Finite Group Databasebeta

← #12628

Subgroups — #12628

Order 256.

314 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 81 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
2C2×#353C212yesmaximal
3C4×#99C212yesmaximal, complemented
4#1393C212yesmaximal, complemented
5C2×#127C2211yescharacteristic
6C2×C4×D4C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8C2×#99C2214yes—
9#390C411yescharacteristic, complemented
10#390C2211yescharacteristic
11#386—22no—
12#386—22no—
13#382C411yescharacteristic, complemented
14C4×C22⋊C4—22no—
15#376C411yescharacteristic, complemented
16#353C418yescomplemented
17#342—22no—
18C22×D4—22no—
19C22×D4C2×C411yescharacteristic
20C22×D4C2311yescharacteristic
21C23×C4—21no—
22C23×C4D412yes—
23#127—22no—
24#127C2×C414yes—
25C4×D4—42no—
26C4×D4—42no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C22⋊C4—24no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4D412yes—
34C2×C42—42no—
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38#99—24no—
39#99C2×C418yescomplemented
40#95—21no—
41#95—22no—
42#95—21no—
43#95—42no—
44C24—21no—
45C24C2×D412yes—
46C2×D4—28no—
47C2×D4—21no—
48C2×D4—41no—
49C2×D4—41no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4—82no—
53C2×D4—82no—
54C2×D4C4212yescomplemented
55C2×D4C22×C412yes—
56C2×D4#4112yes—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—42no—
63C22×C4—44no—
64C22×C4—82no—
65C22×C4—82no—
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C412yes—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22×C411yescharacteristic
70C4⋊C4—41no—
71C4⋊C4—41no—
72C22⋊C4—28no—
73C22⋊C4—42no—
74C22⋊C4—44no—
75C22⋊C4—42no—
76C22⋊C4—82no—
77C22⋊C4—82no—
78C42—22no—
79C42—21no—
80C42—41no—
81C42—84no—
82C42C4212yescomplemented
83C23—22no—
84C23—22no—
85C23—22no—
86C23—42no—
87C23—42no—
88C23—42no—
89C23—42no—
90C23—41no—
91C23—41no—
92C23C2×C22⋊C411yescharacteristic
93C23C4×D414yes—
94D4—42no—
95D4—42no—
96D4—42no—
97D4—42no—
98D4—88no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—41no—
105C2×C4—44no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—48no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—84no—
118C2×C4—84no—
119C2×C4—84no—
120C2×C4#9912yes—
121C2×C4C2×C4211yescharacteristic
122C2×C4#11711yescharacteristic
123C22—21no—
124C22—21no—
125C22—24no—
126C22—44no—
127C22—44no—
128C22—42no—
129C22—42no—
130C22—42no—
131C22—82no—
132C22—82no—
133C22—81no—
134C22—81no—
135C22C4×C22⋊C411yescharacteristic
136C22C2×#9911yescharacteristic
137C22#41011yescharacteristic
138C4—22no—
139C4—21no—
140C4—21no—
141C4—42no—
142C4—42no—
143C4—41no—
144C4—81no—
145C4—88no—
146C4—81no—
147C2—21no—
148C2—21no—
149C2—44no—
150C2—82no—
151C2C4×#9911yescharacteristic
152C2#212711yescharacteristic
153C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC2×C4C2×C42
Frattini, cosocleC22×D4C23
prime radicalC2×#127C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#99C2semidirect
4. #1393C2semidirect
6. C2×C4×D4C4semidirect
9. #390C4semidirect
13. #382C4semidirect
15. #376C4semidirect
16. #353C4semidirect
39. #99C2×C4semidirect
54. C2×D4C42semidirect
82. C42C42semidirect

Maximal subgroups

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

subgroupindex
#23282
C2×#3532
C4×#992
#13932