Finite Group Databasebeta

← #8714

Subgroups — #8714

Order 256.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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#2283C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1434C211yesmaximal, characteristic
4#1294C212yesmaximal
5#1294C212yesmaximal
6C22×C42C2211yescharacteristic
7C2×C22⋊C8C2212yes—
8#383—21no—
9#382C411yescharacteristic, complemented
10#382C411yescharacteristic, complemented
11#381—21no—
12#380C2211yescharacteristic
13#380C2211yescharacteristic
14C4×C4⋊C4C411yescharacteristic, complemented
15C4×C4⋊C4C411yescharacteristic, complemented
16C4×C22⋊C4—21no—
17#376—21no—
18C2×#95—21no—
19C2×#95—21no—
20C2×#95C2212yes—
21#336—21no—
22#336—21no—
23C23×C4—22no—
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C22×C8—22no—
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C22⋊C4—21no—
33C2×C22⋊C4—21no—
34C2×C42—24no—
35C2×C42—21no—
36C2×C42—21no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42D411yescharacteristic
40C2×C42D411yescharacteristic
41C22⋊C8—24no—
42#95—24no—
43#95—21no—
44#95—21no—
45#95—21no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—24no—
52#95—42no—
53C24C2×D411yescharacteristic
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—24no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4#4114yes—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—42no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C22⋊C4—42no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C42—22no—
92C42—22no—
93C42—22no—
94C42—22no—
95C42—42no—
96C42—42no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23C2×C22⋊C411yescharacteristic
102C23#12312yes—
103C23#13512yes—
104C23AGL(1,ℤ/8)11yescharacteristic
105C23#13711yescharacteristic
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—44no—
122C2×C4—44no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—44no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—82no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#10412yes—
145C2×C4#11712yes—
146C2×C4#12312yes—
147C8—82no—
148C22—22no—
149C22—22no—
150C22—22no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22#38111yescharacteristic
157C22#41011yescharacteristic
158C22#41011yescharacteristic
159C22#41611yescharacteristic
160C22#41711yescharacteristic
161C22C2×#10411yescharacteristic
162C22#42111yescharacteristic
163C22#48212yes—
164C22#48512yes—
165C4—22no—
166C4—22no—
167C4—22no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—84no—
172C4—81no—
173C4—81no—
174C4—81no—
175C4—81no—
176C2—22no—
177C2—22no—
178C2#150011yescharacteristic
179C2#150911yescharacteristic
180C2#151211yescharacteristic
181C2#152111yescharacteristic
182C2#182011yescharacteristic
183C2#182511yescharacteristic
184C2#182611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #382C4semidirect
10. #382C4semidirect
14. C4×C4⋊C4C4semidirect
15. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#22832
#18172
#14342
#12942