Finite Group Databasebeta

← #32038

Subgroups — #32038

Order 256.

472 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#2541C211yesmaximal, characteristic
2#2486C211yesmaximal, characteristic, complemented
3#2041C212yesmaximal, complemented
4#2034C212yesmaximal, complemented
5#2034C212yesmaximal, complemented
6#1919C212yesmaximal, complemented
7#1895C212yesmaximal, complemented
8#1874C212yesmaximal, complemented
9#1870C211yesmaximal, characteristic, complemented
10C22×M16C2211yescharacteristic
11C2×#127—21no—
12C2×#127C2211yescharacteristic
13C2×#123—21no—
14C2×#121C2212yes—
15C2×#121C2212yes—
16C2×C4×D4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C2×#106C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24C2×C22⋊C8C2211yescharacteristic
25#398—21no—
26#397—21no—
27#392—21no—
28#392—21no—
29#390—21no—
30#389C2212yes—
31#381—21no—
32#380C2212yes—
33#380C2211yescharacteristic
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—21no—
42C22×D4—21no—
43C22×D4D411yescharacteristic
44C22×D4C2312yes—
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C2×M16—42no—
49C2×M16—41no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52#127—41no—
53#127—41no—
54#127—44no—
55#123—44no—
56#121—24no—
57#121—24no—
58#121—42no—
59#121—42no—
60C4×D4—41no—
61C4×D4—41no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D411yescharacteristic
75C2×C22⋊C4D411yescharacteristic
76C2×C42—21no—
77C2×C42C2311yescharacteristic
78#106—24no—
79#102—42no—
80#102—42no—
81#102—42no—
82#102—42no—
83C22⋊C8—41no—
84C22⋊C8—42no—
85#95—21no—
86#95—22no—
87#95—21no—
88#95—22no—
89C24—22no—
90C24C2×D411yescharacteristic
91C24C2×D411yescharacteristic
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—41no—
101C2×D4—44no—
102C2×D4—44no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—42no—
109C2×D4—82no—
110C2×D4—82no—
111C2×D4—82no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131M16—44no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4#4114yes—
147C22⋊C4—41no—
148C22⋊C4—41no—
149C22⋊C4—44no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C42—21no—
154C42—21no—
155C42—42no—
156C23—21no—
157C23—22no—
158C23—21no—
159C23—22no—
160C23—21no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23—22no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23AGL(1,ℤ/8)12yes—
173C23C22×D411yescharacteristic
174D4—44no—
175D4—42no—
176D4—42no—
177D4—42no—
178D4—42no—
179D4—84no—
180D4—88no—
181D4—88no—
182D4—84no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—84no—
207C2×C4AGL(1,ℤ/8)12yes—
208C2×C4C2×#4112yes—
209C2×C4H(5,𝔽2)11yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C2×C4H(5,𝔽2)11yescharacteristic
212C8—42no—
213C8—42no—
214C22—22no—
215C22—24no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22#53411yescharacteristic
232C22#54612yes—
233C22C2×AGL(1,ℤ/8)11yescharacteristic
234C22C2×AGL(1,ℤ/8)11yescharacteristic
235C22#57712yes—
236C4—22no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—44no—
241C4—82no—
242C4—82no—
243C4#54012yes—
244C4#55911yescharacteristic
245C4#56011yescharacteristic
246C2—42no—
247C2—81no—
248C2—162no—
249C2#270911yescharacteristic
250C2#319211yescharacteristic
251C2#320011yescharacteristic
252C2#331012yes—
253C2#333212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2486C2semidirect
3. #2041C2semidirect
4. #2034C2semidirect
5. #2034C2semidirect
6. #1919C2semidirect
7. #1895C2semidirect
8. #1874C2semidirect
9. #1870C2semidirect

Maximal subgroups

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

subgroupindex
#25412
#24862
#20412
#20342
#19192
#18952
#18742
#18702