Finite Group Databasebeta

← #9023

Subgroups — #9023

Order 256.

308 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#400C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic
3#1824C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×#117—21no—
7C22×C4⋊C4C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10#400—22no—
11#400C414yescomplemented
12#389—24no—
13#380—21no—
14#380C2212yes—
15C2×#95—21no—
16C2×#95—21no—
17C2×#95C2211yescharacteristic
18#340—24no—
19#324—22no—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26#117—42no—
27#117—42no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4C2×C412yes—
36C2×C4⋊C4C2×C412yes—
37C2×C4⋊C4D412yes—
38C2×C4⋊C4D414yes—
39C2×C22⋊C4—22no—
40C2×C42—22no—
41C22⋊C8—22no—
42C22⋊C8—22no—
43#95—22no—
44#95—24no—
45#95—24no—
46#95—21no—
47#95—22no—
48#95C2×C412yes—
49C24C2×D411yescharacteristic
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—24no—
59C22×C4—42no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—44no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4C22⋊C412yes—
69C22×C4C22⋊C412yes—
70C22×C4SD1612yes—
71C22×C4Q1612yes—
72C22×C4C22×C411yescharacteristic
73C22×C4C2×D412yes—
74C22×C4C2×D411yescharacteristic
75C22×C4#4112yes—
76C2×C8—44no—
77C2×C8—42no—
78C4⋊C4—24no—
79C4⋊C4—24no—
80C4⋊C4—24no—
81C4⋊C4—44no—
82C4⋊C4—44no—
83C4⋊C4—42no—
84C4⋊C4—82no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C42—44no—
88C23—22no—
89C23—22no—
90C23C2×C22⋊C411yescharacteristic
91C23#12011yescharacteristic
92C23#12311yescharacteristic
93C23C2×SD1611yescharacteristic
94C23C2×Q1611yescharacteristic
95C23AGL(1,ℤ/8)11yescharacteristic
96C23#13711yescharacteristic
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—24no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—44no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—44no—
116C2×C4—44no—
117C2×C4—44no—
118C2×C4—81no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4—82no—
122C2×C4—82no—
123C2×C4#10314yes—
124C2×C4C4×D412yes—
125C2×C4#12112yes—
126C8—82no—
127C22—22no—
128C22—22no—
129C22—22no—
130C22—22no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22#38611yescharacteristic
136C22C2×#10311yescharacteristic
137C22#41711yescharacteristic
138C22#45011yescharacteristic
139C22#45111yescharacteristic
140C22#45412yes—
141C22#45812yes—
142C22#48311yescharacteristic
143C22#48411yescharacteristic
144C4—24no—
145C4—22no—
146C4—22no—
147C4—42no—
148C4—82no—
149C4—84no—
150C4—82no—
151C2—21no—
152C2—21no—
153C2—21no—
154C2—21no—
155C2#161612yes—
156C2#161712yes—
157C2#187211yescharacteristic
158C2#188811yescharacteristic
159C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #400C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4002
#23162
#18242
C2×#3242
#12942