Finite Group Databasebeta

← #12864

Subgroups — #12864

Order 256.

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

normal subgroups only

This group has 171 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#2922C211yesmaximal, characteristic
2#1827C212yesmaximal
3#1395C212yesmaximal
4#1392C211yesmaximal, characteristic
5#1391C211yesmaximal, characteristic
6C4×#41C2211yescharacteristic
7#433—22no—
8#431—22no—
9#424C411yescharacteristic, complemented
10#424C411yescharacteristic, complemented
11#413C2212yes—
12#410C2212yes—
13#408C2211yescharacteristic
14#408C2211yescharacteristic
15C4×M16C411yescharacteristic, complemented
16C2×C4×C8C411yescharacteristic, complemented
17#339—22no—
18H(3,ℤ/4)—22no—
19C2×#41C2311yescharacteristic
20C2×M16—22no—
21C2×M16C2×C411yescharacteristic
22C2×M16C2×C411yescharacteristic
23C22×C8C2×C411yescharacteristic
24C22×C8C2×C411yescharacteristic
25C4×Q8—21no—
26C4×D4—21no—
27C4×D4—22no—
28#117—22no—
29#117C2×C411yescharacteristic
30#117D412yes—
31C2×C42C2×C411yescharacteristic
32C2×C42D412yes—
33C4⋊C8—22no—
34C4⋊C8—22no—
35#101—22no—
36#100—22no—
37#99—22no—
38#99—22no—
39C22⋊C8—22no—
40C22⋊C8—22no—
41#97—21no—
42C4×C8—21no—
43C4×C8—21no—
44C4×C8—21no—
45#41—41no—
46#41—41no—
47C2×Q8#4111yescharacteristic
48C2×D4#4111yescharacteristic
49C2×D4#4112yes—
50C22×C4—21no—
51C22×C4—22no—
52C22×C4C22×C411yescharacteristic
53C22×C4C2×D412yes—
54M16—41no—
55M16—41no—
56M16—42no—
57M16—42no—
58C2×C8—21no—
59C2×C8—21no—
60C2×C8—21no—
61C2×C8—21no—
62C2×C8—41no—
63C2×C8—41no—
64C2×C8—42no—
65C4⋊C4—21no—
66C4⋊C4—22no—
67C4⋊C4—42no—
68C22⋊C4—21no—
69C22⋊C4—22no—
70C22⋊C4—42no—
71C42—21no—
72C42—22no—
73C42—42no—
74C42C22⋊C411yescharacteristic
75C42C22⋊C412yes—
76C42C22⋊C411yescharacteristic
77C23#11711yescharacteristic
78C23#12312yes—
79Q8—41no—
80D4—42no—
81D4—41no—
82C2×C4—21no—
83C2×C4—22no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—22no—
88C2×C4—21no—
89C2×C4—42no—
90C2×C4—44no—
91C2×C4—82no—
92C2×C4C2×C22⋊C411yescharacteristic
93C2×C4#11711yescharacteristic
94C2×C4#12312yes—
95C8—41no—
96C8—41no—
97C8—41no—
98C8—41no—
99C8—82no—
100C22—22no—
101C22—21no—
102C22—41no—
103C22—42no—
104C22#38111yescharacteristic
105C4—22no—
106C4—22no—
107C4—21no—
108C4—42no—
109C4—41no—
110C4—162no—
111C4#41011yescharacteristic
112C4#41011yescharacteristic
113C2—21no—
114C2—42no—
115C2—41no—
116C2#182011yescharacteristic
subgroupG/H
centerC4#410
commutatorC22×C4C22×C4
Frattini, prime radical, cosocleC2×#41C23
socleC2#1820
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C42 ◁ C4×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #424C4semidirect
10. #424C4semidirect
15. C4×M16C4semidirect
16. C2×C4×C8C4semidirect

Maximal subgroups

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

subgroupindex
#29222
#18272
#13952
#13922
#13912