Finite Group Databasebeta

← C4×#355 #12629

Subgroups — C4×#355 #12629

Order 256.

266 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×#355C212yesmaximal
3C4×#101C212yesmaximal, complemented
4#1394C212yesmaximal, complemented
5C2×#124C2211yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7C2×C4×D4C411yescharacteristic
8C2×#101C2214yes—
9C4×M16—22no—
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#388—22no—
13#382C411yescharacteristic
14C4×C22⋊C4—22no—
15C43C411yescharacteristic
16#355C418yesdirect product factor
17#343—22no—
18C22×Q8C2311yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C2×M16—24no—
22#124—22no—
23#124C2×C414yes—
24C4×Q8—41no—
25C4×Q8—41no—
26C4×D4—41no—
27C4×D4—41no—
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4D412yes—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—21no—
33C2×C22⋊C4—22no—
34C2×C42—21no—
35C2×C42—21no—
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42D412yes—
40#101—24no—
41#101C2×C418yescomplemented
42#97—42no—
43C4×C8—42no—
44#95—22no—
45#95—22no—
46C24—21no—
47C2×Q8—41no—
48C2×Q8—41no—
49C2×Q8C22×C412yes—
50C2×Q8#4112yes—
51C2×D4—21no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4C4212yes—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—44no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4C22⋊C412yes—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D412yes—
66M16—28no—
67M16—44no—
68C2×C8—44no—
69C4⋊C4—41no—
70C4⋊C4—42no—
71C4⋊C4—41no—
72C22⋊C4—28no—
73C22⋊C4—44no—
74C22⋊C4—42no—
75C42—22no—
76C42—21no—
77C42—22no—
78C42—21no—
79C42—21no—
80C42—41no—
81C42—42no—
82C42—42no—
83C42C4212yes—
84C23—22no—
85C23—42no—
86C23—42no—
87C23—41no—
88C23—41no—
89C23C2×C22⋊C411yescharacteristic
90Q8—42no—
91Q8—42no—
92D4—42no—
93D4—42no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—82no—
112C2×C4—82no—
113C2×C4#9911yescharacteristic
114C2×C4#9911yescharacteristic
115C2×C4C2×C4211yescharacteristic
116C2×C4#11711yescharacteristic
117C2×C4C4×D414yes—
118C8—48no—
119C22—21no—
120C22—21no—
121C22—44no—
122C22—42no—
123C22—82no—
124C22—81no—
125C22—81no—
126C22C4×C22⋊C411yescharacteristic
127C22C2×#9911yescharacteristic
128C22#41011yescharacteristic
129C4—21no—
130C4—24no—
131C4—21no—
132C4—21no—
133C4—42no—
134C4—42no—
135C4—41no—
136C4—42no—
137C4—81no—
138C4#35512yesdirect product factor
139C2—21no—
140C2—21no—
141C2—82no—
142C2C4×#9911yescharacteristic
143C2C2×#35511yescharacteristic
144C2#213211yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC2×C4C2×C42
Frattini, cosocleC22×Q8C23
socleC22C2×#99
prime radicalC22×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#101C2semidirect
4. #1394C2semidirect
16. #355C4direct (N × H)
41. #101C2×C4semidirect
138. C4#355direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#1242
C2×#3552
C4×#1012
#13942