Finite Group Databasebeta

← #7496

Subgroups — #7496

Order 256.

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

normal subgroups only

This group has 137 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
1#2132C211yesmaximal, characteristic
2#1884C211yesmaximal, characteristic
3#1560C211yesmaximal, characteristic
4#568C411yescharacteristic
5#532C2211yescharacteristic
6#423—21no—
7#421—21no—
8#413—21no—
9#410C411yescharacteristic
10#356—21no—
11#355C412yescomplemented
12#339—21no—
13#333—21no—
14#331C412yescomplemented
15C2×#41C2×C411yescharacteristic
16C2×#41D411yescharacteristic
17#131—42no—
18C2×M16—21no—
19C2×M16—41no—
20C2×M16—41no—
21C2×M16—41no—
22C4⋊Q8D411yescharacteristic
23#127C2×C411yescharacteristic
24#124C2×C411yescharacteristic
25#124Q811yescharacteristic
26#121—41no—
27C4×D4—41no—
28#117—21no—
29C2×C42D411yescharacteristic
30C4⋊C8—22no—
31C4⋊C8—41no—
32#104—42no—
33#101—22no—
34#100—41no—
35#99—41no—
36#41—41no—
37#41—41no—
38#41—41no—
39#41—41no—
40C2×Q8C22⋊C411yescharacteristic
41C2×Q8C4⋊C411yescharacteristic
42C2×D4—21no—
43C2×D4—21no—
44C2×D4—41no—
45C2×D4C4211yescharacteristic
46C2×D4C4⋊C411yescharacteristic
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4C22⋊C411yescharacteristic
51M16—22no—
52M16—41no—
53M16—42no—
54M16—42no—
55M16—82no—
56M16—81no—
57C2×C8—41no—
58C2×C8—42no—
59C2×C8—42no—
60C2×C8—81no—
61C4⋊C4—41no—
62C4⋊C4—41no—
63C22⋊C4—41no—
64C22⋊C4—41no—
65C22⋊C4—41no—
66C42—21no—
67C42—41no—
68C42C22⋊C411yescharacteristic
69C42C4⋊C411yescharacteristic
70C23—21no—
71C23—21no—
72C23#9911yescharacteristic
73Q8—41no—
74Q8—41no—
75D4—41no—
76D4—41no—
77D4—42no—
78D4—81no—
79D4—81no—
80D4—81no—
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—41no—
92C2×C4—41no—
93C2×C4—81no—
94C2×C4#9511yescharacteristic
95C2×C4#9911yescharacteristic
96C8—42no—
97C8—82no—
98C8—82no—
99C22—21no—
100C22—41no—
101C22—41no—
102C22—41no—
103C22—81no—
104C22—81no—
105C22#34211yescharacteristic
106C4—21no—
107C4—21no—
108C4—21no—
109C4—41no—
110C4—41no—
111C4—41no—
112C4—161no—
113C4AΣL(1,𝔽16)11yescharacteristic
114C4#35211yescharacteristic
115C2—21no—
116C2—41no—
117C2—81no—
118C2—81no—
119C2#139011yescharacteristic
subgroupG/H
center, socleC2#1390
commutatorC2×D4C42
Frattini, prime radical, cosocle#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #355C4semidirect
14. #331C4semidirect

Maximal subgroups

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

subgroupindex
#21322
#18842
#15602