Finite Group Databasebeta

← #11992

Subgroups — #11992

Order 256.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#3151C211yesmaximal, characteristic
2#2165C212yesmaximal, complemented
3#1359C212yesmaximal, complemented
4#1357C212yesmaximal, complemented
5C2×AGL(1,ℤ/8)—22no—
6#532C411yescharacteristic
7C2×#127C411yescharacteristic
8#496—22no—
9#493C411yescharacteristic
10#493C2211yescharacteristic
11#492C411yescharacteristic
12#492C2211yescharacteristic
13C4×M16C2211yescharacteristic
14#364C2214yescomplemented
15#349—22no—
16#347—22no—
17C2×#41—21no—
18C22×D4—21no—
19C22×D4—41no—
20AGL(1,ℤ/8)—48no—
21C2×SD16—22no—
22C2×SD16—22no—
23C2×D8—22no—
24C2×D8—22no—
25C2×M16D412yes—
26C4⋊Q8C2×C411yescharacteristic
27#127—21no—
28#127—41no—
29#127C2×C411yescharacteristic
30#127C2×C411yescharacteristic
31#127C2×C411yescharacteristic
32#124—21no—
33#121—41no—
34C4×D4—41no—
35C2×C42C2×C411yescharacteristic
36M32—24no—
37#97D412yes—
38C4×C8C2×C411yescharacteristic
39C4×C8C2311yescharacteristic
40C24—41no—
41#41—44no—
42C2×Q8—21no—
43C2×D4—21no—
44C2×D4—21no—
45C2×D4—21no—
46C2×D4—44no—
47C2×D4—41no—
48C2×D4—41no—
49C2×D4—41no—
50C2×D4—42no—
51C2×D4—41no—
52C2×D4—41no—
53C2×D4—84no—
54C22×C4—21no—
55C22×C4—41no—
56C22×C4C22⋊C411yescharacteristic
57SD16—44no—
58SD16—44no—
59D8—44no—
60D8—44no—
61M16—24no—
62C2×C8C22⋊C412yes—
63C2×C8C2×D412yes—
64C4⋊C4—41no—
65C22⋊C4—42no—
66C42—21no—
67C42C22⋊C411yescharacteristic
68C42C22×C411yescharacteristic
69C16—44no—
70C23—41no—
71C23—41no—
72C23—41no—
73C23—82no—
74C23—84no—
75C23#10011yescharacteristic
76Q8—42no—
77D4—42no—
78D4—42no—
79D4—42no—
80D4—82no—
81D4—82no—
82D4—82no—
83D4—82no—
84D4—84no—
85D4—82no—
86C2×C4—21no—
87C2×C4—22no—
88C2×C4—21no—
89C2×C4—42no—
90C2×C4—41no—
91C2×C4—82no—
92C2×C4#10011yescharacteristic
93C2×C4C2×C22⋊C411yescharacteristic
94C8—22no—
95C8AGL(1,ℤ/8)14yes—
96C22—21no—
97C22—41no—
98C22—82no—
99C22—82no—
100C22—82no—
101C22—81no—
102C22—81no—
103C22—81no—
104C22—164no—
105C22C2×#10011yescharacteristic
106C4—22no—
107C4—21no—
108C4—41no—
109C4—81no—
110C4#41812yes—
111C2—21no—
112C2—41no—
113C2—161no—
114C2—161no—
115C2—161no—
116C2#155511yescharacteristic
subgroupG/H
center, socleC2#1555
commutatorC42C22×C4
Frattini, cosocleC4×C8C23
prime radical#3151C2
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ 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. #1359C2semidirect
4. #1357C2semidirect
14. #364C22semidirect

Maximal subgroups

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

subgroupindex
#31512
#21652
#13592
#13572