Finite Group Databasebeta

← #11837

Subgroups — #11837

Order 256.

162 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 45 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#3184C211yesmaximal, characteristic
2#2165C211yesmaximal, characteristic, complemented
3#1841C211yesmaximal, characteristic, complemented
4#1341C212yesmaximal, complemented
5#1339C212yesmaximal, complemented
6#532—21no—
7#498C411yescharacteristic, complemented
8#493C411yescharacteristic, complemented
9#486—21no—
10#469—21no—
11#469C2212yes—
12#442—21no—
13#442C412yescomplemented
14C4×M16C2211yescharacteristic
15#364—21no—
16#349C2212yes—
17#347—21no—
18#344—21no—
19#339—21no—
20#338C2211yescharacteristic
21H(3,ℤ/4)C2211yescharacteristic
22C2×#41—41no—
23C2×D8—21no—
24C2×D8C2×C412yes—
25C2×M16—21no—
26C2×M16D411yescharacteristic
27C2×M16C2311yescharacteristic
28C4⋊Q8—21no—
29#127—21no—
30#124—41no—
31#121—22no—
32#121—41no—
33C4×D4—22no—
34C4×D4—41no—
35#117—21no—
36C2×C42D411yescharacteristic
37M32—22no—
38#106—21no—
39#106C2×C412yes—
40#102—22no—
41#102—22no—
42#97C2×C411yescharacteristic
43#97D411yescharacteristic
44C4×C8C2×C411yescharacteristic
45C4×C8D411yescharacteristic
46#41—84no—
47C2×Q8—41no—
48C2×D4—22no—
49C2×D4—41no—
50C2×D4—41no—
51C2×D4—42no—
52C22×C4—21no—
53C22×C4—42no—
54C22×C4C2×D411yescharacteristic
55D8—42no—
56D8—42no—
57M16—41no—
58M16—41no—
59M16—41no—
60M16—41no—
61C2×C8—41no—
62C2×C8C22⋊C411yescharacteristic
63C2×C8C22⋊C411yescharacteristic
64C2×C8C22×C411yescharacteristic
65C2×C8C2×D411yescharacteristic
66C4⋊C4—22no—
67C4⋊C4—41no—
68C4⋊C4—41no—
69C22⋊C4—42no—
70C22⋊C4—42no—
71C22⋊C4—41no—
72C42—21no—
73C42—41no—
74C42C22⋊C411yescharacteristic
75C42C22⋊C411yescharacteristic
76C16—42no—
77C23—42no—
78C23AGL(1,ℤ/8)11yescharacteristic
79Q8—82no—
80D4—42no—
81D4—42no—
82D4—82no—
83D4—82no—
84D4—84no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—42no—
90C2×C4—42no—
91C2×C4—84no—
92C2×C4—81no—
93C2×C4C2×C22⋊C411yescharacteristic
94C2×C4AGL(1,ℤ/8)11yescharacteristic
95C8—21no—
96C8—21no—
97C8—21no—
98C8—21no—
99C8—81no—
100C22—21no—
101C22—41no—
102C22—82no—
103C22—82no—
104C22—82no—
105C22#41811yescharacteristic
106C4—22no—
107C4—21no—
108C4—41no—
109C4—82no—
110C4—161no—
111C4#41011yescharacteristic
112C4#42111yescharacteristic
113C2—21no—
114C2—41no—
115C2—162no—
116C2#151411yescharacteristic
subgroupG/H
center, socleC2#1514
commutatorC2×C8C22×C4
Frattini, cosocleC2×M16C23
prime radical#3184C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2165C2semidirect
3. #1841C2semidirect
4. #1341C2semidirect
5. #1339C2semidirect
7. #498C4semidirect
8. #493C4semidirect
13. #442C4semidirect

Maximal subgroups

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

subgroupindex
#31842
#21652
#18412
#13412
#13392