Finite Group Databasebeta

← #8717

Subgroups — #8717

Order 256.

294 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 81 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#2285C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1434C211yesmaximal, characteristic
4#1294C214yesmaximal
5C2×#117—21no—
6C22×C42C2211yescharacteristic
7C2×C22⋊C8C2212yes—
8#384C411yescharacteristic, complemented
9#384C411yescharacteristic, complemented
10#382—21no—
11#380—21no—
12#380C2212yes—
13C4×C4⋊C4—21no—
14C4×C4⋊C4C412yescomplemented
15C2×#95—21no—
16C2×#95—21no—
17C2×#95C2212yes—
18#336—21no—
19#336—21no—
20C23×C4—22no—
21C23×C4D411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4C2311yescharacteristic
24C22×C8—22no—
25#117—44no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4C2×C412yes—
31C2×C4⋊C4C2×C412yes—
32C2×C22⋊C4—22no—
33C2×C42—24no—
34C2×C42—22no—
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C411yescharacteristic
37C2×C42D411yescharacteristic
38C2×C42D411yescharacteristic
39C22⋊C8—24no—
40#95—24no—
41#95—22no—
42#95—22no—
43#95—24no—
44#95—42no—
45C24C2×D411yescharacteristic
46C22×C4—22no—
47C22×C4—22no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—24no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—41no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—42no—
57C22×C4—42no—
58C22×C4—42no—
59C22×C4—42no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4SD1612yes—
67C22×C4Q1612yes—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4#4114yes—
71C2×C8—42no—
72C2×C8—42no—
73C2×C8—42no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C4⋊C4—42no—
80C22⋊C4—44no—
81C42—22no—
82C42—22no—
83C42—22no—
84C42—22no—
85C42—44no—
86C23—21no—
87C23—21no—
88C23—21no—
89C23—21no—
90C23C2×C22⋊C411yescharacteristic
91C23#12312yes—
92C23C2×SD1611yescharacteristic
93C23C2×Q1611yescharacteristic
94C23AGL(1,ℤ/8)12yes—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—24no—
106C2×C4—24no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—44no—
111C2×C4—44no—
112C2×C4—44no—
113C2×C4—44no—
114C2×C4—44no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—82no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4—82no—
125C2×C4#10312yes—
126C2×C4#10312yes—
127C2×C4#10412yes—
128C2×C4#11712yes—
129C2×C4#12312yes—
130C8—82no—
131C22—22no—
132C22—22no—
133C22—21no—
134C22—21no—
135C22—22no—
136C22—22no—
137C22—21no—
138C22—21no—
139C22#38111yescharacteristic
140C22#41011yescharacteristic
141C22#41011yescharacteristic
142C22C2×#10311yescharacteristic
143C22#41811yescharacteristic
144C22C2×#10411yescharacteristic
145C22#42111yescharacteristic
146C22#48112yes—
147C22#48412yes—
148C4—22no—
149C4—22no—
150C4—22no—
151C4—22no—
152C4—22no—
153C4—41no—
154C4—84no—
155C4—82no—
156C4—82no—
157C2—22no—
158C2—22no—
159C2#150311yescharacteristic
160C2#150511yescharacteristic
161C2#151411yescharacteristic
162C2#151811yescharacteristic
163C2#182011yescharacteristic
164C2#182411yescharacteristic
165C2#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
8. #384C4semidirect
9. #384C4semidirect
14. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#22852
#18172
#14342
#12942