Finite Group Databasebeta

← #7819

Subgroups — #7819

Order 256.

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

normal subgroups only

This group has 217 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#2724C211yesmaximal, characteristic
2#2093C211yesmaximal, characteristic
3#2084C212yesmaximal, complemented
4#1271C212yesmaximal, complemented
5#1270C211yesmaximal, characteristic
6C2×#123—21no—
7C2×#107—21no—
8C2×#106—22no—
9C2×#102—21no—
10#398—21no—
11#397—22no—
12#390C2211yescharacteristic
13#389C2212yes—
14#385—22no—
15#384C2211yescharacteristic
16#340—22no—
17#336C2212yes—
18#336C2211yescharacteristic
19C22×D4D411yescharacteristic
20C23×C4—21no—
21C22×C8—22no—
22C22×C8—21no—
23#123—42no—
24#123—42no—
25C2×C4⋊C4—22no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4D412yes—
30C2×C4⋊C4D412yes—
31C2×C4⋊C4D411yescharacteristic
32C2×C22⋊C4—22no—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—22no—
35C2×C42C2311yescharacteristic
36#107—42no—
37#106—44no—
38#102—42no—
39#95—22no—
40#95—21no—
41#95—22no—
42C24—21no—
43C2×D4—21no—
44C2×D4—41no—
45C2×D4—41no—
46C2×D4SD1612yes—
47C22×C4—21no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—42no—
51C22×C4—41no—
52C22×C4—81no—
53C22×C4—81no—
54C22×C4C2×D411yescharacteristic
55C22×C4C2×D412yes—
56C2×C8—22no—
57C2×C8—24no—
58C2×C8—42no—
59C2×C8—41no—
60C2×C8—42no—
61C2×C8—41no—
62C4⋊C4—22no—
63C4⋊C4—22no—
64C4⋊C4—41no—
65C4⋊C4—42no—
66C4⋊C4—41no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C4⋊C4—42no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C4⋊C4—81no—
73C22⋊C4—42no—
74C22⋊C4—44no—
75C22⋊C4—42no—
76C22⋊C4—42no—
77C42—41no—
78C23—22no—
79C23—42no—
80C23—41no—
81C23—42no—
82C23—41no—
83C23#12011yescharacteristic
84D4—42no—
85D4—81no—
86C2×C4—21no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4—41no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—84no—
99C2×C4—82no—
100C2×C4—81no—
101C2×C4C2×SD1611yescharacteristic
102C2×C4#13512yes—
103C2×C4AGL(1,ℤ/8)12yes—
104C2×C4AGL(1,ℤ/8)11yescharacteristic
105C8—44no—
106C8—42no—
107C22—42no—
108C22—44no—
109C22—82no—
110C22—81no—
111C22—81no—
112C22#44912yes—
113C22#45011yescharacteristic
114C22#45311yescharacteristic
115C22#45612yes—
116C22#45811yescharacteristic
117C4—22no—
118C4—22no—
119C4—21no—
120C4—21no—
121C4—82no—
122C4—81no—
123C4—82no—
124C4—161no—
125C2—82no—
126C2#160912yes—
127C2#161311yescharacteristic
128C2#162011yescharacteristic
129C2#165412yes—
130C2#166011yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2084C2semidirect
4. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#27242
#20932
#20842
#12712
#12702