Finite Group Databasebeta

← #12870

Subgroups — #12870

Order 256.

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

normal subgroups only

This group has 210 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
1C2×#427C211yesmaximal, characteristic
2#2112C211yesmaximal, characteristic
3#1807C211yesmaximal, characteristic
4#1386C214yesmaximal, complemented
5C23×C8C2211yescharacteristic
6C2×#117—21no—
7C2×M32C2212yes—
8#427—21no—
9#427—21no—
10#427C412yescomplemented
11#427C2212yes—
12C2×#107C411yescharacteristic, complemented
13C2×#107C411yescharacteristic, complemented
14C2×#106—21no—
15C2×#106C2212yes—
16#380—21no—
17C22⋊C16—21no—
18C22⋊C16—21no—
19#340—21no—
20C23×C4D411yescharacteristic
21C22×C8—24no—
22C22×C8C2×C411yescharacteristic
23C22×C8C2×C411yescharacteristic
24C22×C8D411yescharacteristic
25C22×C8D411yescharacteristic
26C22×C8D411yescharacteristic
27C22×C8C2311yescharacteristic
28#117—22no—
29#117—22no—
30#117—41no—
31#117—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C22⋊C4—41no—
36C2×C22⋊C4—41no—
37C2×C42—41no—
38M32—24no—
39C2×C16—22no—
40#107—21no—
41#107—21no—
42#107C2×C412yes—
43#107C2×C412yes—
44#106—22no—
45#106—22no—
46#106—24no—
47#95—41no—
48C24D811yescharacteristic
49C22×C4—22no—
50C22×C4—22no—
51C22×C4—41no—
52C22×C4—41no—
53C22×C4—42no—
54C22×C4C22⋊C411yescharacteristic
55C22×C4C22⋊C411yescharacteristic
56C22×C4D811yescharacteristic
57C22×C4SD1611yescharacteristic
58C22×C4SD1611yescharacteristic
59C22×C4C2×D411yescharacteristic
60C2×C8—22no—
61C2×C8—22no—
62C2×C8—22no—
63C2×C8—22no—
64C2×C8—22no—
65C2×C8C22⋊C411yescharacteristic
66C2×C8C22⋊C411yescharacteristic
67C2×C8C22×C411yescharacteristic
68C2×C8C2×D411yescharacteristic
69C2×C8#4114yes—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C22⋊C4—82no—
79C42—42no—
80C42—42no—
81C16—42no—
82C23—22no—
83C23—22no—
84C23—41no—
85C23#10211yescharacteristic
86C23#10211yescharacteristic
87C23C2×D811yescharacteristic
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—82no—
98C2×C4—82no—
99C2×C4—82no—
100C2×C4—82no—
101C2×C4—82no—
102C2×C4#10211yescharacteristic
103C2×C4#10211yescharacteristic
104C2×C4C2×C22⋊C411yescharacteristic
105C2×C4#12312yes—
106C2×C4C2×SD1611yescharacteristic
107C2×C4#13712yes—
108C8—21no—
109C8—21no—
110C8#11712yes—
111C8#12312yes—
112C22—21no—
113C22—21no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—44no—
120C22#36012yes—
121C22C2×#10211yescharacteristic
122C22#48012yes—
123C4—21no—
124C4—21no—
125C4—82no—
126C4—82no—
127C4—162no—
128C4#38111yescharacteristic
129C4#41911yescharacteristic
130C4#48312yes—
131C2—22no—
132C2—22no—
133C2—41no—
134C2#182311yescharacteristic
135C2C2×#36011yescharacteristic
136C2#214511yescharacteristic
subgroupG/H
center, socleC22C2×#102
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
prime radicalC24D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1386C2semidirect
10. #427C4semidirect
12. C2×#107C4semidirect
13. C2×#107C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4272
#21122
#18072
#13862