Finite Group Databasebeta

← #9103

Subgroups — #9103

Order 256.

269 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 71 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#2975C211yesmaximal, characteristic
2#2326C211yesmaximal, characteristic
3#1479C212yesmaximal, complemented
4#1460C211yesmaximal, characteristic
5#1274C212yesmaximal, complemented
6C2×#124C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8C2×C4×D4C2211yescharacteristic
9#436—24no—
10#431C2212yes—
11#431C2212yes—
12#407—22no—
13#391C411yescharacteristic
14#390C411yescharacteristic
15#388—22no—
16#385—22no—
17#384C411yescharacteristic
18#376C2211yescharacteristic
19Q8⋊C8—24no—
20#323—22no—
21C22×Q8C2×C411yescharacteristic
22C22×D4C2×C411yescharacteristic
23C23×C4D412yes—
24C2×M16—24no—
25#124—22no—
26#124—22no—
27C4×Q8—22no—
28C4×Q8—22no—
29C4×D4—22no—
30C4×D4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4D412yes—
38C2×C42—21no—
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42C2311yescharacteristic
42C4⋊C8—22no—
43C4⋊C8—22no—
44C22⋊C8—24no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47C4×C8—22no—
48C4×C8D414yes—
49#95—22no—
50#95—21no—
51#95—21no—
52C24C22⋊C412yes—
53C2×Q8—21no—
54C2×Q8—22no—
55C2×Q8—22no—
56C2×Q8M1612yescomplemented
57C2×D4—22no—
58C2×D4—21no—
59C2×D4—21no—
60C2×D4—22no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—82no—
68C22×C4C22⋊C412yes—
69C22×C4C22×C411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D411yescharacteristic
72M16—84no—
73C2×C8—24no—
74C2×C8—42no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—82no—
80C22⋊C4—28no—
81C22⋊C4—24no—
82C22⋊C4—82no—
83C42—21no—
84C42—22no—
85C42—42no—
86C42M1612yescomplemented
87C42C2×D412yes—
88C42#4112yes—
89C23—22no—
90C23—22no—
91C23—22no—
92C23—42no—
93C23—42no—
94C23C2×C22⋊C411yescharacteristic
95Q8—24no—
96Q8—42no—
97D4—42no—
98D4—42no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—44no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—82no—
116C2×C4C4×D412yes—
117C2×C4#12011yescharacteristic
118C2×C4#12112yes—
119C2×C4#12311yescharacteristic
120C2×C4C2×M1611yescharacteristic
121C2×C4C2×M1611yescharacteristic
122C2×C4#13112yes—
123C8—44no—
124C8—82no—
125C22—21no—
126C22—21no—
127C22—44no—
128C22—42no—
129C22—42no—
130C22—42no—
131C22—82no—
132C22#38611yescharacteristic
133C22#40711yescharacteristic
134C22#40811yescharacteristic
135C4—22no—
136C4—21no—
137C4—21no—
138C4—42no—
139C4—42no—
140C4—42no—
141C4—81no—
142C4—81no—
143C4#43512yes—
144C4#43612yes—
145C2—21no—
146C2—21no—
147C2—82no—
148C2#187011yescharacteristic
149C2#190911yescharacteristic
150C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1479C2semidirect
5. #1274C2semidirect
56. C2×Q8M16semidirect
86. C42M16semidirect

Maximal subgroups

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

subgroupindex
#29752
#23262
#14792
#14602
#12742