Finite Group Databasebeta

← #7491

Subgroups — #7491

Order 256.

138 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 35 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
3C2×#333C211yesmaximal, characteristic
4C2×C4⋊Q8C2211yescharacteristic
5#519C411yescharacteristic
6C2×C4⋊C8—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—
19C22×C8—41no—
20C4⋊Q8—41no—
21C4⋊Q8C2×C411yescharacteristic
22C4⋊Q8C2×C411yescharacteristic
23C4⋊Q8D411yescharacteristic
24C4⋊Q8Q811yescharacteristic
25C4×Q8—42no—
26#117—21no—
27#117—41no—
28C2×C4⋊C4—41no—
29C2×C42D411yescharacteristic
30C4⋊C8—22no—
31C4⋊C8—41no—
32#103—42no—
33#101—22no—
34#101—41no—
35#101—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—41no—
55C2×C8—42no—
56C2×C8—82no—
57C2×C8—81no—
58C4⋊C4—22no—
59C4⋊C4—41no—
60C4⋊C4—41no—
61C4⋊C4—41no—
62C4⋊C4—41no—
63C4⋊C4—42no—
64C22⋊C4—41no—
65C42—21no—
66C42—41no—
67C42—42no—
68C42C22⋊C411yescharacteristic
69C42C4⋊C411yescharacteristic
70C23#9911yescharacteristic
71Q8—41no—
72Q8—41no—
73Q8—41no—
74Q8—41no—
75Q8—42no—
76Q8—81no—
77C2×C4—22no—
78C2×C4—22no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—42no—
92C2×C4#9511yescharacteristic
93C2×C4#9911yescharacteristic
94C8—82no—
95C8—81no—
96C8—82no—
97C22—22no—
98C22#34211yescharacteristic
99C22AΣL(1,𝔽16)11yescharacteristic
100C22#35211yescharacteristic
101C4—21no—
102C4—22no—
103C4—21no—
104C4—21no—
105C4—21no—
106C4—41no—
107C4—41no—
108C4—41no—
109C4—41no—
110C4—82no—
111C2—21no—
112C2—21no—
113C2#139011yescharacteristic
114C2#140512yes—
subgroupG/H
center, socleC22AΣL(1,𝔽16)
commutatorC2×Q8C42
Frattini, cosocleC2×C4⋊Q8C22
prime radicalC23#99
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
C2×#3332