Finite Group Databasebeta

← #13428

Subgroups — #13428

Order 256.

222 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 45 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#2734C211yesmaximal, characteristic
2C2×#356C212yesmaximal
3#1886C212yesmaximal
4#1857C211yesmaximal, characteristic
5#1394C211yesmaximal, characteristic
6C2×C4⋊Q8C2212yes—
7#519C2211yescharacteristic
8#428—21no—
9#419—22no—
10C2×#101C2212yes—
11C2×#101C2211yescharacteristic
12#398—21no—
13#395—22no—
14#391C2211yescharacteristic
15#389—21no—
16#387—22no—
17#356—24no—
18#339—21no—
19#339—22no—
20C22×Q8D412yes—
21C22×Q8C2311yescharacteristic
22C2×M16—21no—
23C2×M16—22no—
24C4⋊Q8—22no—
25C4⋊Q8—22no—
26C4⋊Q8—22no—
27C4⋊Q8—22no—
28C4×Q8—41no—
29C4×Q8—41no—
30#117—22no—
31#117—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C42D412yes—
39C2×C42D411yescharacteristic
40#107—41no—
41#106—41no—
42#103—42no—
43#103—42no—
44#101—21no—
45#101—22no—
46#101—22no—
47#101D412yes—
48#95—21no—
49#95—22no—
50#95—22no—
51#95—22no—
52C2×Q8—22no—
53C2×Q8—22no—
54C2×Q8—41no—
55C2×Q8—41no—
56C2×Q8—42no—
57C2×Q8—42no—
58C2×Q8C2×D411yescharacteristic
59C2×Q8#4111yescharacteristic
60C2×Q8#4112yes—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4C2×D412yes—
66C22×C4C2×D411yescharacteristic
67M16—22no—
68M16—41no—
69M16—42no—
70M16—42no—
71C2×C8—41no—
72C2×C8—42no—
73C4⋊C4—24no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—21no—
77C4⋊C4—21no—
78C4⋊C4—41no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C4⋊C4—42no—
82C4⋊C4—42no—
83C4⋊C4—81no—
84C4⋊C4—81no—
85C22⋊C4—41no—
86C42—22no—
87C42—22no—
88C42—41no—
89C42—41no—
90C42—42no—
91C23#12011yescharacteristic
92Q8—41no—
93Q8—41no—
94Q8—42no—
95Q8—44no—
96Q8—82no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—24no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4#12112yes—
115C2×C4#12111yescharacteristic
116C2×C4#12312yes—
117C2×C4#12411yescharacteristic
118C8—42no—
119C8—82no—
120C22—21no—
121C22—21no—
122C22#39411yescharacteristic
123C22B(4,𝔽2)11yescharacteristic
124C22#45811yescharacteristic
125C4—21no—
126C4—22no—
127C4—22no—
128C4—21no—
129C4—44no—
130C4—42no—
131C4—81no—
132C4—81no—
133C4—81no—
134C4—82no—
135C2—21no—
136C2—21no—
137C2#204411yescharacteristic
138C2#220311yescharacteristic
139C2#220511yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23
prime radicalC23#120
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#27342
C2×#3562
#18862
#18572
#13942