Finite Group Databasebeta

← #7236

Subgroups — #7236

Order 256.

186 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 47 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#1869C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic
3#1799C211yesmaximal, characteristic
4C23×C8C411yescharacteristic, complemented
5C2×C4×D4C2211yescharacteristic
6#423—21no—
7C2×#100—21no—
8C2×#99—21no—
9C2×C22⋊C8—21no—
10C2×C22⋊C8C411yescharacteristic, complemented
11#380—21no—
12#343C412yescomplemented
13#342C412yes—
14#336—21no—
15C22×D4D411yescharacteristic
16C23×C4C2×C411yescharacteristic
17C23×C4D411yescharacteristic
18C2×M16—22no—
19C22×C8—21no—
20C22×C8—21no—
21C22×C8—42no—
22C22×C8—41no—
23C4×D4—41no—
24C4×D4—41no—
25C2×C4⋊C4—41no—
26C2×C4⋊C4C2×C411yescharacteristic
27C2×C22⋊C4—22no—
28C2×C22⋊C4C2×C411yescharacteristic
29C2×C22⋊C4Q811yescharacteristic
30C2×C42D411yescharacteristic
31C4⋊C8—41no—
32C4⋊C8—41no—
33#100—41no—
34#100—41no—
35#99—41no—
36#99—41no—
37C22⋊C8—42no—
38C22⋊C8—41no—
39#95—41no—
40C24C22⋊C411yescharacteristic
41C24C4⋊C411yescharacteristic
42C2×D4—41no—
43C2×D4—41no—
44C2×D4D811yescharacteristic
45C2×D4SD1611yescharacteristic
46C2×D4SD1611yescharacteristic
47C2×D4Q1611yescharacteristic
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—41no—
53C22×C4—41no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—42no—
57C22×C4C4211yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C4⋊C411yescharacteristic
61C22×C4C4⋊C411yescharacteristic
62M16—42no—
63M16—42no—
64C2×C8—22no—
65C2×C8—41no—
66C2×C8—42no—
67C2×C8—41no—
68C2×C8—42no—
69C2×C8—41no—
70C2×C8—42no—
71C2×C8—41no—
72C2×C8—41no—
73C4⋊C4—41no—
74C4⋊C4—81no—
75C4⋊C4—81no—
76C22⋊C4—41no—
77C22⋊C4—41no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C42—21no—
81C42—21no—
82C23—21no—
83C23—21no—
84C23—21no—
85C23—21no—
86C23—41no—
87C23—41no—
88C23—41no—
89C23—41no—
90C23#9511yescharacteristic
91C23#10211yescharacteristic
92C23#10311yescharacteristic
93C23#10611yescharacteristic
94C23#10711yescharacteristic
95D4—41no—
96D4—41no—
97D4—41no—
98D4—41no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—81no—
115C2×C4—82no—
116C2×C4—82no—
117C2×C4—82no—
118C2×C4#9911yescharacteristic
119C2×C4#9911yescharacteristic
120C2×C4#10211yescharacteristic
121C2×C4#10311yescharacteristic
122C8—42no—
123C8—81no—
124C8—82no—
125C22—21no—
126C22—21no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—42no—
136C22—41no—
137C22—41no—
138C22—41no—
139C22—41no—
140C22—81no—
141C22H(3,ℤ/4)11yescharacteristic
142C22#34011yescharacteristic
143C22#34211yescharacteristic
144C4—21no—
145C4—21no—
146C4—21no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—162no—
151C4AΣL(1,𝔽16)11yescharacteristic
152C4#35211yescharacteristic
153C2—21no—
154C2—21no—
155C2—41no—
156C2—41no—
157C2—41no—
158C2—41no—
159C2#128911yescharacteristic
160C2#139011yescharacteristic
161C2#139511yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC22×C4C42
Frattini, cosocleC2×C4×D4C22
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C23×C8C4semidirect
10. C2×C22⋊C8C4semidirect
12. #343C4semidirect

Maximal subgroups

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

subgroupindex
#18692
#18002
#17992