Finite Group Databasebeta

← #12121

Subgroups — #12121

Order 256.

235 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 37 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#2607C211yesmaximal, characteristic, complemented
2#2022C211yesmaximal, characteristic
3#1601C212yesmaximal
4#1351C212yesmaximal, complemented
5#1317C211yesmaximal, characteristic, complemented
6#534C2211yescharacteristic
7C2×#123—21no—
8#448—22no—
9C2×#99—21no—
10#399—21no—
11#393C2212yes—
12#391—22no—
13#387—22no—
14#381C2211yescharacteristic
15C4×C22⋊C4—21no—
16#342C2211yescharacteristic
17#327—22no—
18#323C2212yes—
19C22×Q8D412yes—
20C22×D4—21no—
21C23×C4D411yescharacteristic
22C2×SD16—42no—
23#123—42no—
24#123—41no—
25#123—42no—
26#121—22no—
27#120—41no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—41no—
31C2×C22⋊C4—21no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—41no—
34C2×C22⋊C4D411yescharacteristic
35C2×C22⋊C4C2311yescharacteristic
36C2×C42—21no—
37C2×C42—21no—
38#103—42no—
39#99—42no—
40C22⋊C8—22no—
41#95—22no—
42#95—21no—
43#95—22no—
44#95—22no—
45#95D412yes—
46C24—21no—
47C24C2×D411yescharacteristic
48C2×Q8—44no—
49C2×Q8—42no—
50C2×D4—22no—
51C2×D4—41no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—41no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4SD1612yescomplemented
67C22×C4C2×D412yes—
68SD16—84no—
69C2×C8—42no—
70C4⋊C4—22no—
71C4⋊C4—44no—
72C4⋊C4—44no—
73C4⋊C4—81no—
74C22⋊C4—22no—
75C22⋊C4—41no—
76C22⋊C4—44no—
77C22⋊C4—42no—
78C22⋊C4—44no—
79C22⋊C4—41no—
80C22⋊C4—41no—
81C42—81no—
82C42—81no—
83C23—21no—
84C23—21no—
85C23—42no—
86C23—41no—
87C23—41no—
88C23—41no—
89C23—82no—
90C23#12011yescharacteristic
91C23C2×SD1611yescharacteristic
92C23#13711yescharacteristic
93Q8—44no—
94Q8—82no—
95D4—82no—
96D4—81no—
97D4—81no—
98D4—81no—
99D4—81no—
100C2×C4—24no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—81no—
119C2×C4—82no—
120C8—82no—
121C22—22no—
122C22—21no—
123C22—21no—
124C22—41no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22—81no—
130C22—82no—
131C22—161no—
132C22#44811yescharacteristic
133C22#45311yescharacteristic
134C22B(4,𝔽2)11yescharacteristic
135C4—42no—
136C4—42no—
137C4—44no—
138C4—82no—
139C4—81no—
140C4—82no—
141C4—81no—
142C2—21no—
143C2—21no—
144C2—41no—
145C2—41no—
146C2—161no—
147C2#159611yescharacteristic
148C2#219712yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2607C2semidirect
4. #1351C2semidirect
5. #1317C2semidirect
66. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#26072
#20222
#16012
#13512
#13172