Finite Group Databasebeta

← #12130

Subgroups — #12130

Order 256.

221 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 37 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#2811C211yesmaximal, characteristic, complemented
2#2007C211yesmaximal, characteristic
3#1604C212yesmaximal
4#1354C212yesmaximal, complemented
5#1319C211yesmaximal, characteristic, complemented
6C2×#137—21no—
7#537C2211yescharacteristic
8#448—22no—
9C2×#104—21no—
10#395—22no—
11#395C2212yes—
12#391—21no—
13#391—21no—
14#391—22no—
15#391C2211yescharacteristic
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8—21no—
21C22×Q8D411yescharacteristic
22C22×Q8D412yes—
23#137—42no—
24#137—42no—
25C2×Q16—41no—
26C2×SD16—41no—
27C2×SD16—42no—
28C2×M16—21no—
29C22⋊Q8—41no—
30#121—22no—
31#121—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4C2311yescharacteristic
36C2×C42—21no—
37C2×C42D411yescharacteristic
38#104—42no—
39#103—42no—
40C22⋊C8—22no—
41#95—22no—
42#95—22no—
43#95—22no—
44#95—22no—
45#95D412yes—
46#41—41no—
47#41—41no—
48#41—41no—
49#41—41no—
50C2×Q8—24no—
51C2×Q8—21no—
52C2×Q8—21no—
53C2×Q8—42no—
54C2×Q8—41no—
55C2×Q8—41no—
56C2×Q8—82no—
57C2×Q8—81no—
58C2×Q8—81no—
59C2×Q8SD1612yescomplemented
60C2×D4—21no—
61C2×D4—22no—
62C2×D4—41no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—41no—
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D412yes—
70Q16—44no—
71SD16—44no—
72SD16—84no—
73M16—22no—
74M16—41no—
75C2×C8—41no—
76C2×C8—42no—
77C4⋊C4—22no—
78C4⋊C4—41no—
79C4⋊C4—82no—
80C4⋊C4—81no—
81C4⋊C4—81no—
82C22⋊C4—42no—
83C42—41no—
84C42—81no—
85C23—41no—
86C23#12011yescharacteristic
87Q8—42no—
88Q8—41no—
89Q8—41no—
90Q8—44no—
91Q8—81no—
92Q8—84no—
93Q8—82no—
94D4—41no—
95D4—41no—
96D4—82no—
97D4—81no—
98D4—81no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—24no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—81no—
117C2×C4—81no—
118C2×C4C2×SD1611yescharacteristic
119C2×C4#13711yescharacteristic
120C8—42no—
121C8—82no—
122C22—21no—
123C22—21no—
124C22—81no—
125C22—81no—
126C22—81no—
127C22#44811yescharacteristic
128C22#45311yescharacteristic
129C22B(4,𝔽2)11yescharacteristic
130C4—21no—
131C4—21no—
132C4—42no—
133C4—44no—
134C4—42no—
135C4—42no—
136C4—81no—
137C4—82no—
138C2—21no—
139C2—21no—
140C2—161no—
141C2#159611yescharacteristic
142C2#219911yescharacteristic
143C2#220511yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2811C2semidirect
4. #1354C2semidirect
5. #1319C2semidirect
59. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#28112
#20072
#16042
#13542
#13192