Finite Group Databasebeta

← #7320

Subgroups — #7320

Order 256.

171 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 39 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#1870C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1466C211yesmaximal, characteristic
4C22×M16C411yescharacteristic, complemented
5C2×C4×D4C2211yescharacteristic
6#432—21no—
7C2×#99C412yes—
8C2×C22⋊C8—21no—
9C2×C22⋊C8C411yescharacteristic, complemented
10#380—21no—
11#342—21no—
12#336—21no—
13#323—21no—
14#323C412yescomplemented
15C22×D4C2×C411yescharacteristic
16C23×C4C2×C411yescharacteristic
17C23×C4D411yescharacteristic
18C2×M16—42no—
19C2×M16—41no—
20C22×C8—21no—
21C22×C8—21no—
22C4×D4—41no—
23C4×D4—41no—
24C2×C4⋊C4—41no—
25C2×C4⋊C4Q811yescharacteristic
26C2×C22⋊C4—22no—
27C2×C22⋊C4C2×C411yescharacteristic
28C2×C22⋊C4D411yescharacteristic
29C2×C42D411yescharacteristic
30C4⋊C8—41no—
31#99—42no—
32C22⋊C8—22no—
33C22⋊C8—42no—
34C22⋊C8—41no—
35#97—41no—
36#95—41no—
37C24C4211yescharacteristic
38C24C22⋊C411yescharacteristic
39C2×D4—21no—
40C2×D4—21no—
41C2×D4—41no—
42C2×D4—41no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—41no—
48C22×C4—41no—
49C22×C4—41no—
50C22×C4—41no—
51C22×C4—42no—
52C22×C4C22⋊C411yescharacteristic
53C22×C4C22⋊C411yescharacteristic
54C22×C4C4⋊C411yescharacteristic
55C22×C4C4⋊C411yescharacteristic
56C22×C4C4⋊C411yescharacteristic
57M16—42no—
58M16—42no—
59C2×C8—22no—
60C2×C8—41no—
61C2×C8—41no—
62C2×C8—42no—
63C2×C8—41no—
64C2×C8—41no—
65C2×C8—41no—
66C4⋊C4—41no—
67C4⋊C4—81no—
68C4⋊C4—81no—
69C22⋊C4—41no—
70C22⋊C4—41no—
71C22⋊C4—42no—
72C22⋊C4—42no—
73C42—21no—
74C42—21no—
75C23—21no—
76C23—21no—
77C23—21no—
78C23—21no—
79C23—21no—
80C23—41no—
81C23—41no—
82C23—41no—
83C23—41no—
84C23#9511yescharacteristic
85C23#10011yescharacteristic
86C23#10111yescharacteristic
87D4—42no—
88D4—42no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—81no—
106C2×C4—82no—
107C2×C4—82no—
108C2×C4—82no—
109C2×C4#9911yescharacteristic
110C2×C4#9911yescharacteristic
111C8—42no—
112C8—81no—
113C8—82no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—41no—
124C22—41no—
125C22—81no—
126C22—81no—
127C22—81no—
128C22#34211yescharacteristic
129C22#34311yescharacteristic
130C22#34411yescharacteristic
131C4—21no—
132C4—41no—
133C4—41no—
134C4—41no—
135C4—41no—
136C4—162no—
137C4AΣL(1,𝔽16)11yescharacteristic
138C4#35211yescharacteristic
139C2—21no—
140C2—21no—
141C2—41no—
142C2—41no—
143C2—81no—
144C2#130511yescharacteristic
145C2#139011yescharacteristic
146C2#139111yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC24C42
Frattini, cosocleC2×C4×D4C22
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×M16C4semidirect
9. C2×C22⋊C8C4semidirect
14. #323C4semidirect

Maximal subgroups

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

subgroupindex
#18702
#17992
#14662