Finite Group Databasebeta

← #7755

Subgroups — #7755

Order 256.

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

normal subgroups only

This group has 240 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#2804C211yesmaximal, characteristic
2#2024C212yesmaximal, complemented
3#2013C211yesmaximal, characteristic
4#1272C212yesmaximal, complemented
5#1270C211yesmaximal, characteristic
6C22×SD16—21no—
7C2×C4⋊Q8C2211yescharacteristic
8C2×#103—22no—
9C2×#103—21no—
10#400—21no—
11#393—22no—
12#391C2212yes—
13#390C2211yescharacteristic
14#386—21no—
15#385—22no—
16#340—22no—
17#336C2211yescharacteristic
18#336C2212yes—
19C22×Q8—21no—
20C22×Q8D412yes—
21C22×D4D411yescharacteristic
22C23×C4—21no—
23C2×SD16—42no—
24C2×SD16—42no—
25C2×SD16—42no—
26C22×C8—21no—
27C22×C8—22no—
28C4⋊Q8—42no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D412yes—
32C2×C4⋊C4D411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—21no—
36C2×C42C2311yescharacteristic
37#103—44no—
38#103—42no—
39#95—22no—
40#95—21no—
41#95—22no—
42#95—22no—
43C24—21no—
44C2×Q8—22no—
45C2×Q8—22no—
46C2×Q8—22no—
47C2×Q8—41no—
48C2×Q8—42no—
49C2×Q8—42no—
50C2×Q8—41no—
51C2×Q8—41no—
52C2×Q8—41no—
53C2×Q8—41no—
54C2×D4—21no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4SD1612yes—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—41no—
62C22×C4—81no—
63C22×C4—82no—
64C22×C4C2×D411yescharacteristic
65C22×C4C2×D412yes—
66SD16—48no—
67C2×C8—22no—
68C2×C8—41no—
69C2×C8—41no—
70C2×C8—42no—
71C2×C8—42no—
72C2×C8—42no—
73C4⋊C4—24no—
74C4⋊C4—41no—
75C4⋊C4—42no—
76C4⋊C4—42no—
77C4⋊C4—81no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—82no—
81C22⋊C4—81no—
82C42—41no—
83C23—22no—
84C23—42no—
85C23—41no—
86C23—42no—
87C23—41no—
88C23#12011yescharacteristic
89Q8—42no—
90Q8—42no—
91Q8—44no—
92Q8—82no—
93Q8—81no—
94Q8—81no—
95D4—42no—
96D4—81no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—84no—
111C2×C4—81no—
112C2×C4—81no—
113C2×C4C2×SD1611yescharacteristic
114C2×C4#13512yes—
115C2×C4#13711yescharacteristic
116C2×C4#13712yes—
117C8—42no—
118C8—82no—
119C22—42no—
120C22—44no—
121C22—82no—
122C22—81no—
123C22—81no—
124C22#44811yescharacteristic
125C22#45212yes—
126C22#45311yescharacteristic
127C22#45512yes—
128C22B(4,𝔽2)11yescharacteristic
129C4—21no—
130C4—21no—
131C4—22no—
132C4—22no—
133C4—82no—
134C4—82no—
135C4—81no—
136C4—82no—
137C2—82no—
138C2#159611yescharacteristic
139C2#160612yes—
140C2#163011yescharacteristic
141C2#164312yes—
142C2#165911yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2024C2semidirect
4. #1272C2semidirect

Maximal subgroups

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

subgroupindex
#28042
#20242
#20132
#12722
#12702