Finite Group Databasebeta

← #11558

Subgroups — #11558

Order 256.

223 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 53 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#2486C211yesmaximal, characteristic
2#1466C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1317C214yesmaximal, complemented
5C2×#123C412yes—
6C2×C4×D4C2211yescharacteristic
7C22×C4⋊C4C411yescharacteristic
8#432—21no—
9C2×C22⋊C8—21no—
10#398—21no—
11#397—21no—
12#389—21no—
13#381C2212yes—
14#380—21no—
15#380C411yescharacteristic
16#324—21no—
17#323—21no—
18#323C2212yes—
19#323C2212yes—
20C22×D4C2×C411yescharacteristic
21C23×C4C2×C412yes—
22C23×C4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C22×C8—41no—
25#123—44no—
26C4×D4—41no—
27C4×D4—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4D411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4C2×C412yes—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4C2311yescharacteristic
39C2×C42D411yescharacteristic
40C4⋊C8—41no—
41C22⋊C8—22no—
42C22⋊C8—22no—
43C22⋊C8—41no—
44#97—41no—
45#95—21no—
46#95—22no—
47#95—22no—
48C24C22⋊C411yescharacteristic
49C24C22×C411yescharacteristic
50C2×D4—22no—
51C2×D4—22no—
52C2×D4—41no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—24no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69C2×C8—42no—
70C2×C8—42no—
71C2×C8—82no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—44no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C22⋊C4—41no—
79C22⋊C4—41no—
80C22⋊C4—44no—
81C22⋊C4—42no—
82C42—21no—
83C42—21no—
84C23—21no—
85C23—21no—
86C23—21no—
87C23—21no—
88C23—41no—
89C23—41no—
90C23—41no—
91C23—41no—
92C23#10012yes—
93C23C2×C22⋊C411yescharacteristic
94C23#13112yes—
95D4—44no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—81no—
116C2×C4—84no—
117C2×C4#9911yescharacteristic
118C2×C4#9911yescharacteristic
119C8—82no—
120C8—82no—
121C22—21no—
122C22—21no—
123C22—22no—
124C22—22no—
125C22—42no—
126C22—42no—
127C22—42no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—41no—
132C22—81no—
133C22#40811yescharacteristic
134C22C2×#9911yescharacteristic
135C22C2×#10011yescharacteristic
136C4—21no—
137C4—41no—
138C4—41no—
139C4—41no—
140C4—41no—
141C4—82no—
142C4—82no—
143C4#35212yes—
144C2—21no—
145C2—21no—
146C2—42no—
147C2—42no—
148C2#147111yescharacteristic
149C2C2×#35211yescharacteristic
150C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1317C2semidirect

Maximal subgroups

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

subgroupindex
#24862
#14662
#14592
#13172