Finite Group Databasebeta

← #9434

Subgroups — #9434

Order 256.

200 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 63 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#2986C211yesmaximal, characteristic
3#1561C211yesmaximal, characteristic
4#1548C211yesmaximal, characteristic
5#1546C211yesmaximal, characteristic
6#1299C211yesmaximal, characteristic
7#1299C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#501C414yes—
11#500—21no—
12#498—21no—
13#446—21no—
14#446—21no—
15C8×Q8—21no—
16C8×Q8—21no—
17#433—22no—
18#431C2211yescharacteristic
19#431C2211yescharacteristic
20#428—22no—
21#423C2211yescharacteristic
22#423C2211yescharacteristic
23C4×M16C2211yescharacteristic
24#333—21no—
25#333—21no—
26#332—21no—
27#332—21no—
28#332—21no—
29#332—21no—
30C22×Q8—21no—
31C2×M16—21no—
32C2×M16—21no—
33C2×M16—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8C2×C412yes—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#125C2×C412yes—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45#117D412yes—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C42C2311yescharacteristic
49#107—24no—
50#106—24no—
51C4⋊C8—22no—
52C4⋊C8—22no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8D411yescharacteristic
58C4⋊C8D411yescharacteristic
59C4⋊C8D411yescharacteristic
60C4⋊C8D411yescharacteristic
61C22⋊C8—21no—
62C22⋊C8—21no—
63#97—22no—
64#97C2×C412yes—
65C4×C8—22no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—22no—
72C2×Q8—41no—
73C2×Q8—81no—
74C2×Q8—81no—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77C22×C4#4111yescharacteristic
78C22×C4#4111yescharacteristic
79M16—22no—
80M16—22no—
81M16—81no—
82M16—81no—
83C2×C8—22no—
84C2×C8—21no—
85C2×C8—21no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4C22⋊C412yes—
99C4⋊C4C22⋊C412yes—
100C22⋊C4—22no—
101C42—22no—
102C42C22×C411yescharacteristic
103C42C2×D411yescharacteristic
104C42C2×D411yescharacteristic
105C42C2×D411yescharacteristic
106C23#12311yescharacteristic
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—41no—
111Q8—82no—
112Q8—81no—
113Q8—81no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—42no—
123C2×C4—82no—
124C2×C4C2×C22⋊C411yescharacteristic
125C2×C4C4×D411yescharacteristic
126C2×C4C4×D411yescharacteristic
127C2×C4#12011yescharacteristic
128C2×C4#12111yescharacteristic
129C2×C4#12111yescharacteristic
130C8—24no—
131C8—41no—
132C8—41no—
133C8—41no—
134C8—41no—
135C8—81no—
136C8—81no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22#38611yescharacteristic
141C4—21no—
142C4—21no—
143C4—21no—
144C4—42no—
145C4—42no—
146C4—82no—
147C4#45611yescharacteristic
148C4#45611yescharacteristic
149C4#47111yescharacteristic
150C4#47111yescharacteristic
151C4#47211yescharacteristic
152C4#47311yescharacteristic
153C2—41no—
154C2#188611yescharacteristic
155C2#190711yescharacteristic
156C2#190811yescharacteristic
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
#29862
#15612
#15482
#15462
#12992