Finite Group Databasebeta

← #9420

Subgroups — #9420

Order 256.

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

normal subgroups only

This group has 192 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#2987C211yesmaximal, characteristic
3#1558C211yesmaximal, characteristic
4#1556C211yesmaximal, characteristic
5#1537C211yesmaximal, characteristic
6#1291C211yesmaximal, characteristic
7#1291C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#501—22no—
11#500C412yes—
12#498C412yes—
13#446—22no—
14#446—22no—
15#433—22no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#428—22no—
21C4×M16C2211yescharacteristic
22#333—22no—
23#332—22no—
24#330—22no—
25C22×Q8—21no—
26C2×M16—21no—
27C4⋊Q8—21no—
28C4⋊Q8—21no—
29C4⋊Q8—21no—
30C4⋊Q8C2×C412yes—
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33#125D412yes—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38#117D412yes—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C42C2311yescharacteristic
42#107—24no—
43#106—24no—
44C4⋊C8—24no—
45C4⋊C8—21no—
46C4⋊C8—21no—
47C4⋊C8D412yes—
48C4⋊C8D412yes—
49C22⋊C8—21no—
50C22⋊C8—21no—
51C22⋊C8—21no—
52C22⋊C8—21no—
53#97—22no—
54#97—21no—
55#97—21no—
56#97—21no—
57#97—21no—
58#97—21no—
59C4×C8—22no—
60C4×C8C2×C412yes—
61C2×Q8—21no—
62C2×Q8—21no—
63C2×Q8—22no—
64C2×Q8—41no—
65C2×Q8—82no—
66C22×C4—41no—
67C22×C4C2×D411yescharacteristic
68C22×C4#4111yescharacteristic
69C22×C4#4111yescharacteristic
70M16—24no—
71C2×C8—22no—
72C2×C8—22no—
73C2×C8—22no—
74C2×C8—41no—
75C2×C8—41no—
76C4⋊C4—21no—
77C4⋊C4—21no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—22no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4C22⋊C412yes—
85C4⋊C4C22⋊C412yes—
86C22⋊C4—22no—
87C42—22no—
88C42C22×C411yescharacteristic
89C42C2×D411yescharacteristic
90C42C2×D411yescharacteristic
91C42C2×D411yescharacteristic
92C23#12311yescharacteristic
93Q8—81no—
94Q8—81no—
95Q8—82no—
96Q8—82no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—82no—
104C2×C4C2×C22⋊C411yescharacteristic
105C2×C4C4×D411yescharacteristic
106C2×C4C4×D411yescharacteristic
107C2×C4#12011yescharacteristic
108C2×C4#12111yescharacteristic
109C2×C4#12111yescharacteristic
110C8—24no—
111C8—42no—
112C8—42no—
113C8—81no—
114C8—81no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22#38611yescharacteristic
119C4—21no—
120C4—21no—
121C4—21no—
122C4—21no—
123C4—21no—
124C4—42no—
125C4—42no—
126C4—82no—
127C4#45612yes—
128C2—41no—
129C2#188611yescharacteristic
130C2#190211yescharacteristic
131C2#190311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#31652
#29872
#15582
#15562
#15372
#12912