Finite Group Databasebeta

← #7204

Subgroups — #7204

Order 256.

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

normal subgroups only

This group has 190 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
1#1869C211yesmaximal, characteristic
2C4×#99C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4C23×C8C411yescharacteristic, complemented
5C2×C4×D4C2211yescharacteristic
6#431—21no—
7C2×#99C412yes—
8C2×C22⋊C8—21no—
9C2×C22⋊C8C411yescharacteristic, complemented
10C4×C22⋊C4—21no—
11#342—21no—
12#336—21no—
13#323—21no—
14#323C412yescomplemented
15C22×D4C2×C411yescharacteristic
16C23×C4C2×C411yescharacteristic
17C23×C4D411yescharacteristic
18C22×C8—21no—
19C22×C8—21no—
20C22×C8—42no—
21C22×C8—41no—
22C4×D4—41no—
23C4×D4—41no—
24C2×C4⋊C4Q811yescharacteristic
25C2×C22⋊C4—22no—
26C2×C22⋊C4C2×C411yescharacteristic
27C2×C22⋊C4D411yescharacteristic
28C2×C42—41no—
29C2×C42D411yescharacteristic
30C4⋊C8—41no—
31#99—22no—
32#99C814yescomplemented
33C22⋊C8—22no—
34C22⋊C8—42no—
35C22⋊C8—41no—
36C4×C8—41no—
37#95—41no—
38C24C4211yescharacteristic
39C24C22⋊C411yescharacteristic
40C2×D4—41no—
41C2×D4—41no—
42C2×D4C2×C812yes—
43C2×D4M1612yes—
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—41no—
49C22×C4—41no—
50C22×C4—41no—
51C22×C4—41no—
52C22×C4—42no—
53C22×C4C22⋊C411yescharacteristic
54C22×C4C22⋊C411yescharacteristic
55C22×C4C4⋊C411yescharacteristic
56C22×C4C4⋊C411yescharacteristic
57C22×C4C4⋊C411yescharacteristic
58C2×C8—22no—
59C2×C8—41no—
60C2×C8—42no—
61C2×C8—41no—
62C2×C8—42no—
63C2×C8—42no—
64C2×C8—41no—
65C2×C8—41no—
66C2×C8—41no—
67C4⋊C4—41no—
68C22⋊C4—24no—
69C22⋊C4—41no—
70C22⋊C4—41no—
71C22⋊C4—42no—
72C42—21no—
73C42—21no—
74C42—81no—
75C42—81no—
76C23—21no—
77C23—21no—
78C23—21no—
79C23—21no—
80C23—41no—
81C23—41no—
82C23—41no—
83C23—41no—
84C23#9511yescharacteristic
85C23C4×C811yescharacteristic
86C23#9711yescharacteristic
87C23C22⋊C812yes—
88D4—42no—
89D4—42no—
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—44no—
106C2×C4—81no—
107C2×C4—82no—
108C2×C4—82no—
109C2×C4#9911yescharacteristic
110C2×C4#9911yescharacteristic
111C2×C4C4⋊C812yes—
112C8—42no—
113C8—81no—
114C8—82no—
115C22—21no—
116C22—21no—
117C22—22no—
118C22—21no—
119C22—21no—
120C22—41no—
121C22—41no—
122C22—42no—
123C22—42no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22—81no—
129C22#33611yescharacteristic
130C22#34211yescharacteristic
131C22#34411yescharacteristic
132C4—22no—
133C4—21no—
134C4—41no—
135C4—41no—
136C4—41no—
137C4—84no—
138C4AΣL(1,𝔽16)11yescharacteristic
139C4#35211yescharacteristic
140C2—21no—
141C2—21no—
142C2—42no—
143C2—41no—
144C2—41no—
145C2#128211yescharacteristic
146C2#139011yescharacteristic
147C2#139111yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC23C4×C8
Frattini, cosocleC2×C4×D4C22
socleC22#342
prime radicalC22×D4C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. C23×C8C4semidirect
9. C2×C22⋊C8C4semidirect
14. #323C4semidirect
32. #99C8semidirect

Maximal subgroups

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

subgroupindex
#18692
C4×#992
#14602