Finite Group Databasebeta

← #9528

Subgroups — #9528

Order 256.

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

normal subgroups only

This group has 198 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#3113C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic
3#1559C211yesmaximal, characteristic
4#1487C211yesmaximal, characteristic
5#1467C211yesmaximal, characteristic
6#1307C211yesmaximal, characteristic
7#1287C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#483C412yes—
11#475—21no—
12#462—21no—
13#451C412yes—
14#446—22no—
15#446—21no—
16C8×Q8—21no—
17#441—22no—
18#433—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24#333—21no—
25#332—21no—
26Q8⋊C8—21no—
27Q8⋊C8—21no—
28#324—22no—
29C22×Q8C2×C411yescharacteristic
30C2×Q16—22no—
31C2×M16—21no—
32C4⋊Q8D411yescharacteristic
33C4⋊Q8D411yescharacteristic
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8C2×C412yes—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8D412yes—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4C2×C411yescharacteristic
46C2×C42—21no—
47C2×C42C2311yescharacteristic
48#106—22no—
49C4⋊C8—22no—
50C4⋊C8—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57#103—22no—
58#103—22no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C22⋊C8C2×C412yes—
62C22⋊C8D412yes—
63#97—22no—
64#97—21no—
65#97—21no—
66#97—21no—
67C4×C8—22no—
68C4×C8D411yescharacteristic
69C4×C8D411yescharacteristic
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—42no—
74C2×Q8C22⋊C412yes—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4C22×C411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4#4111yescharacteristic
80Q16—44no—
81M16—81no—
82C2×C8—22no—
83C2×C8—21no—
84C2×C8—22no—
85C2×C8—21no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—22no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—22no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4C22⋊C412yes—
98C22⋊C4—22no—
99C42—22no—
100C42—41no—
101C42—41no—
102C42C2×D411yescharacteristic
103C42C2×D411yescharacteristic
104C42C2×D411yescharacteristic
105C42#4111yescharacteristic
106C23C4×D411yescharacteristic
107Q8—22no—
108Q8—22no—
109Q8—42no—
110Q8—41no—
111Q8—41no—
112Q8—41no—
113Q8—41no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—81no—
125C2×C4C2×C22⋊C411yescharacteristic
126C2×C4C4×D411yescharacteristic
127C2×C4#12011yescharacteristic
128C2×C4#12111yescharacteristic
129C2×C4#12111yescharacteristic
130C2×C4#12311yescharacteristic
131C8—41no—
132C8—42no—
133C8—41no—
134C8—42no—
135C8—81no—
136C8—81no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22#38611yescharacteristic
141C4—21no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—42no—
149C4—81no—
150C4#47111yescharacteristic
151C4#47311yescharacteristic
152C2—41no—
153C2#189111yescharacteristic
154C2#190511yescharacteristic
155C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23C4×D4
LCSG ▷ C22×C4 ▷ 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
#31132
#29852
#15592
#14872
#14672
#13072
#12872