Finite Group Databasebeta

← #7508

Subgroups — #7508

Order 256.

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

normal subgroups only

This group has 145 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#1959C211yesmaximal, characteristic
4C2×#124C2211yescharacteristic
5C2×#103—21no—
6C2×#102—21no—
7C2×#101—21no—
8C2×#99—21no—
9C2×#97C411yescharacteristic, complemented
10#384C411yescharacteristic, 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×C4⋊C4—41no—
26C2×C22⋊C4—41no—
27C2×C22⋊C4—41no—
28C2×C42C2×C411yescharacteristic
29#103—42no—
30#102—42no—
31#101—22no—
32#101—41no—
33#99—22no—
34#99—41no—
35#97—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—
60C4⋊C4—82no—
61C22⋊C4—41no—
62C22⋊C4—41no—
63C22⋊C4—41no—
64C22⋊C4—41no—
65C22⋊C4—42no—
66C22⋊C4—81no—
67C42—21no—
68C42C4211yescharacteristic
69C42C4⋊C411yescharacteristic
70C23—21no—
71C23—21no—
72C23—41no—
73C23—41no—
74C23—81no—
75C23—81no—
76C23#9911yescharacteristic
77Q8—41no—
78Q8—41no—
79Q8—81no—
80D4—41no—
81D4—41no—
82D4—81no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—82no—
94C2×C4—82no—
95C2×C4—81no—
96C2×C4—161no—
97C2×C4—161no—
98C2×C4#9511yescharacteristic
99C2×C4#9911yescharacteristic
100C8—42no—
101C8—82no—
102C22—21no—
103C22—21no—
104C22—41no—
105C22—41no—
106C22—81no—
107C22—81no—
108C22—81no—
109C22—81no—
110C22—161no—
111C22#34211yescharacteristic
112C22#35311yescharacteristic
113C22#35411yescharacteristic
114C4—21no—
115C4—21no—
116C4—41no—
117C4—41no—
118C4—41no—
119C4—41no—
120C4—162no—
121C4—161no—
122C2—21no—
123C2—21no—
124C2—81no—
125C2—81no—
126C2#139311yescharacteristic
127C2#140611yescharacteristic
128C2#140711yescharacteristic
subgroupG/H
center, socleC22#353
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×#97C4semidirect
10. #384C4semidirect
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
#19592