Finite Group Databasebeta

← #7736

Subgroups — #7736

Order 256.

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

normal subgroups only

This group has 280 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, complemented
2#2041C212yesmaximal, complemented
3#2011C211yesmaximal, characteristic
4#1271C212yesmaximal, complemented
5#1270C211yesmaximal, characteristic, complemented
6C22×D8—21no—
7C2×#127C2211yescharacteristic
8C2×#123—21no—
9C2×#102—21no—
10C2×#102—22no—
11C2×#102—22no—
12#398—21no—
13#397—22no—
14#390C2211yescharacteristic
15#389C2212yes—
16#385—22no—
17#336C2211yescharacteristic
18#336C2212yes—
19C22×D4—21no—
20C22×D4—22no—
21C22×D4D411yescharacteristic
22C23×C4—21no—
23C2×D8—42no—
24C2×D8—42no—
25C2×D8—42no—
26C22×C8—21no—
27C22×C8—22no—
28#127—42no—
29#123—42no—
30#123—42no—
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#102—44no—
41#102—42no—
42#102—44no—
43#95—22no—
44#95—21no—
45#95—22no—
46C24—21no—
47C24—41no—
48C2×D4—24no—
49C2×D4—22no—
50C2×D4—21no—
51C2×D4—41no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—41no—
63C2×D4D812yes—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—42no—
68C22×C4—81no—
69C22×C4—81no—
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D412yes—
72D8—48no—
73C2×C8—22no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—41no—
82C4⋊C4—42no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C4⋊C4—81no—
86C22⋊C4—42no—
87C22⋊C4—44no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C42—41no—
91C23—22no—
92C23—42no—
93C23—41no—
94C23—42no—
95C23—41no—
96C23—42no—
97C23—81no—
98C23—82no—
99C23—81no—
100C23—82no—
101C23#12011yescharacteristic
102D4—42no—
103D4—42no—
104D4—42no—
105D4—44no—
106D4—44no—
107D4—81no—
108D4—81no—
109D4—81no—
110D4—82no—
111D4—82no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—84no—
125C2×C4C2×D811yescharacteristic
126C2×C4#13512yes—
127C2×C4AGL(1,ℤ/8)11yescharacteristic
128C2×C4AGL(1,ℤ/8)12yes—
129C8—42no—
130C8—82no—
131C22—42no—
132C22—44no—
133C22—82no—
134C22—81no—
135C22—81no—
136C22—82no—
137C22—84no—
138C22—161no—
139C22—162no—
140C22—161no—
141C22#44711yescharacteristic
142C22#44912yes—
143C22#45311yescharacteristic
144C22#45412yes—
145C22B(4,𝔽2)11yescharacteristic
146C4—21no—
147C4—21no—
148C4—22no—
149C4—22no—
150C4—82no—
151C4—81no—
152C4—82no—
153C2—82no—
154C2—162no—
155C2#159511yescharacteristic
156C2#159712yes—
157C2#162011yescharacteristic
158C2#162512yes—
159C2#165711yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radical#2011C2
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
1. #2724C2semidirect
2. #2041C2semidirect
4. #1271C2semidirect
5. #1270C2semidirect

Maximal subgroups

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

subgroupindex
#27242
#20412
#20112
#12712
#12702