Finite Group Databasebeta

← #33004

Subgroups — #33004

Order 256.

904 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×C4×D4C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4#2854C211yesmaximal, characteristic, complemented
5#1879C214yesmaximal, complemented
6#1864C214yesmaximal, complemented
7#1824C211yesmaximal, characteristic, complemented
8#1817C211yesmaximal, characteristic, complemented
9#1786C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C24×C4—21no—
12C22×D8C2212yes—
13C2×#123—21no—
14C2×#121C2211yescharacteristic
15C2×C4×D4—24no—
16C2×C4×D4—24no—
17C2×C4×D4—22no—
18C2×C4×D4C2214yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4—21no—
21C22×C42C2211yescharacteristic
22#481—24no—
23#447—24no—
24C2×#106C2212yescomplemented
25C2×#102C2212yes—
26C2×#102C2214yes—
27C2×#102C2212yescomplemented
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#390C2212yescomplemented
31#389C2212yescomplemented
32#388—22no—
33#386—21no—
34#380C2211yescharacteristic
35#340C2214yescomplemented
36#336C2214yescomplemented
37C25—21no—
38C22×D4—28no—
39C22×D4—21no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4D414yes—
43C22×D4C2311yescharacteristic
44C22×D4C2312yes—
45C23×C4—28no—
46C23×C4—21no—
47C23×C4—24no—
48C23×C4—48no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×D8—24no—
53C2×D8—28no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56#123—44no—
57#121—22no—
58#121—22no—
59C4×D4—216no—
60C4×D4—28no—
61C4×D4—28no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D414yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—28no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42D412yes—
78C2×C42C2312yes—
79#106—24no—
80#102—24no—
81#102—28no—
82#102—24no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#95—21no—
86#95—22no—
87C24—28no—
88C24—24no—
89C24—22no—
90C24—21no—
91C24—44no—
92C24—42no—
93C24—42no—
94C24C2×D411yescharacteristic
95C2×D4—28no—
96C2×D4—216no—
97C2×D4—216no—
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4C2×D414yes—
108C2×D4#4118yes—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—216no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—432no—
121C22×C4—48no—
122C22×C4—44no—
123C22×C4—48no—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2411yescharacteristic
130D8—48no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—24no—
134C2×C8—24no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—24no—
140C4⋊C4—44no—
141C4⋊C4C2×D414yes—
142C22⋊C4—216no—
143C22⋊C4—44no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C42—24no—
148C42—22no—
149C42—22no—
150C23—22no—
151C23—22no—
152C23—216no—
153C23—24no—
154C23—28no—
155C23—48no—
156C23—416no—
157C23—416no—
158C23—42no—
159C23—44no—
160C23—42no—
161C23—48no—
162C23—41no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23—42no—
167C23#12012yes—
168C23#13512yes—
169C23AGL(1,ℤ/8)12yes—
170C23C22×D411yescharacteristic
171D4—216no—
172D4—24no—
173D4—28no—
174D4—42no—
175D4—42no—
176D4—84no—
177D4#12318yescomplemented
178C2×C4—28no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—432no—
189C2×C4—48no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4#12012yes—
199C2×C4AGL(1,ℤ/8)14yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C22×D411yescharacteristic
202C2×C4C2×#4114yes—
203C8—42no—
204C8—42no—
205C22—22no—
206C22—24no—
207C22—24no—
208C22—22no—
209C22—28no—
210C22—432no—
211C22—48no—
212C22—48no—
213C22—44no—
214C22—44no—
215C22—81no—
216C22—82no—
217C22—82no—
218C22—81no—
219C22—81no—
220C22#44914yes—
221C22C2×#12011yescharacteristic
222C22#52512yes—
223C22C2×#13511yescharacteristic
224C22C2×AGL(1,ℤ/8)12yes—
225C22C2×AGL(1,ℤ/8)11yescharacteristic
226C4—22no—
227C4—24no—
228C4—42no—
229C4—44no—
230C4—82no—
231C4—81no—
232C4C2×#12312yes—
233C4#52512yes—
234C2—24no—
235C2—48no—
236C2—161no—
237C2#284911yescharacteristic
238C2C2×#44911yescharacteristic
239C2#300311yescharacteristic
240C2#310512yes—
241C2#310912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×D4C2semidirect
2. C2×#481C2semidirect
4. #2854C2semidirect
5. #1879C2semidirect
6. #1864C2semidirect
7. #1824C2semidirect
8. #1817C2semidirect
9. #1786C2semidirect
24. C2×#106C22semidirect
27. C2×#102C22semidirect
30. #390C22semidirect
31. #389C22semidirect
35. #340C22semidirect
36. #336C22semidirect
177. D4#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4812
C2×#4472
#28542
#18792
#18642
#18242
#18172
#17862