Finite Group Databasebeta

← #7489

Subgroups — #7489

Order 256.

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

normal subgroups only

This group has 151 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×#354C211yesmaximal, characteristic
2#1888C211yesmaximal, characteristic
3C2×#329C211yesmaximal, characteristic
4C2×#124C2211yescharacteristic
5#519C411yescharacteristic, complemented
6#418—21no—
7C2×#101C411yescharacteristic, complemented
8C2×#99—21no—
9C2×#97—21no—
10#354—21no—
11#354C412yescomplemented
12#339—21no—
13#329—21no—
14#329C412yescomplemented
15C22×Q8C2×C411yescharacteristic
16C22×D4D411yescharacteristic
17C2×M16—21no—
18C22×C8—41no—
19#124—41no—
20#124C2×C411yescharacteristic
21#124C2×C411yescharacteristic
22#124D411yescharacteristic
23#124Q811yescharacteristic
24C4×Q8—42no—
25#117—21no—
26#117—41no—
27C2×C22⋊C4—41no—
28C2×C22⋊C4—41no—
29C2×C42D411yescharacteristic
30#102—42no—
31#101—41no—
32#99—22no—
33#99—41no—
34#97—22no—
35#97—41no—
36C24—21no—
37C2×Q8—21no—
38C2×Q8—41no—
39C2×Q8—41no—
40C2×Q8C4211yescharacteristic
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—81no—
50C22×C4C22⋊C411yescharacteristic
51M16—41no—
52M16—41no—
53C2×C8—42no—
54C2×C8—44no—
55C2×C8—41no—
56C4⋊C4—22no—
57C4⋊C4—42no—
58C22⋊C4—41no—
59C22⋊C4—41no—
60C22⋊C4—41no—
61C22⋊C4—41no—
62C22⋊C4—41no—
63C22⋊C4—42no—
64C22⋊C4—81no—
65C42—21no—
66C42—41no—
67C42—42no—
68C42C22⋊C411yescharacteristic
69C42C4⋊C411yescharacteristic
70C23—21no—
71C23—21no—
72C23—41no—
73C23—41no—
74C23—82no—
75C23#9911yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—42no—
79D4—41no—
80D4—41no—
81D4—81no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—42no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—42no—
92C2×C4—82no—
93C2×C4—162no—
94C2×C4#9511yescharacteristic
95C2×C4#9911yescharacteristic
96C8—44no—
97C8—81no—
98C22—22no—
99C22—41no—
100C22—41no—
101C22—81no—
102C22—81no—
103C22—82no—
104C22—161no—
105C22#34211yescharacteristic
106C22AΣL(1,𝔽16)11yescharacteristic
107C22#35211yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—41no—
113C4—41no—
114C4—82no—
115C4—162no—
116C2—21no—
117C2—21no—
118C2—81no—
119C2—81no—
120C2#139011yescharacteristic
121C2#140412yes—
subgroupG/H
center, socleC22AΣL(1,𝔽16)
commutatorC2×Q8C42
Frattini, cosocleC2×#124C22
prime radicalC22×D4D4
LCSG ▷ C2×Q8 ▷ 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
5. #519C4semidirect
7. C2×#101C4semidirect
11. #354C4semidirect
14. #329C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3542
#18882
C2×#3292