Finite Group Databasebeta

← #13220

Subgroups — #13220

Order 256.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#2486C211yesmaximal, characteristic
2C2×#352C214yesmaximal
3#1800C211yesmaximal, characteristic
4C4×#99C211yesmaximal, characteristic
5C2×#123C2212yes—
6C2×C4×D4C2211yescharacteristic
7C22×C4⋊C4C411yescharacteristic, complemented
8#423—21no—
9C2×#100C2212yes—
10C2×#99C2212yes—
11#398—21no—
12#397—21no—
13#389—21no—
14#381C412yescomplemented
15#380—21no—
16#380C411yescharacteristic, complemented
17C4×C22⋊C4—21no—
18#352—28no—
19#343—21no—
20#342—21no—
21C22×D4C2311yescharacteristic
22C23×C4C2×C412yes—
23C23×C4C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C2×M16—22no—
26#123—24no—
27#123—24no—
28C4×D4—41no—
29C4×D4—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—41no—
35C2×C4⋊C4—41no—
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4C2×C412yes—
40C2×C22⋊C4C2×C411yescharacteristic
41C2×C22⋊C4D411yescharacteristic
42C2×C42—41no—
43C2×C42D411yescharacteristic
44C4⋊C8—41no—
45C4⋊C8—41no—
46#100—24no—
47#99—22no—
48#99D414yes—
49#95—22no—
50#95—21no—
51#95—22no—
52#95—41no—
53C24C22×C411yescharacteristic
54C24C2×D411yescharacteristic
55C2×D4—41no—
56C2×D4—41no—
57C2×D4C2×D412yes—
58C2×D4#4112yes—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—24no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C2×D411yescharacteristic
76M16—42no—
77M16—42no—
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—42no—
81C4⋊C4—42no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C22⋊C4—24no—
87C22⋊C4—24no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C22⋊C4—44no—
92C42—21no—
93C42—21no—
94C42—82no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23C2×C22⋊C411yescharacteristic
104C23C4×D412yes—
105C23#12112yes—
106D4—42no—
107D4—42no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—44no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—81no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—84no—
131C2×C4#9911yescharacteristic
132C2×C4#9911yescharacteristic
133C2×C4#12411yescharacteristic
134C2×C4#12711yescharacteristic
135C8—82no—
136C22—21no—
137C22—21no—
138C22—22no—
139C22—22no—
140C22—42no—
141C22—42no—
142C22—42no—
143C22—41no—
144C22—41no—
145C22—41no—
146C22—41no—
147C22—81no—
148C22#39011yescharacteristic
149C22C2×#9911yescharacteristic
150C22#41011yescharacteristic
151C4—21no—
152C4—21no—
153C4—21no—
154C4—41no—
155C4—41no—
156C4—41no—
157C4—82no—
158C4—84no—
159C4—82no—
160C4#35212yescomplemented
161C2—21no—
162C2—21no—
163C2—42no—
164C2—42no—
165C2#196411yescharacteristic
166C2C2×#35211yescharacteristic
167C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C22×C4⋊C4C4semidirect
14. #381C4semidirect
16. #380C4semidirect
160. C4#352semidirect

Maximal subgroups

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

subgroupindex
#24862
C2×#3522
#18002
C4×#992