Finite Group Databasebeta

← #9410

Subgroups — #9410

Order 256.

322 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 65 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#2274C211yesmaximal, characteristic, complemented
3#1507C212yesmaximal
4#1463C211yesmaximal, characteristic, complemented
5#1297C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×#117—21no—
8C22×C4⋊C4C2211yescharacteristic
9#407C2212yes—
10#400C412yescomplemented
11#394C412yescomplemented
12#388—22no—
13#386—21no—
14#386—21no—
15C4×C4⋊C4—24no—
16C4×C22⋊C4—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—
29#117—28no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4D412yes—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C42—22no—
42C2×C42—22no—
43C2×C42—22no—
44C2×C42—21no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#95—22no—
48#95—22no—
49#95—22no—
50#95C2×C412yes—
51#95D414yes—
52C24—21no—
53C24C2×D411yescharacteristic
54C2×D4—22no—
55C2×D4—44no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4—81no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—24no—
62C22×C4—21no—
63C22×C4—24no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—81no—
68C22×C4—81no—
69C22×C4C22⋊C412yes—
70C22×C4C22⋊C412yes—
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D412yes—
73C22×C4C2×D411yescharacteristic
74C22×C4#4112yes—
75M16—82no—
76C2×C8—42no—
77C4⋊C4—28no—
78C4⋊C4—24no—
79C4⋊C4—24no—
80C4⋊C4—22no—
81C22⋊C4—28no—
82C22⋊C4—42no—
83C22⋊C4—44no—
84C22⋊C4—42no—
85C22⋊C4—82no—
86C42—28no—
87C42—82no—
88C42—81no—
89C42—82no—
90C23—22no—
91C23—22no—
92C23—41no—
93C23—42no—
94C23—42no—
95C23—41no—
96C23—42no—
97C23C2×C22⋊C411yescharacteristic
98C23#12011yescharacteristic
99C23#12311yescharacteristic
100D4—82no—
101D4—82no—
102D4—82no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—24no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—28no—
109C2×C4—22no—
110C2×C4—41no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—41no—
114C2×C4—44no—
115C2×C4—42no—
116C2×C4—44no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—81no—
124C2×C4#10414yes—
125C2×C4C4×D412yes—
126C2×C4#12112yes—
127C8—82no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—44no—
136C22—84no—
137C22—81no—
138C22—81no—
139C22—81no—
140C22#38611yescharacteristic
141C22C2×#10412yes—
142C22#45412yes—
143C4—24no—
144C4—22no—
145C4—22no—
146C4—48no—
147C4—42no—
148C4—82no—
149C4—81no—
150C2—22no—
151C2—22no—
152C2—41no—
153C2—161no—
154C2#188811yescharacteristic
155C2#189611yescharacteristic
156C2#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. #2274C2semidirect
4. #1463C2semidirect
5. #1297C2semidirect
10. #400C4semidirect
11. #394C4semidirect

Maximal subgroups

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

subgroupindex
#25912
#22742
#15072
#14632
#12972