Finite Group Databasebeta

← #7214

Subgroups — #7214

Order 256.

145 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 37 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#1820C211yesmaximal, characteristic
2#1750C211yesmaximal, characteristic
3#1464C211yesmaximal, characteristic
4C2×#117C2211yescharacteristic
5#432—21no—
6C2×C22⋊C8—21no—
7C2×#97C411yescharacteristic, complemented
8C2×C4×C8C411yescharacteristic, complemented
9#381—21no—
10#342—21no—
11#342C412yes—
12#336—21no—
13#324C412yescomplemented
14#323—21no—
15C23×C4D411yescharacteristic
16C22×C8—21no—
17C22×C8—21no—
18#117—41no—
19#117—41no—
20C2×C4⋊C4C2×C411yescharacteristic
21C2×C4⋊C4Q811yescharacteristic
22C2×C22⋊C4—22no—
23C2×C22⋊C4C2×C411yescharacteristic
24C2×C22⋊C4D411yescharacteristic
25C2×C42C2×C411yescharacteristic
26C2×C42D411yescharacteristic
27C4⋊C8—41no—
28C22⋊C8—24no—
29C22⋊C8—22no—
30#97—41no—
31#97—41no—
32C4×C8—41no—
33#95—41no—
34#95—41no—
35C24C22⋊C411yescharacteristic
36C22×C4—22no—
37C22×C4—21no—
38C22×C4—21no—
39C22×C4—41no—
40C22×C4—42no—
41C22×C4C4211yescharacteristic
42C22×C4C22⋊C411yescharacteristic
43C22×C4C22⋊C411yescharacteristic
44C22×C4C4⋊C411yescharacteristic
45C22×C4C4⋊C411yescharacteristic
46C22×C4C4⋊C411yescharacteristic
47C2×C8—22no—
48C2×C8—22no—
49C2×C8—41no—
50C2×C8—41no—
51C2×C8—41no—
52C2×C8—41no—
53C2×C8—42no—
54C4⋊C4—41no—
55C4⋊C4—41no—
56C22⋊C4—41no—
57C22⋊C4—41no—
58C22⋊C4—42no—
59C22⋊C4—42no—
60C42—21no—
61C42—21no—
62C42—21no—
63C42—21no—
64C23—21no—
65C23—21no—
66C23—41no—
67C23—41no—
68C23#9511yescharacteristic
69C23#10412yes—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—21no—
74C2×C4—21no—
75C2×C4—21no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—82no—
86C2×C4—82no—
87C2×C4—82no—
88C2×C4#9911yescharacteristic
89C2×C4#9911yescharacteristic
90C8—42no—
91C8—42no—
92C8—82no—
93C22—21no—
94C22—21no—
95C22—22no—
96C22—42no—
97C22—41no—
98C22—41no—
99C22—81no—
100C22H(3,ℤ/4)11yescharacteristic
101C22#33911yescharacteristic
102C22#34211yescharacteristic
103C4—21no—
104C4—21no—
105C4—41no—
106C4—41no—
107C4—41no—
108C4—41no—
109C4—41no—
110C4—162no—
111C2—21no—
112C2—21no—
113C2—42no—
114C2#128411yescharacteristic
115C2#139111yescharacteristic
116C2#139511yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC22×C4C42
Frattini, cosocleC2×#117C22
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#97C4semidirect
8. C2×C4×C8C4semidirect
13. #324C4semidirect

Maximal subgroups

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

subgroupindex
#18202
#17502
#14642