Finite Group Databasebeta

← #7503

Subgroups — #7503

Order 256.

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

normal subgroups only

This group has 135 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#2126C211yesmaximal, characteristic
3#1963C211yesmaximal, characteristic
4#532C2211yescharacteristic
5#423C411yescharacteristic, complemented
6C2×#104—21no—
7C2×#104—21no—
8#413—21no—
9#410—21no—
10#376C411yescharacteristic, complemented
11#356C412yescomplemented
12#355—21no—
13#354—21no—
14#353C412yescomplemented
15C2×#41D411yescharacteristic
16C2×#41D411yescharacteristic
17C2×M16—21no—
18C2×M16—41no—
19C4⋊Q8C2×C411yescharacteristic
20#127C2×C411yescharacteristic
21#124D411yescharacteristic
22#124Q811yescharacteristic
23#121—41no—
24C4×D4—41no—
25#117—41no—
26C2×C42—21no—
27C2×C42C2×C411yescharacteristic
28C4⋊C8—41no—
29#104—42no—
30#104—42no—
31#101—22no—
32#100—41no—
33#99—22no—
34#99—41no—
35#95—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×D4C22⋊C411yescharacteristic
46C2×D4C4⋊C411yescharacteristic
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—41no—
51C22×C4C22⋊C411yescharacteristic
52M16—42no—
53M16—41no—
54M16—41no—
55M16—81no—
56C2×C8—41no—
57C2×C8—81no—
58C4⋊C4—41no—
59C4⋊C4—81no—
60C22⋊C4—41no—
61C22⋊C4—41no—
62C22⋊C4—42no—
63C42—21no—
64C42—22no—
65C42—41no—
66C42—81no—
67C42C4211yescharacteristic
68C42C4⋊C411yescharacteristic
69C23—21no—
70C23—21no—
71C23#9911yescharacteristic
72Q8—41no—
73Q8—41no—
74D4—41no—
75D4—41no—
76D4—81no—
77D4—81no—
78D4—82no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4—81no—
91C2×C4—82no—
92C2×C4—82no—
93C2×C4#9511yescharacteristic
94C2×C4#9911yescharacteristic
95C8—82no—
96C8—81no—
97C22—21no—
98C22—41no—
99C22—41no—
100C22—41no—
101C22—81no—
102C22—81no—
103C22#34211yescharacteristic
104C4—21no—
105C4—41no—
106C4—41no—
107C4—41no—
108C4—41no—
109C4—82no—
110C4—162no—
111C4#35311yescharacteristic
112C4#35411yescharacteristic
113C2—21no—
114C2—41no—
115C2—81no—
116C2—81no—
117C2#139311yescharacteristic
subgroupG/H
centerC4#354
commutatorC42C42
Frattini, prime radical, cosocle#532C22
socleC2#1393
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ 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
5. #423C4semidirect
10. #376C4semidirect
11. #356C4semidirect
14. #353C4semidirect

Maximal subgroups

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

subgroupindex
#21322
#21262
#19632