Finite Group Databasebeta

← #10370

Subgroups — #10370

Order 256.

227 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 57 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#2399C211yesmaximal, characteristic, complemented
2#1963C211yesmaximal, characteristic
3#1502C212yesmaximal
4#1303C212yesmaximal, complemented
5#1274C211yesmaximal, characteristic, complemented
6#532C2211yescharacteristic
7C2×C4×Q8—21no—
8#431C2212yes—
9#423C2211yescharacteristic
10C2×#104—21no—
11C2×#104—21no—
12#398—22no—
13#395—22no—
14C4×C4⋊C4—21no—
15C4×C4⋊C4C2212yes—
16#376C2211yescharacteristic
17#327—22no—
18#327—22no—
19C2×#41—21no—
20C22×Q8—21no—
21C2×M16—21no—
22C4⋊Q8Q811yescharacteristic, complemented
23#127Q811yescharacteristic, complemented
24#124—21no—
25#121—22no—
26C4×Q8—42no—
27C4×Q8—42no—
28C4×D4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C42—22no—
34C2×C42D411yescharacteristic
35C2×C42D411yescharacteristic
36C2×C42C2311yescharacteristic
37C4⋊C8—22no—
38C4⋊C8—22no—
39#104—42no—
40#104—42no—
41C22⋊C8—22no—
42C4×C8—22no—
43#95—21no—
44#95—21no—
45#95—22no—
46#95D412yes—
47#95D412yes—
48#41—41no—
49#41—41no—
50#41—41no—
51#41—41no—
52C2×Q8—21no—
53C2×Q8—42no—
54C2×Q8—44no—
55C2×D4—21no—
56C2×D4—22no—
57C2×D4—41no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—41no—
63C22×C4C2×D412yes—
64C22×C4C2×D411yescharacteristic
65M16—41no—
66M16—41no—
67C2×C8—42no—
68C2×C8—41no—
69C4⋊C4—22no—
70C4⋊C4—44no—
71C4⋊C4—44no—
72C4⋊C4—44no—
73C4⋊C4—42no—
74C22⋊C4—42no—
75C42—21no—
76C42—21no—
77C42—42no—
78C42—42no—
79C42D812yes—
80C42SD1612yes—
81C42C2×Q811yescharacteristic
82C42#4111yescharacteristic
83C42#4112yes—
84C23—41no—
85C23#12011yescharacteristic
86Q8—48no—
87Q8—41no—
88Q8—41no—
89D4—41no—
90D4—41no—
91D4—82no—
92D4—81no—
93D4—81no—
94C2×C4—24no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4C22⋊Q812yes—
109C2×C4C22⋊Q811yescharacteristic
110C2×C4#12412yes—
111C2×C4#12411yescharacteristic
112C2×C4C2×D811yescharacteristic
113C2×C4C2×SD1611yescharacteristic
114C2×C4AGL(1,ℤ/8)11yescharacteristic
115C2×C4#13711yescharacteristic
116C8—82no—
117C8—81no—
118C22—22no—
119C22—81no—
120C22—82no—
121C22#39311yescharacteristic
122C22#44711yescharacteristic
123C22#44811yescharacteristic
124C4—22no—
125C4—22no—
126C4—42no—
127C4—44no—
128C4—42no—
129C4—82no—
130C4—81no—
131C4#47411yescharacteristic
132C4#47611yescharacteristic
133C4#48611yescharacteristic
134C4#48911yescharacteristic
135C2—21no—
136C2—21no—
137C2—161no—
138C2#202311yescharacteristic
139C2#203112yes—
subgroupG/H
centerC2×C4#124
commutator, Frattini, cosocleC2×C42C23
socleC22#393
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2399C2semidirect
4. #1303C2semidirect
5. #1274C2semidirect
22. C4⋊Q8Q8semidirect
23. #127Q8semidirect

Maximal subgroups

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

subgroupindex
#23992
#19632
#15022
#13032
#12742