Finite Group Databasebeta

← #7322

Subgroups — #7322

Order 256.

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

normal subgroups only

This group has 169 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#1955C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic
4C2×C4×D4C2211yescharacteristic
5C2×C4⋊C8C411yescharacteristic, complemented
6C2×#99C412yes—
7#407—21no—
8C2×C22⋊C8—21no—
9C2×C22⋊C8—21no—
10C2×#97C411yescharacteristic, complemented
11#380—21no—
12#342—21no—
13#324C412yescomplemented
14#323—21no—
15C22×D4C2×C411yescharacteristic
16C23×C4D411yescharacteristic
17C23×C4D411yescharacteristic
18C2×M16—41no—
19C22×C8—21no—
20C22×C8—21no—
21C4×D4—41no—
22C4×D4—41no—
23C2×C4⋊C4—41no—
24C2×C4⋊C4C2×C411yescharacteristic
25C2×C22⋊C4—22no—
26C2×C22⋊C4D411yescharacteristic
27C2×C22⋊C4Q811yescharacteristic
28C2×C42C2×C411yescharacteristic
29C4⋊C8—41no—
30#99—42no—
31C22⋊C8—22no—
32C22⋊C8—42no—
33C22⋊C8—42no—
34C22⋊C8—41no—
35#97—41no—
36#95—41no—
37C24C22⋊C411yescharacteristic
38C24C4⋊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×C4C4211yescharacteristic
53C22×C4C22⋊C411yescharacteristic
54C22×C4C22⋊C411yescharacteristic
55C22×C4C4⋊C411yescharacteristic
56C22×C4C4⋊C411yescharacteristic
57M16—81no—
58M16—81no—
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—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—81no—
105C2×C4—81no—
106C2×C4—82no—
107C2×C4—82no—
108C2×C4—82no—
109C2×C4#9911yescharacteristic
110C2×C4#9911yescharacteristic
111C8—42no—
112C8—82no—
113C8—81no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22—41no—
124C22—41no—
125C22—41no—
126C22—81no—
127C22—81no—
128C22#34211yescharacteristic
129C22#34311yescharacteristic
130C22#34411yescharacteristic
131C4—21no—
132C4—41no—
133C4—41no—
134C4—41no—
135C4—41no—
136C4—162no—
137C4#35311yescharacteristic
138C4#35411yescharacteristic
139C2—21no—
140C2—21no—
141C2—41no—
142C2—41no—
143C2—81no—
144C2#130511yescharacteristic
145C2#139311yescharacteristic
146C2#139511yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC22×C4C42
Frattini, cosocleC2×C4×D4C22
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ 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
5. C2×C4⋊C8C4semidirect
10. C2×#97C4semidirect
13. #324C4semidirect

Maximal subgroups

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

subgroupindex
#19552
#17992
#14652