Finite Group Databasebeta

← #7502

Subgroups — #7502

Order 256.

143 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 143 conjugacy classes of subgroups — too many to draw. Only its 35 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×#355C211yesmaximal, characteristic
2C2×#354C211yesmaximal, characteristic
3#1958C211yesmaximal, characteristic
4C2×#124C2211yescharacteristic
5C2×#103—21no—
6C2×#102—21no—
7C2×#101—21no—
8C2×#99—21no—
9C2×C4×C8C411yescharacteristic, complemented
10#382C411yescharacteristic, complemented
11#355—21no—
12#355C412yescomplemented
13#354—21no—
14#354C412yescomplemented
15C22×Q8D411yescharacteristic
16C22×D4D411yescharacteristic
17C2×M16—41no—
18C22×C8—21no—
19#124—41no—
20#124C2×C411yescharacteristic
21#124C2×C411yescharacteristic
22#124D411yescharacteristic
23#124Q811yescharacteristic
24C2×C4⋊C4—21no—
25C2×C22⋊C4—41no—
26C2×C22⋊C4—41no—
27C2×C42C2×C411yescharacteristic
28#103—42no—
29#102—42no—
30#101—22no—
31#101—41no—
32#99—22no—
33#99—41no—
34C4×C8—41no—
35#95—41no—
36C24—21no—
37C2×Q8—21no—
38C2×Q8—41no—
39C2×Q8—41no—
40C2×Q8C22⋊C411yescharacteristic
41C2×Q8C4⋊C411yescharacteristic
42C2×D4—21no—
43C2×D4—41no—
44C2×D4—41no—
45C2×D4C22⋊C411yescharacteristic
46C2×D4C4⋊C411yescharacteristic
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—41no—
50C22×C4—81no—
51C22×C4C22⋊C411yescharacteristic
52M16—42no—
53M16—81no—
54C2×C8—22no—
55C2×C8—41no—
56C2×C8—41no—
57C2×C8—81no—
58C4⋊C4—41no—
59C4⋊C4—41no—
60C22⋊C4—41no—
61C22⋊C4—41no—
62C22⋊C4—41no—
63C22⋊C4—41no—
64C22⋊C4—42no—
65C22⋊C4—81no—
66C42—21no—
67C42C4211yescharacteristic
68C42C4⋊C411yescharacteristic
69C23—21no—
70C23—21no—
71C23—41no—
72C23—41no—
73C23—82no—
74C23#9911yescharacteristic
75Q8—41no—
76Q8—41no—
77Q8—81no—
78D4—41no—
79D4—41no—
80D4—81no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—42no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—82no—
91C2×C4—82no—
92C2×C4—81no—
93C2×C4—162no—
94C2×C4#9511yescharacteristic
95C2×C4#9911yescharacteristic
96C8—42no—
97C8—82no—
98C22—22no—
99C22—41no—
100C22—41no—
101C22—81no—
102C22—81no—
103C22—82no—
104C22—161no—
105C22#34211yescharacteristic
106C22#35311yescharacteristic
107C22#35411yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—41no—
113C4—41no—
114C4—161no—
115C4—162no—
116C2—21no—
117C2—21no—
118C2—81no—
119C2—81no—
120C2#139311yescharacteristic
121C2#141412yes—
subgroupG/H
center, socleC22#354
commutatorC42C42
Frattini, cosocleC2×#124C22
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G
LECSG ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×C4×C8C4semidirect
10. #382C4semidirect
12. #355C4semidirect
14. #354C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3552
C2×#3542
#19582