Finite Group Databasebeta

← #9158

Subgroups — #9158

Order 256.

490 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 79 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#2847C211yesmaximal, characteristic
2C2×#381C211yesmaximal, characteristic, complemented
3#1779C211yesmaximal, characteristic, complemented
4C2×#327C212yesmaximal, complemented
5#1294C212yesmaximal, complemented
6C24×C4C2211yescharacteristic
7C2×#121C412yescomplemented
8C2×#121C2211yescharacteristic
9C2×#120—21no—
10C22×C22⋊C4—21no—
11C2×C22⋊C8—22no—
12C2×C22⋊C8C2212yes—
13#386—21no—
14#386—22no—
15#381—24no—
16#380C412yescomplemented
17#380C2211yescharacteristic
18C2×#95—21no—
19C2×#95C2212yes—
20#327—24no—
21C25D411yescharacteristic
22C22×D4—22no—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—22no—
26C23×C4—22no—
27C23×C4—22no—
28C23×C4—44no—
29C23×C4C2×C412yes—
30C23×C4D412yes—
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C22×C8—22no—
34#121—22no—
35#121—22no—
36#121—44no—
37#120—44no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C412yes—
40C2×C22⋊C4—24no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—44no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—44no—
49#95—24no—
50#95—22no—
51#95—22no—
52#95—21no—
53#95D414yescomplemented
54C24—24no—
55C24—21no—
56C24—21no—
57C24—22no—
58C24—22no—
59C24—21no—
60C24—42no—
61C24C22⋊C412yes—
62C24C2×D411yescharacteristic
63C2×D4—22no—
64C2×D4—42no—
65C2×D4—44no—
66C2×D4—44no—
67C2×D4—44no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4—42no—
72C22×C4—24no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—24no—
77C22×C4—24no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—48no—
86C22×C4—48no—
87C22×C4—44no—
88C22×C4—44no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4C22⋊C412yes—
92C22×C4C22×C411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D412yes—
95C22×C4#4112yes—
96C2×C8—44no—
97C2×C8—42no—
98C4⋊C4—22no—
99C4⋊C4—42no—
100C22⋊C4—44no—
101C22⋊C4—48no—
102C22⋊C4—44no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C23—22no—
106C23—22no—
107C23—24no—
108C23—24no—
109C23—22no—
110C23—22no—
111C23—22no—
112C23—22no—
113C23—22no—
114C23—22no—
115C23—48no—
116C23—48no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—41no—
122C23—42no—
123C23—41no—
124C23—41no—
125C23—42no—
126C23#9914yes—
127C23C2×C22⋊C411yescharacteristic
128C23#12011yescharacteristic
129C23#12311yescharacteristic
130C23#13512yes—
131C23AGL(1,ℤ/8)12yes—
132D4—82no—
133D4—88no—
134D4—82no—
135D4—82no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—48no—
144C2×C4—48no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—84no—
157C2×C4—84no—
158C2×C4C4×D412yes—
159C2×C4#12112yes—
160C8—82no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—24no—
166C22—24no—
167C22—24no—
168C22—24no—
169C22—24no—
170C22—48no—
171C22—48no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—82no—
180C22—82no—
181C22#38611yescharacteristic
182C22C2×#9912yes—
183C22#41611yescharacteristic
184C22#41811yescharacteristic
185C22#42112yes—
186C22#44912yes—
187C22#48212yes—
188C4—22no—
189C4—22no—
190C4—42no—
191C4—84no—
192C4—81no—
193C4—82no—
194C2—22no—
195C2—22no—
196C2—44no—
197C2—161no—
198C2#151012yes—
199C2#151612yes—
200C2#187511yescharacteristic
201C2#189711yescharacteristic
202C2#189811yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#381C2semidirect
3. #1779C2semidirect
4. C2×#327C2semidirect
5. #1294C2semidirect
7. C2×#121C4semidirect
16. #380C4semidirect
53. #95D4semidirect

Maximal subgroups

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

subgroupindex
#28472
C2×#3812
#17792
C2×#3272
#12942