Finite Group Databasebeta

← #7751

Subgroups — #7751

Order 256.

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

normal subgroups only

This group has 246 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#2057C212yesmaximal, complemented
3#2013C211yesmaximal, characteristic
4#1271C212yesmaximal, complemented
5#1270C211yesmaximal, characteristic
6C22×SD16—21no—
7C2×C4⋊Q8C2211yescharacteristic
8C2×#123—21no—
9C2×#107—22no—
10C2×#103—21no—
11#398—21no—
12#397—22no—
13#390C2211yescharacteristic
14#389C2212yes—
15#385—22no—
16#340—22no—
17#336C2212yes—
18#336C2211yescharacteristic
19C22×Q8—21no—
20C22×D4D411yescharacteristic
21C23×C4—21no—
22C2×SD16—42no—
23C2×SD16—42no—
24C2×SD16—42no—
25C22×C8—21no—
26C22×C8—22no—
27C4⋊Q8—42no—
28#123—42no—
29#123—42no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4D412yes—
34C2×C4⋊C4D412yes—
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C42C2311yescharacteristic
40#107—44no—
41#103—42no—
42#95—22no—
43#95—21no—
44#95—22no—
45C24—21no—
46C2×Q8—22no—
47C2×Q8—41no—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8—41no—
51C2×Q8—41no—
52C2×D4—21no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4D812yes—
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—42no—
60C22×C4—41no—
61C22×C4—81no—
62C22×C4—81no—
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D412yes—
65SD16—48no—
66C2×C8—22no—
67C2×C8—24no—
68C2×C8—42no—
69C2×C8—41no—
70C2×C8—41no—
71C2×C8—42no—
72C4⋊C4—24no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—41no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C4⋊C4—42no—
80C4⋊C4—81no—
81C22⋊C4—42no—
82C22⋊C4—44no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C42—41no—
86C23—22no—
87C23—42no—
88C23—41no—
89C23—42no—
90C23—41no—
91C23#12011yescharacteristic
92Q8—42no—
93Q8—42no—
94Q8—81no—
95Q8—81no—
96D4—42no—
97D4—81no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—84no—
112C2×C4—81no—
113C2×C4—81no—
114C2×C4C2×D811yescharacteristic
115C2×C4#13512yes—
116C2×C4AGL(1,ℤ/8)11yescharacteristic
117C2×C4#13712yes—
118C8—44no—
119C8—42no—
120C22—42no—
121C22—44no—
122C22—82no—
123C22—81no—
124C22—81no—
125C22#44711yescharacteristic
126C22#45212yes—
127C22#45311yescharacteristic
128C22#45612yes—
129C22B(4,𝔽2)11yescharacteristic
130C4—21no—
131C4—21no—
132C4—22no—
133C4—22no—
134C4—82no—
135C4—82no—
136C4—81no—
137C4—82no—
138C2—82no—
139C2#159511yescharacteristic
140C2#160312yes—
141C2#163011yescharacteristic
142C2#164712yes—
143C2#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
2. #2057C2semidirect
4. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#27242
#20572
#20132
#12712
#12702