Finite Group Databasebeta

← #7483

Subgroups — #7483

Order 256.

131 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 131 conjugacy classes of subgroups — too many to draw. Only its 33 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×#356C211yesmaximal, characteristic
2#1886C211yesmaximal, characteristic
3#1558C211yesmaximal, characteristic
4C2×C4⋊Q8C2211yescharacteristic
5#519C411yescharacteristic
6#432—21no—
7#419—21no—
8C2×#101—21no—
9C2×#101C411yescharacteristic
10#356—21no—
11#356C412yescomplemented
12#339—21no—
13#333—21no—
14#333C412yescomplemented
15C22×Q8C2×C411yescharacteristic
16C22×Q8D411yescharacteristic
17C2×M16—21no—
18C2×M16—41no—
19C4⋊Q8—41no—
20C4⋊Q8C2×C411yescharacteristic
21C4⋊Q8C2×C411yescharacteristic
22C4⋊Q8D411yescharacteristic
23C4⋊Q8Q811yescharacteristic
24C4×Q8—42no—
25#117—21no—
26#117—41no—
27C2×C4⋊C4—41no—
28C2×C42D411yescharacteristic
29C4⋊C8—22no—
30#103—42no—
31#101—22no—
32#101—41no—
33#101—41no—
34C22⋊C8—41no—
35#97—41no—
36C2×Q8—21no—
37C2×Q8—21no—
38C2×Q8—41no—
39C2×Q8—41no—
40C2×Q8—41no—
41C2×Q8—41no—
42C2×Q8C4211yescharacteristic
43C2×Q8C22⋊C411yescharacteristic
44C2×Q8C4⋊C411yescharacteristic
45C2×Q8C4⋊C411yescharacteristic
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—21no—
49C22×C4C22⋊C411yescharacteristic
50M16—42no—
51M16—41no—
52M16—41no—
53M16—81no—
54C2×C8—42no—
55C2×C8—41no—
56C2×C8—81no—
57C4⋊C4—22no—
58C4⋊C4—41no—
59C4⋊C4—41no—
60C4⋊C4—41no—
61C4⋊C4—41no—
62C4⋊C4—42no—
63C22⋊C4—41no—
64C42—21no—
65C42—41no—
66C42—42no—
67C42C22⋊C411yescharacteristic
68C42C4⋊C411yescharacteristic
69C23#9911yescharacteristic
70Q8—41no—
71Q8—41no—
72Q8—41no—
73Q8—41no—
74Q8—42no—
75Q8—81no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4#9511yescharacteristic
91C2×C4#9911yescharacteristic
92C8—82no—
93C8—81no—
94C8—82no—
95C22—41no—
96C22#34211yescharacteristic
97C22AΣL(1,𝔽16)11yescharacteristic
98C22#35211yescharacteristic
99C4—22no—
100C4—21no—
101C4—21no—
102C4—41no—
103C4—41no—
104C4—41no—
105C4—41no—
106C4—41no—
107C4—82no—
108C2—21no—
109C2—21no—
110C2—21no—
111C2#139011yescharacteristic
subgroupG/H
center, socleC2#1390
commutatorC2×Q8C42
Frattini, cosocleC2×C4⋊Q8C22
prime radicalC23#99
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #356C4semidirect
14. #333C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3562
#18862
#15582