Finite Group Databasebeta

← #9102

Subgroups — #9102

Order 256.

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

normal subgroups only

This group has 273 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
1#2982C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#1478C212yesmaximal
4#1461C211yesmaximal, characteristic, complemented
5#1274C212yesmaximal, complemented
6C2×#124C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8C2×C4×D4C2211yescharacteristic
9C8×Q8—24no—
10#431—22no—
11#431C2212yes—
12#431C2212yes—
13#391C411yescharacteristic
14#390C411yescharacteristic
15#388—22no—
16#385—22no—
17#384C411yescharacteristic
18#376C2211yescharacteristic
19D4⋊C8—24no—
20#324—22no—
21C22×Q8C2×C411yescharacteristic
22C22×D4C2×C411yescharacteristic
23C23×C4—21no—
24#124—22no—
25#124C814yescomplemented
26C4×Q8—22no—
27C4×Q8—22no—
28C4×D4—22no—
29C4×D4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4D412yes—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—22no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42D412yes—
40C2×C42C2311yescharacteristic
41C4⋊C8—24no—
42C4⋊C8—22no—
43C4⋊C8—22no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46C4×C8—24no—
47C4×C8—22no—
48C4×C8D414yes—
49#95—22no—
50#95—21no—
51#95—21no—
52C24—21no—
53C2×Q8—22no—
54C2×Q8—22no—
55C2×Q8C2×C812yes—
56C2×Q8M1612yes—
57C2×D4—22no—
58C2×D4—21no—
59C2×D4—22no—
60C2×D4C2×C812yes—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—42no—
65C22×C4—82no—
66C22×C4C22⋊C412yes—
67C22×C4C22⋊C412yes—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C2×C8—24no—
72C2×C8—42no—
73C4⋊C4—22no—
74C4⋊C4—24no—
75C4⋊C4—22no—
76C4⋊C4—82no—
77C22⋊C4—28no—
78C22⋊C4—41no—
79C22⋊C4—41no—
80C22⋊C4—82no—
81C42—22no—
82C42—22no—
83C42—21no—
84C42—22no—
85C42C2×C812yes—
86C42C2×D412yes—
87C42#4112yes—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—41no—
92C23—41no—
93C23C2×C22⋊C411yescharacteristic
94Q8—24no—
95Q8—24no—
96D4—24no—
97D4—42no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—84no—
113C2×C4C22⋊C814yes—
114C2×C4C4×D412yes—
115C2×C4#12011yescharacteristic
116C2×C4#12112yes—
117C2×C4#12311yescharacteristic
118C2×C4C22×C811yescharacteristic
119C2×C4C2×M1611yescharacteristic
120C2×C4#13112yes—
121C8—44no—
122C8—82no—
123C22—21no—
124C22—21no—
125C22—44no—
126C22—42no—
127C22—82no—
128C22—81no—
129C22—81no—
130C22#38611yescharacteristic
131C22C2×C22⋊C811yescharacteristic
132C22#40811yescharacteristic
133C4—24no—
134C4—22no—
135C4—21no—
136C4—21no—
137C4—42no—
138C4—42no—
139C4—81no—
140C4—81no—
141C4C8×D412yes—
142C4#43512yes—
143C2—21no—
144C2—21no—
145C2—82no—
146C2#186911yescharacteristic
147C2#191011yescharacteristic
148C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C2×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
1. #2982C2semidirect
4. #1461C2semidirect
5. #1274C2semidirect
25. #124C8semidirect

Maximal subgroups

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

subgroupindex
#29822
#23262
#14782
#14612
#12742