Finite Group Databasebeta

← #7254

Subgroups — #7254

Order 256.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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×#349C211yesmaximal, characteristic
2#1809C211yesmaximal, characteristic
3C2×H(3,ℤ/4)C211yesmaximal, characteristic
4C22×M16C2211yescharacteristic
5C2×#117—21no—
6C2×M32—21no—
7C2×#107C411yescharacteristic, complemented
8C2×#106C411yescharacteristic, complemented
9#380—21no—
10#349—21no—
11#349C412yescomplemented
12#340—21no—
13H(3,ℤ/4)—21no—
14H(3,ℤ/4)C412yescomplemented
15C23×C4D411yescharacteristic
16C2×M16—41no—
17C2×M16—41no—
18C2×M16C2×C411yescharacteristic
19C2×M16C2×C411yescharacteristic
20C2×M16D411yescharacteristic
21C2×M16Q811yescharacteristic
22C22×C8C2×C411yescharacteristic
23C22×C8D411yescharacteristic
24#117—22no—
25#117—22no—
26#117—41no—
27#117—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—41no—
31C2×C22⋊C4—41no—
32C2×C22⋊C4—41no—
33C2×C42—22no—
34M32—22no—
35M32—41no—
36C2×C16—41no—
37#107—41no—
38#106—41no—
39#95—41no—
40C24D811yescharacteristic
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—42no—
46C22×C4—41no—
47C22×C4—41no—
48C22×C4C22⋊C411yescharacteristic
49C22×C4C22⋊C411yescharacteristic
50C22×C4C4⋊C411yescharacteristic
51C22×C4SD1611yescharacteristic
52C22×C4SD1611yescharacteristic
53C22×C4Q1611yescharacteristic
54M16—41no—
55M16—41no—
56M16—41no—
57M16—41no—
58C2×C8—21no—
59C2×C8—21no—
60C2×C8—21no—
61C2×C8—21no—
62C2×C8C4211yescharacteristic
63C2×C8C22⋊C411yescharacteristic
64C2×C8C4⋊C411yescharacteristic
65C2×C8C4⋊C411yescharacteristic
66C4⋊C4—42no—
67C4⋊C4—42no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—41no—
72C22⋊C4—42no—
73C22⋊C4—42no—
74C22⋊C4—82no—
75C42—24no—
76C42—42no—
77C16—42no—
78C23—21no—
79C23—21no—
80C23—41no—
81C23—41no—
82C23#10211yescharacteristic
83C23#10211yescharacteristic
84C23#10711yescharacteristic
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—44no—
96C2×C4—82no—
97C2×C4—82no—
98C2×C4—81no—
99C2×C4—81no—
100C2×C4—81no—
101C2×C4—81no—
102C2×C4—81no—
103C2×C4—81no—
104C2×C4#9511yescharacteristic
105C2×C4#9911yescharacteristic
106C2×C4#9911yescharacteristic
107C2×C4#10311yescharacteristic
108C2×C4#10311yescharacteristic
109C2×C4#10411yescharacteristic
110C2×C4#10411yescharacteristic
111C2×C4#10611yescharacteristic
112C8—21no—
113C8—21no—
114C8—21no—
115C8—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—21no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—41no—
124C22—81no—
125C22#32711yescharacteristic
126C22#32711yescharacteristic
127C22#34011yescharacteristic
128C4—21no—
129C4—21no—
130C4—84no—
131C4—161no—
132C4—161no—
133C4#32811yescharacteristic
134C4#32811yescharacteristic
135C4#33911yescharacteristic
136C4#34211yescharacteristic
137C2—21no—
138C2—21no—
139C2—41no—
140C2—41no—
141C2#129411yescharacteristic
142C2#133911yescharacteristic
143C2#134011yescharacteristic
subgroupG/H
center, socleC22#327
commutatorC2×C8C42
Frattini, cosocleC22×M16C22
prime radicalC24D8
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#107C4semidirect
8. C2×#106C4semidirect
11. #349C4semidirect
14. H(3,ℤ/4)C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3492
#18092
C2×H(3,ℤ/4)2