Finite Group Databasebeta

← #12260

Subgroups — #12260

Order 256.

171 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 37 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#2844C211yesmaximal, characteristic
2#2061C211yesmaximal, characteristic
3#1605C212yesmaximal
4#1354C212yesmaximal
5#1324C211yesmaximal, characteristic
6#544C2211yescharacteristic
7#451—22no—
8#428—21no—
9#400—21no—
10#400—22no—
11#395—22no—
12#395C2212yes—
13#382—21no—
14#382C2211yescharacteristic
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×Q8D412yes—
20C2×Q16—42no—
21C2×M16—21no—
22C4⋊Q8—41no—
23#125—41no—
24C22⋊Q8—22no—
25#117—21no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4D411yescharacteristic
31C2×C4⋊C4C2311yescharacteristic
32C2×C42D411yescharacteristic
33#107—41no—
34#106—41no—
35#103—42no—
36C22⋊C8—22no—
37#95—22no—
38#95—22no—
39#95—21no—
40#95—22no—
41#95—22no—
42#95D412yes—
43C2×Q8—22no—
44C2×Q8—42no—
45C2×Q8—82no—
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4C2×D411yescharacteristic
51C22×C4C2×D412yes—
52Q16—84no—
53M16—22no—
54M16—41no—
55C2×C8—41no—
56C2×C8—42no—
57C4⋊C4—22no—
58C4⋊C4—21no—
59C4⋊C4—21no—
60C4⋊C4—41no—
61C4⋊C4—41no—
62C4⋊C4—42no—
63C4⋊C4—81no—
64C4⋊C4—81no—
65C4⋊C4—82no—
66C4⋊C4—81no—
67C22⋊C4—41no—
68C42—21no—
69C42—41no—
70C42Q1612yescomplemented
71C23#12011yescharacteristic
72Q8—44no—
73Q8—82no—
74Q8—82no—
75C2×C4—21no—
76C2×C4—21no—
77C2×C4—22no—
78C2×C4—22no—
79C2×C4—41no—
80C2×C4—41no—
81C2×C4—42no—
82C2×C4—42no—
83C2×C4—42no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—42no—
92C2×C4C2×Q1611yescharacteristic
93C2×C4#13711yescharacteristic
94C8—42no—
95C8—82no—
96C22—21no—
97C22—21no—
98C22#45111yescharacteristic
99C22#45611yescharacteristic
100C22B(4,𝔽2)11yescharacteristic
101C4—21no—
102C4—21no—
103C4—42no—
104C4—42no—
105C4—82no—
106C4—82no—
107C4—81no—
108C4—81no—
109C4—81no—
110C2—21no—
111C2—21no—
112C2#160511yescharacteristic
113C2#219911yescharacteristic
114C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC23#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
70. C42Q16semidirect

Maximal subgroups

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

subgroupindex
#28442
#20612
#16052
#13542
#13242