Finite Group Databasebeta

← #8531

Subgroups — #8531

Order 256.

364 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 75 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#3550C211yesmaximal, characteristic
2#1994C212yesmaximal, complemented
3#1962C211yesmaximal, characteristic
4#1561C212yesmaximal, complemented
5#1560C211yesmaximal, characteristic
6#563C414yescomplemented
7#559—21no—
8Q82—21no—
9#556—22no—
10D4×Q8—22no—
11#542—24no—
12#536—22no—
13#533C2212yes—
14#532C2211yescharacteristic
15#423C2212yes—
16#423C2211yescharacteristic
17C2×#104—22no—
18C43C2211yescharacteristic
19#339—24no—
20#333—21no—
21#332—22no—
22#332—22no—
23#331—21no—
24C2×#41D412yes—
25C22×Q8—22no—
26C2×M16—22no—
27C2×M16—21no—
28C4⋊Q8—24no—
29C4⋊Q8—24no—
30C4⋊Q8D411yescharacteristic
31C4⋊Q8D412yes—
32C4⋊Q8D412yes—
33#127D411yescharacteristic
34#126—28no—
35#125C2×C414yes—
36#124—24no—
37#124—24no—
38#124C2×C412yes—
39#123—24no—
40#123—24no—
41C22⋊Q8—24no—
42C22⋊Q8—22no—
43C22⋊Q8—22no—
44C22⋊Q8—28no—
45#121—22no—
46C4×Q8—24no—
47C4×Q8—22no—
48C4×Q8—22no—
49C4×D4—24no—
50C4×D4—22no—
51#117—22no—
52#117D414yes—
53C2×C42—22no—
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—22no—
57C4⋊C8—22no—
58C4⋊C8—22no—
59#104—44no—
60#41—42no—
61#41—42no—
62C2×Q8—24no—
63C2×Q8—22no—
64C2×Q8—22no—
65C2×Q8—82no—
66C2×Q8—82no—
67C2×Q8C22⋊C412yes—
68C2×D4—22no—
69C2×D4—22no—
70C2×D4C22⋊C412yes—
71C22×C4—24no—
72C22×C4—22no—
73C22×C4—41no—
74C22×C4C2×D412yes—
75C22×C4C2×D411yescharacteristic
76M16—42no—
77M16—44no—
78C2×C8—41no—
79C2×C8—42no—
80C4⋊C4—28no—
81C4⋊C4—24no—
82C4⋊C4—24no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4C22⋊C414yes—
87C4⋊C4C22⋊C414yes—
88C22⋊C4—24no—
89C22⋊C4—28no—
90C22⋊C4—24no—
91C22⋊C4—24no—
92C42—24no—
93C42—24no—
94C42—22no—
95C42—24no—
96C42—42no—
97C42—41no—
98C42C22×C411yescharacteristic
99C42C2×D412yes—
100C42C2×D411yescharacteristic
101C23—22no—
102C23#12011yescharacteristic
103Q8—42no—
104Q8—42no—
105Q8—42no—
106Q8—42no—
107Q8—42no—
108Q8—84no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—82no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—24no—
118C2×C4—22no—
119C2×C4—24no—
120C2×C4—42no—
121C2×C4—44no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—84no—
125C2×C4C2×C22⋊C412yes—
126C2×C4C2×C22⋊C411yescharacteristic
127C2×C4#12011yescharacteristic
128C2×C4#12012yes—
129C8—81no—
130C8—82no—
131C22—21no—
132C22—22no—
133C22—42no—
134C22—44no—
135C22#37911yescharacteristic
136C4—21no—
137C4—22no—
138C4—44no—
139C4—42no—
140C4—44no—
141C4—44no—
142C4#45311yescharacteristic
143C4#45412yes—
144C4#45511yescharacteristic
145C4#45612yes—
146C2—41no—
147C2—82no—
148C2#179211yescharacteristic
149C2#179312yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1994C2semidirect
4. #1561C2semidirect
6. #563C4semidirect

Maximal subgroups

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

subgroupindex
#35502
#19942
#19622
#15612
#15602