Finite Group Databasebeta

← #7215

Subgroups — #7215

Order 256.

211 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 41 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#1821C211yesmaximal, characteristic
2#1463C211yesmaximal, characteristic
3#1436C211yesmaximal, characteristic
4C22×C4⋊C4C2211yescharacteristic
5C22×C42C411yescharacteristic
6#407—21no—
7#407—21no—
8C2×#95—21no—
9C2×#95—21no—
10C2×#95C411yescharacteristic
11#343—21no—
12#343C412yescomplemented
13#324—21no—
14#324C412yescomplemented
15C23×C4—21no—
16C23×C4—21no—
17C23×C4C2×C411yescharacteristic
18C23×C4D411yescharacteristic
19C23×C4D411yescharacteristic
20C2×M16—22no—
21C2×M16—41no—
22C2×C4⋊C4—41no—
23C2×C4⋊C4—41no—
24C2×C4⋊C4C2×C411yescharacteristic
25C2×C4⋊C4C2×C411yescharacteristic
26C2×C4⋊C4D411yescharacteristic
27C2×C4⋊C4Q811yescharacteristic
28C2×C42—41no—
29C2×C42—42no—
30C22⋊C8—22no—
31C22⋊C8—41no—
32C22⋊C8—41no—
33C22⋊C8—41no—
34#95—24no—
35#95—41no—
36#95—42no—
37C24C22⋊C411yescharacteristic
38C22×C4—22no—
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—22no—
48C22×C4—22no—
49C22×C4—41no—
50C22×C4—41no—
51C22×C4—41no—
52C22×C4—41no—
53C22×C4—41no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4C4211yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C4⋊C411yescharacteristic
68C22×C4C4⋊C411yescharacteristic
69C22×C4C4⋊C411yescharacteristic
70M16—42no—
71M16—42no—
72M16—81no—
73M16—81no—
74C2×C8—42no—
75C2×C8—42no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C42—42no—
81C42—42no—
82C23—21no—
83C23—21no—
84C23—21no—
85C23—21no—
86C23—21no—
87C23—21no—
88C23#9511yescharacteristic
89C23#9911yescharacteristic
90C23#9911yescharacteristic
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—24no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—81no—
122C2×C4—81no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4—81no—
126C2×C4#10412yes—
127C8—82no—
128C8—82no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22H(3,ℤ/4)11yescharacteristic
144C22#33911yescharacteristic
145C22#34211yescharacteristic
146C22AΣL(1,𝔽16)11yescharacteristic
147C22#35211yescharacteristic
148C22#35511yescharacteristic
149C22#35611yescharacteristic
150C4—22no—
151C4—44no—
152C4—41no—
153C4—41no—
154C4—41no—
155C4—41no—
156C4—41no—
157C4—82no—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—21no—
162C2—21no—
163C2—21no—
164C2#128411yescharacteristic
165C2#139011yescharacteristic
166C2#139411yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC22×C4C42
Frattini, cosocleC22×C4⋊C4C22
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #343C4semidirect
14. #324C4semidirect

Maximal subgroups

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

subgroupindex
#18212
#14632
#14362