Finite Group Databasebeta

← #9414

Subgroups — #9414

Order 256.

276 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 61 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#2591C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic, complemented
3#1507C212yesmaximal
4#1463C211yesmaximal, characteristic, complemented
5#1297C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#407C2212yes—
9#400C412yescomplemented
10#394C412yescomplemented
11#388—22no—
12#387—24no—
13#386—21no—
14#386—21no—
15#381—21no—
16#380—21no—
17C4×C22⋊C4C2211yescharacteristic
18C4×C22⋊C4C2212yes—
19#327—24no—
20#324—22no—
21C22×D4—21no—
22C22×D4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C2×M16—22no—
27#121—22no—
28#121—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C412yes—
34C2×C4⋊C4D412yes—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C42—22no—
41C2×C42—21no—
42C22⋊C8—22no—
43C22⋊C8—22no—
44#95—22no—
45#95—24no—
46#95—22no—
47#95—22no—
48#95C2×C412yes—
49#95D414yes—
50C24—21no—
51C24C2×D411yescharacteristic
52C2×D4—22no—
53C2×D4—44no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—81no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—24no—
64C22×C4—42no—
65C22×C4—81no—
66C22×C4—81no—
67C22×C4C22⋊C412yes—
68C22×C4C22⋊C412yes—
69C22×C4C22×C411yescharacteristic
70C22×C4C2×D412yes—
71C22×C4C2×D411yescharacteristic
72C22×C4#4112yes—
73M16—82no—
74C2×C8—42no—
75C4⋊C4—24no—
76C4⋊C4—24no—
77C4⋊C4—22no—
78C4⋊C4—82no—
79C22⋊C4—44no—
80C22⋊C4—44no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C22⋊C4—82no—
84C42—81no—
85C42—82no—
86C23—22no—
87C23—22no—
88C23—41no—
89C23—42no—
90C23—42no—
91C23—41no—
92C23—42no—
93C23C2×C22⋊C411yescharacteristic
94C23#12011yescharacteristic
95C23#12311yescharacteristic
96D4—82no—
97D4—82no—
98D4—82no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
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—44no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—44no—
119C2×C4—81no—
120C2×C4C4×D412yes—
121C2×C4#12112yes—
122C8—82no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—44no—
131C22—84no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22#38611yescharacteristic
136C22#42112yes—
137C22#45412yes—
138C4—22no—
139C4—22no—
140C4—42no—
141C4—42no—
142C4—84no—
143C4—82no—
144C4—81no—
145C2—22no—
146C2—22no—
147C2—41no—
148C2—161no—
149C2#188811yescharacteristic
150C2#189711yescharacteristic
151C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2313C2semidirect
4. #1463C2semidirect
5. #1297C2semidirect
9. #400C4semidirect
10. #394C4semidirect

Maximal subgroups

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

subgroupindex
#25912
#23132
#15072
#14632
#12972