Finite Group Databasebeta

← C3×#1533 #72248

Subgroups — C3×#1533 #72248

Order 384.

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×C3×#124C211yesmaximal, characteristic
2C3×#432C212yesmaximal, complemented
3C3×#329C214yesmaximal, complemented
4#1533C311yesmaximal, characteristic, direct product factor, Sylow
5C22×C3×Q8C411yescharacteristic
6C22×C3×D4C411yescharacteristic
7C3×#124—22no—
8C3×#124C412yes—
9C3×#124C2212yes—
10C2×C3×C22⋊C4—22no—
11C2×C3×C42C2211yescharacteristic
12C3×C4⋊C8—22no—
13C3×C22⋊C8—22no—
14C3×#97C2214yescomplemented
15C2×#124C2×C311yescharacteristic
16#432C2×C312yescomplemented
17#329C2×C314yescomplemented
18C24×C3—21no—
19C2×C3×Q8—21no—
20C2×C3×Q8—42no—
21C2×C3×Q8C2×C412yes—
22C2×C3×D4—21no—
23C2×C3×D4—42no—
24C2×C3×D4C2×C412yes—
25C22×C3×C4—21no—
26C22×C3×C4C2×C411yescharacteristic
27C22×C3×C4D412yes—
28C2×C3×C8—24no—
29C3×C22⋊C4—24no—
30C3×C22⋊C4—24no—
31C3×C42C2×C411yescharacteristic
32C3×C42D412yes—
33C3×C42C2311yescharacteristic
34C22×Q8C3×C411yescharacteristic
35C22×D4C3×C411yescharacteristic
36#124—22no—
37#124C3×C412yes—
38#124C22×C312yes—
39C2×C22⋊C4—22no—
40C2×C42C22×C311yescharacteristic
41C4⋊C8—22no—
42C22⋊C8—22no—
43#97C22×C314yescomplemented
44C23×C3—22no—
45C23×C3—44no—
46C23×C3—42no—
47C23×C3C22⋊C411yescharacteristic
48C3×Q8—42no—
49C3×Q8—42no—
50C3×D4—42no—
51C3×D4—42no—
52C2×C3×C4—22no—
53C2×C3×C4—22no—
54C2×C3×C4—42no—
55C2×C3×C4—42no—
56C2×C3×C4C22⋊C412yes—
57C2×C3×C4C22⋊C411yescharacteristic
58C2×C3×C4C22×C411yescharacteristic
59C2×C3×C4C2×D412yes—
60C3×C8—44no—
61C24—21no—
62C2×Q8—21no—
63C2×Q8—42no—
64C2×Q8C2×C3×C412yes—
65C2×D4—21no—
66C2×D4—42no—
67C2×D4C2×C3×C412yes—
68C22×C4—21no—
69C22×C4C2×C3×C411yescharacteristic
70C22×C4C3×D412yes—
71C2×C8—24no—
72C22⋊C4—24no—
73C22⋊C4—24no—
74C42C2×C3×C411yescharacteristic
75C42C3×D412yes—
76C42C23×C311yescharacteristic
77C22×C3—22no—
78C22×C3—42no—
79C22×C3—44no—
80C22×C3—84no—
81C22×C3#10012yes—
82C22×C3C2×C22⋊C411yescharacteristic
83C3×C4—21no—
84C3×C4—21no—
85C3×C4—22no—
86C3×C4—22no—
87C3×C4—42no—
88C23—22no—
89C23—44no—
90C23—42no—
91C23C3×C22⋊C411yescharacteristic
92Q8—42no—
93Q8—42no—
94D4—42no—
95D4—42no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4C3×C22⋊C412yes—
101C2×C4C3×C22⋊C411yescharacteristic
102C2×C4C22×C3×C411yescharacteristic
103C2×C4C2×C3×D412yes—
104C8—44no—
105C2×C3—22no—
106C2×C3—82no—
107C2×C3C2×#10011yescharacteristic
108C2×C3#42112yes—
109C22—22no—
110C22—42no—
111C22—44no—
112C22—84no—
113C22C3×#10012yes—
114C22C2×C3×C22⋊C411yescharacteristic
115C4—21no—
116C4—21no—
117C4—22no—
118C4—22no—
119C4—42no—
120C3#153311yescharacteristic, direct product factor, Sylow
121C2—22no—
122C2—82no—
123C2C2×C3×#10011yescharacteristic
124C2C3×#42112yes—
subgroupG/H
center, socleC22×C3C2×C22⋊C4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC42C23×C3
prime radicalC22×C3×D4C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#432C2semidirect
3. C3×#329C2semidirect
4. #1533C3direct (N × H)
14. C3×#97C22semidirect
16. #432C2×C3semidirect
17. #329C2×C3semidirect
43. #97C22×C3semidirect
120. C3#1533direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C3×#4322
C3×#3292
#15333