Finite Group Databasebeta

← #9427

Subgroups — #9427

Order 256.

211 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 59 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#3165C211yesmaximal, characteristic
2#2381C211yesmaximal, characteristic
3#1556C211yesmaximal, characteristic
4#1515C211yesmaximal, characteristic
5#1515C211yesmaximal, characteristic
6#1288C211yesmaximal, characteristic
7#1288C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#501—22no—
11#500C412yescomplemented
12#498C412yescomplemented
13#432C2211yescharacteristic
14#432C2211yescharacteristic
15#428—22no—
16C4×M16C2211yescharacteristic
17#391—22no—
18#387—22no—
19#387—22no—
20#383C2211yescharacteristic
21#383C2211yescharacteristic
22#332—22no—
23#328—22no—
24#328—22no—
25C22×Q8—21no—
26C2×M16—21no—
27C4⋊Q8—21no—
28C4⋊Q8—21no—
29C4⋊Q8C2×C411yescharacteristic
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C412yes—
32C4⋊Q8D412yes—
33#125—21no—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38#117—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4D412yes—
42C2×C42—22no—
43C2×C42C2311yescharacteristic
44#107—24no—
45#106—24no—
46C4⋊C8—21no—
47C4⋊C8—21no—
48C22⋊C8—21no—
49C22⋊C8—21no—
50#97—21no—
51#97—21no—
52#97—21no—
53C4×C8C2×C412yes—
54#95—24no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95D412yes—
60#95D412yes—
61C2×Q8—22no—
62C2×Q8—22no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—81no—
66C2×Q8—81no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73M16—24no—
74C2×C8—22no—
75C2×C8—41no—
76C2×C8—41no—
77C4⋊C4—22no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—82no—
87C4⋊C4C22⋊C414yes—
88C22⋊C4—41no—
89C42—41no—
90C42—82no—
91C42C22×C411yescharacteristic
92C42C2×D411yescharacteristic
93C42#4111yescharacteristic
94C42#4111yescharacteristic
95C23#12011yescharacteristic
96Q8—82no—
97Q8—82no—
98Q8—81no—
99Q8—81no—
100C2×C4—22no—
101C2×C4—24no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4C2×C22⋊C411yescharacteristic
115C2×C4C4×D411yescharacteristic
116C2×C4C4×D411yescharacteristic
117C2×C4#12111yescharacteristic
118C2×C4#12111yescharacteristic
119C2×C4#12311yescharacteristic
120C8—24no—
121C8—81no—
122C8—81no—
123C22—21no—
124C22—21no—
125C22#38611yescharacteristic
126C22#45612yes—
127C4—21no—
128C4—21no—
129C4—21no—
130C4—21no—
131C4—21no—
132C4—21no—
133C4—44no—
134C4—82no—
135C4—82no—
136C4—81no—
137C4—81no—
138C2—22no—
139C2#188611yescharacteristic
140C2#190411yescharacteristic
141C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#120
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
11. #500C4semidirect
12. #498C4semidirect

Maximal subgroups

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

subgroupindex
#31652
#23812
#15562
#15152
#12882