Finite Group Databasebeta

← #8521

Subgroups — #8521

Order 256.

345 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 71 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×#541C211yesmaximal, characteristic
2#1822C211yesmaximal, characteristic
3#1820C211yesmaximal, characteristic
4#1511C214yesmaximal
5#541—28no—
6C2×C4⋊Q8C411yescharacteristic, complemented
7C2×#126—22no—
8C2×#125—22no—
9C2×#125C411yescharacteristic, complemented
10C2×C22⋊Q8C2212yes—
11C2×C4×Q8C412yescomplemented
12C2×#117C2211yescharacteristic
13#407C2212yes—
14#381C2212yes—
15#343—22no—
16#342—22no—
17#328—28no—
18C22×Q8C2×C412yes—
19C23×C4C2311yescharacteristic
20C2×M16—22no—
21C4⋊Q8—24no—
22#126—216no—
23#125—216no—
24#125—24no—
25C22⋊Q8—24no—
26C22⋊Q8D418yes—
27C4×Q8—24no—
28C4×Q8—24no—
29#117—24no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4D412yes—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4D412yes—
39C2×C42—22no—
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C411yescharacteristic
42C22⋊C8—24no—
43#95—24no—
44C24C2×D411yescharacteristic
45C2×Q8—22no—
46C2×Q8—44no—
47C2×Q8C22⋊C414yes—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—42no—
51C22×C4—42no—
52C22×C4C22⋊C412yes—
53C22×C4C22⋊C412yes—
54C22×C4C22×C411yescharacteristic
55C22×C4C2×D411yescharacteristic
56C22×C4C2×D414yes—
57M16—44no—
58C2×C8—42no—
59C4⋊C4—28no—
60C4⋊C4—28no—
61C4⋊C4—28no—
62C4⋊C4—22no—
63C4⋊C4—28no—
64C4⋊C4—24no—
65C4⋊C4C22⋊C414yes—
66C22⋊C4—28no—
67C22⋊C4—24no—
68C22⋊C4—44no—
69C42—28no—
70C42—22no—
71C42—22no—
72C23—22no—
73C23—42no—
74C23C2×C22⋊C411yescharacteristic
75C23#12012yes—
76Q8—44no—
77Q8—44no—
78C2×C4—22no—
79C2×C4—22no—
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—24no—
83C2×C4—24no—
84C2×C4—44no—
85C2×C4—44no—
86C2×C4—44no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—82no—
90C2×C4—84no—
91C2×C4C2×C22⋊C412yes—
92C2×C4#12012yes—
93C8—82no—
94C22—22no—
95C22—24no—
96C22—22no—
97C22—82no—
98C22#37911yescharacteristic
99C22#41012yes—
100C4—22no—
101C4—22no—
102C4—42no—
103C4—42no—
104C4—44no—
105C4—44no—
106C4—162no—
107C2—22no—
108C2—42no—
109C2#178211yescharacteristic
110C2#179612yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C4⋊Q8C4semidirect
9. C2×#125C4semidirect
11. C2×C4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
#18222
#18202
#15112