Finite Group Databasebeta

← #11479

Subgroups — #11479

Order 256.

262 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 65 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#2328C211yesmaximal, characteristic
2#1459C212yesmaximal, complemented
3#1324C212yesmaximal, complemented
4#1318C212yesmaximal, complemented
5C2×#127C411yescharacteristic
6C2×C4×D4C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C2×C22⋊C8—22no—
9#390C411yescharacteristic
10#390C2211yescharacteristic
11#386—22no—
12#386—22no—
13#382C2211yescharacteristic
14#376C411yescharacteristic
15#324C2214yes—
16#323—22no—
17C22×D4—22no—
18C22×D4C2×C411yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C23×C4D412yes—
22C22×C8—42no—
23#127—22no—
24#127C814yescomplemented
25C4×D4—42no—
26C4×D4—42no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2311yescharacteristic
29C2×C22⋊C4—21no—
30C2×C22⋊C4—22no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4D412yes—
33C2×C42C2×C411yescharacteristic
34C2×C42C2×C411yescharacteristic
35C2×C42C2×C411yescharacteristic
36C22⋊C8—24no—
37C22⋊C8—42no—
38#95—21no—
39#95—21no—
40#95—22no—
41C24—21no—
42C24C22⋊C412yes—
43C2×D4—28no—
44C2×D4—21no—
45C2×D4—41no—
46C2×D4—41no—
47C2×D4—41no—
48C2×D4—41no—
49C2×D4—82no—
50C2×D4—82no—
51C2×D4C2×C812yes—
52C2×D4C2×C812yescomplemented
53C2×D4M1612yes—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—42no—
60C22×C4—82no—
61C22×C4—82no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D412yes—
66C2×C8—44no—
67C2×C8—84no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C22⋊C4—42no—
71C22⋊C4—42no—
72C22⋊C4—82no—
73C22⋊C4—82no—
74C42—22no—
75C42—21no—
76C42—41no—
77C42C2×C812yescomplemented
78C23—22no—
79C23—22no—
80C23—22no—
81C23—42no—
82C23—42no—
83C23—42no—
84C23—42no—
85C23—41no—
86C23—41no—
87C23C22⋊C814yes—
88C23C2×C22⋊C411yescharacteristic
89D4—42no—
90D4—42no—
91D4—42no—
92D4—42no—
93D4—88no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—41no—
100C2×C4—44no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—82no—
111C2×C4—84no—
112C2×C4#9912yes—
113C2×C4C22×C811yescharacteristic
114C2×C4C2×M1611yescharacteristic
115C8—84no—
116C22—21no—
117C22—21no—
118C22—24no—
119C22—44no—
120C22—44no—
121C22—42no—
122C22—42no—
123C22—42no—
124C22—82no—
125C22—82no—
126C22—81no—
127C22—81no—
128C22C2×C22⋊C811yescharacteristic
129C22C2×#9911yescharacteristic
130C22#41311yescharacteristic
131C4—22no—
132C4—21no—
133C4—21no—
134C4—42no—
135C4—42no—
136C4—41no—
137C4—81no—
138C4—81no—
139C2—21no—
140C2—21no—
141C2—44no—
142C2—82no—
143C2#145911yescharacteristic
144C2#212711yescharacteristic
145C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC2×C4C22×C8
Frattini, cosocleC2×C4⋊C4C23
prime radicalC2×#127C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1459C2semidirect
3. #1324C2semidirect
4. #1318C2semidirect
24. #127C8semidirect
52. C2×D4C2×C8semidirect
77. C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#23282
#14592
#13242
#13182