Finite Group Databasebeta

← #12220

Subgroups — #12220

Order 256.

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

normal subgroups only

This group has 230 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#2683C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic
3#1605C212yesmaximal, complemented
4#1354C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic
6#544C2211yescharacteristic
7C2×C22⋊Q8—22no—
8#451—22no—
9#418—21no—
10#396—22no—
11#395C2212yes—
12#390C2211yescharacteristic
13#389—21no—
14#386—21no—
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×Q8D412yes—
20C22×D4D411yescharacteristic
21C23×C4—21no—
22C2×Q16—42no—
23C2×M16—21no—
24C4⋊Q8—41no—
25#125—41no—
26C22⋊Q8—22no—
27C22⋊Q8—48no—
28#117—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2311yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C42D411yescharacteristic
38#103—42no—
39#102—41no—
40#102—41no—
41C22⋊C8—22no—
42#95—22no—
43#95D412yes—
44C24—21no—
45C2×Q8—22no—
46C2×Q8—44no—
47C2×Q8—42no—
48C2×D4—21no—
49C2×D4—41no—
50C2×D4—41no—
51C2×D4Q1612yescomplemented
52C22×C4—22no—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—44no—
56C22×C4—81no—
57C22×C4C2×D411yescharacteristic
58C22×C4C2×D412yes—
59Q16—84no—
60M16—41no—
61M16—41no—
62C2×C8—42no—
63C2×C8—41no—
64C4⋊C4—24no—
65C4⋊C4—21no—
66C4⋊C4—21no—
67C4⋊C4—22no—
68C4⋊C4—41no—
69C4⋊C4—44no—
70C4⋊C4—44no—
71C4⋊C4—42no—
72C4⋊C4—81no—
73C22⋊C4—44no—
74C22⋊C4—44no—
75C22⋊C4—41no—
76C22⋊C4—81no—
77C42—41no—
78C42—41no—
79C23—22no—
80C23—42no—
81C23—42no—
82C23—41no—
83C23—41no—
84C23#12011yescharacteristic
85Q8—44no—
86Q8—82no—
87Q8—82no—
88D4—42no—
89D4—81no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—24no—
94C2×C4—22no—
95C2×C4—42no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—84no—
106C2×C4C2×Q1611yescharacteristic
107C2×C4#13711yescharacteristic
108C8—81no—
109C8—82no—
110C22—21no—
111C22—21no—
112C22—44no—
113C22—42no—
114C22—82no—
115C22—81no—
116C22—81no—
117C22#45111yescharacteristic
118C22#45511yescharacteristic
119C22B(4,𝔽2)11yescharacteristic
120C4—21no—
121C4—21no—
122C4—42no—
123C4—42no—
124C4—44no—
125C4—82no—
126C4—81no—
127C4—81no—
128C2—21no—
129C2—21no—
130C2—82no—
131C2#160211yescharacteristic
132C2#219611yescharacteristic
133C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
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
3. #1605C2semidirect
4. #1354C2semidirect
51. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#26832
#20272
#16052
#13542
#13182