D4—416no—
105C2×D4—48no—
106C2×D4—48no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—82no—
110C2×D4—82no—
111C2×D4C2×D414yes—
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—44no—
120C22×C4—81no—
121C22×C4—81no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2411yescharacteristic
127SD16—84no—
128C2×C8—24no—
129C2×C8—82no—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—81no—
135C4⋊C4—82no—
136C4⋊C4C2×D414yes—
137C22⋊C4—28no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C23—22no—
148C23—24no—
149C23—22no—
150C23—24no—
151C23—24no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—48no—
159C23—44no—
160C23—44no—
161C23—44no—
162C23—81no—
163C23—81no—
164C23—84no—
165C23—84no—
166C23—84no—
167C23—84no—
168C23—816no—
169C23#12011yescharacteristic
170C23#12011yescharacteristic
171C23C2×SD1614yes—
172C23C2×SD1612yes—
173C23AGL(1,ℤ/8)12yes—
174C23C22×D411yescharacteristic
175Q8—44no—
176D4—44no—
177D4—416no—
178D4—44no—
179D4—44no—
180D4—48no—
181D4—82no—
182D4—84no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—48no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4#12012yes—
202C2×C4C22×D412yes—
203C8—44no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—42no—
212C22—42no—
213C22—44no—
214C22—48no—
215C22—42no—
216C22—42no—
217C22—44no—
218C22—44no—
219C22—44no—
220C22—82no—
221C22—84no—
222C22—88no—
223C22—816no—
224C22#45012yes—
225C22#45012yes—
226C22C2×#12011yescharacteristic
227C22C22×SD1611yescharacteristic
228C22C2×AGL(1,ℤ/8)11yescharacteristic
229C4—22no—
230C4—22no—
231C4—44no—
232C4—42no—
233C4—82no—
234C4—81no—
235C4—81no—
236C2—21no—
237C2—22no—
238C2—21no—
239C2—22no—
240C2—44no—
241C2—84no—
242C2C2×#45011yescharacteristic
243C2#301911yescharacteristic
244C2#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. #3187C2semidirect
3. #2626C2semidirect
4. #1613C2semidirect
5. #1601C2semidirect
6. #1519C2semidirect
7. C2×#323C2semidirect
13. C2×C22⋊Q8C22semidirect
19. #481C22semidirect
22. #448C22semidirect
23. C2×C22⋊C8C22semidirect
24. #399C22semidirect
26. #393C22semidirect
28. #380C22semidirect
30. #327C22semidirect
31. #323C22semidirect
34. C22×Q8C23semidirect
65. C2×C4⋊C4C23semidirect
80. C22⋊C8C23semidirect
82. #95C23semidirect
93. C24SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5342
#31872
#26262
#16132
#16012
#15192
C2×#3232