Finite Group Databasebeta

← #11669

Subgroups — #11669

Order 256.

180 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 53 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#2956C211yesmaximal, characteristic
2#2165C211yesmaximal, characteristic, complemented
3#1455C211yesmaximal, characteristic, complemented
4#1336C212yesmaximal, complemented
5#1335C212yesmaximal, complemented
6#575C411yescharacteristic
7#568—21no—
8#444—21no—
9#444C412yes—
10#443—21no—
11#443C2212yes—
12#430C411yescharacteristic
13C2×#104—21no—
14C4×M16—21no—
15C4×M16C2211yescharacteristic
16#364C2212yescomplemented
17#349—21no—
18#347—21no—
19#322—21no—
20#322C2211yescharacteristic, complemented
21C82C2211yescharacteristic, complemented
22C2×#41—21no—
23#137—22no—
24AGL(1,ℤ/8)—22no—
25#135—21no—
26#135C2×C412yes—
27#131—22no—
28#131—22no—
29#131—42no—
30C2×M16—22no—
31C2×M16—21no—
32C2×M16—42no—
33C2×M16C2×C411yescharacteristic
34C2×M16D411yescharacteristic
35C2×C42D411yescharacteristic
36M32—22no—
37#108—21no—
38#108C2×C412yes—
39#104—22no—
40#104—22no—
41#97—22no—
42#97C2×C411yescharacteristic
43#97D411yescharacteristic
44C4×C8—21no—
45C4×C8—21no—
46C4×C8D411yescharacteristic
47C4×C8C2311yescharacteristic
48#41—22no—
49#41—22no—
50#41—41no—
51#41—41no—
52C2×Q8—21no—
53C2×D4—21no—
54C2×D4—41no—
55C22×C4—21no—
56C22×C4—41no—
57C22×C4C22⋊C411yescharacteristic
58Q16—21no—
59Q16M1612yescomplemented
60SD16—22no—
61SD16—22no—
62D8—21no—
63D8M1612yescomplemented
64M16—22no—
65M16—22no—
66M16—22no—
67M16—44no—
68M16—42no—
69M16—41no—
70M16—42no—
71C2×C8—22no—
72C2×C8—22no—
73C2×C8—42no—
74C2×C8—41no—
75C2×C8—42no—
76C2×C8C22⋊C411yescharacteristic
77C2×C8C22⋊C411yescharacteristic
78C2×C8C22×C411yescharacteristic
79C2×C8C2×D411yescharacteristic
80C42—21no—
81C42C22⋊C411yescharacteristic
82C42C2×D411yescharacteristic
83C16—42no—
84C23—41no—
85C23#10411yescharacteristic
86Q8—22no—
87Q8—41no—
88D4—22no—
89D4—41no—
90D4—42no—
91D4—42no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4#10411yescharacteristic
101C2×C4C2×C22⋊C411yescharacteristic
102C8—24no—
103C8—21no—
104C8—21no—
105C8—42no—
106C8—42no—
107C8C2×M1612yes—
108C8AGL(1,ℤ/8)12yes—
109C22—21no—
110C22—41no—
111C22—42no—
112C22—161no—
113C22C2×#10411yescharacteristic
114C4—22no—
115C4—21no—
116C4—41no—
117C4—42no—
118C4#40711yescharacteristic
119C4#41811yescharacteristic
120C2—21no—
121C2—41no—
122C2—82no—
123C2#149011yescharacteristic
subgroupG/H
centerC4#418
commutatorC2×C8C22×C4
Frattini, cosocleC4×C8C23
socleC2#1490
prime radical#575C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C4×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2165C2semidirect
3. #1455C2semidirect
4. #1336C2semidirect
5. #1335C2semidirect
16. #364C22semidirect
20. #322C22semidirect
21. C82C22semidirect
59. Q16M16semidirect
63. D8M16semidirect

Maximal subgroups

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

subgroupindex
#29562
#21652
#14552
#13362
#13352