Finite Group Databasebeta

← C4×#329 #8222

Subgroups — C4×#329 #8222

Order 256.

308 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C4×#124C211yesmaximal, characteristic
2C4×#97C212yesmaximal, complemented
3C2×#329C212yesmaximal
4#1295C212yesmaximal, complemented
5C2×#124C2211yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×D4C411yescharacteristic
8C2×#97—22no—
9C2×#97C2214yes—
10C2×C4×C8—22no—
11#391C411yescharacteristic
12#390C411yescharacteristic
13#388—22no—
14#382C2211yescharacteristic
15C4×C22⋊C4—22no—
16C43C2211yescharacteristic
17#329C418yesdirect product factor
18C22×Q8C2×C411yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C22×C8—24no—
22#124—22no—
23#124C2×C414yes—
24C4×Q8—42no—
25C4×D4—42no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C22⋊C4—22no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—22no—
32C2×C42—21no—
33C2×C42C2×C411yescharacteristic
34C2×C42C2×C411yescharacteristic
35C2×C42D412yes—
36C2×C42D412yes—
37C2×C42C2311yescharacteristic
38#97—28no—
39#97—24no—
40#97C2×C418yescomplemented
41C4×C8—28no—
42#95—22no—
43#95—22no—
44C24—21no—
45C2×Q8—21no—
46C2×Q8—41no—
47C2×Q8—41no—
48C2×Q8C4212yes—
49C2×D4—21no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4C4212yes—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—42no—
57C22×C4—44no—
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C412yes—
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D412yes—
63C2×C8—28no—
64C2×C8—28no—
65C2×C8—28no—
66C4⋊C4—41no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C22⋊C4—28no—
70C22⋊C4—42no—
71C22⋊C4—44no—
72C42—22no—
73C42—24no—
74C42—24no—
75C42—41no—
76C42—42no—
77C42C22×C412yes—
78C42#4112yes—
79C23—22no—
80C23—41no—
81C23—41no—
82C23—42no—
83C23—42no—
84C23C2×C22⋊C411yescharacteristic
85Q8—42no—
86Q8—42no—
87D4—42no—
88D4—42no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—22no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—84no—
105C2×C4#10012yes—
106C2×C4#10414yes—
107C2×C4#10414yes—
108C2×C4C2×C4211yescharacteristic
109C2×C4#11711yescharacteristic
110C2×C4C4×D414yes—
111C8—216no—
112C22—42no—
113C22—44no—
114C22—81no—
115C22—82no—
116C22—81no—
117C22C4×C22⋊C411yescharacteristic
118C22C2×#10011yescharacteristic
119C22#41311yescharacteristic
120C22C2×#10412yes—
121C22C2×#10412yes—
122C4—21no—
123C4—21no—
124C4—24no—
125C4—24no—
126C4—42no—
127C4—41no—
128C4—81no—
129C4—81no—
130C4#32914yesdirect product factor
131C2—82no—
132C2C2×#32911yescharacteristic
133C2#152412yes—
134C2#152511yescharacteristic
135C2C4×#10011yescharacteristic
136C2C4×#10412yes—
subgroupG/H
centerC22×C4C22⋊C4
commutatorC2×C4C2×C42
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC22×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#97C2semidirect
4. #1295C2semidirect
17. #329C4direct (N × H)
40. #97C2×C4semidirect
130. C4#329direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#1242
C4×#972
C2×#3292
#12952