Finite Group Databasebeta

← #9983

Subgroups — #9983

Order 256.

301 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 49 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#3094C212yesmaximal, complemented
2#2815C211yesmaximal, characteristic
3#1820C211yesmaximal, characteristic, complemented
4#1510C212yesmaximal, complemented
5#1306C211yesmaximal, characteristic, complemented
6C2×#121C2212yes—
7C2×#117C2211yescharacteristic
8#482—24no—
9#481—22no—
10#480—22no—
11#428—22no—
12#418—22no—
13#407C2212yes—
14#397—22no—
15#392—22no—
16#392—22no—
17#381C2211yescharacteristic
18#381C2211yescharacteristic
19#342—22no—
20#327—24no—
21C22×D4—22no—
22C22×D4—22no—
23C22×D4D412yes—
24C23×C4C2311yescharacteristic
25C2×M16—22no—
26#121—22no—
27#121—22no—
28#121—22no—
29#121—22no—
30#117—22no—
31#117—22no—
32C2×C4⋊C4D412yes—
33C2×C4⋊C4D412yescomplemented
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—24no—
38C2×C22⋊C4D412yes—
39C2×C42—21no—
40#107—24no—
41#106—24no—
42#102—24no—
43#102—24no—
44C22⋊C8—22no—
45C22⋊C8D414yescomplemented
46#95—22no—
47#95—22no—
48#95—22no—
49C24—22no—
50C24C2×D411yescharacteristic
51C2×D4—22no—
52C2×D4—22no—
53C2×D4—44no—
54C2×D4—42no—
55C2×D4—42no—
56C2×D4—84no—
57C2×D4—84no—
58C2×D4—82no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—42no—
63C22×C4—41no—
64C22×C4C2×D412yes—
65C22×C4C2×D411yescharacteristic
66C22×C4C2×D412yes—
67C22×C4#4111yescharacteristic
68M16—44no—
69C2×C8—24no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C22⋊C4—41no—
75C22⋊C4—41no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—84no—
81C42—42no—
82C23—22no—
83C23—42no—
84C23—42no—
85C23—42no—
86C23—44no—
87C23—42no—
88C23—44no—
89C23#12011yescharacteristic
90C23#12111yescharacteristic
91C23#12711yescharacteristic
92D4—44no—
93D4—82no—
94D4—84no—
95D4—84no—
96D4—88no—
97C2×C4—24no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—44no—
102C2×C4—44no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—82no—
107C2×C4—81no—
108C2×C4—82no—
109C2×C4#12012yes—
110C2×C4#12112yes—
111C8—44no—
112C22—22no—
113C22—22no—
114C22—22no—
115C22—21no—
116C22—21no—
117C22—82no—
118C22—84no—
119C22—84no—
120C22—82no—
121C22—82no—
122C22—82no—
123C22#39211yescharacteristic
124C22#45812yes—
125C4—22no—
126C4—22no—
127C4—82no—
128C4—82no—
129C4—82no—
130C4—161no—
131C2—22no—
132C2—41no—
133C2—41no—
134C2—162no—
135C2#200812yes—
136C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#2815C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3094C2semidirect
3. #1820C2semidirect
4. #1510C2semidirect
5. #1306C2semidirect
33. C2×C4⋊C4D4semidirect
45. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#30942
#28152
#18202
#15102
#13062