Finite Group Databasebeta

← C3×#2129 #72844

Subgroups — C3×#2129 #72844

Order 384.

238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3×#536C211yesmaximal, characteristic
2C3×#410C212yesmaximal
3C3×#354C214yesmaximal, complemented
4#2129C311yesmaximal, characteristic, direct product factor, Sylow
5C2×C3×#41C2211yescharacteristic
6C22×C3×Q8C411yescharacteristic, complemented
7C3×C4⋊Q8C412yescomplemented
8C3×#124C2212yes—
9C3×#123—22no—
10C3×C22⋊Q8—22no—
11C3×#117—22no—
12C3×#117C411yescharacteristic, complemented
13C3×#99—22no—
14C3×#99C2214yes—
15#536C2×C311yescharacteristic
16#410C2×C312yes—
17#354C2×C314yescomplemented
18C3×#41—42no—
19C2×C3×Q8—21no—
20C2×C3×Q8—42no—
21C2×C3×Q8C2×C412yes—
22C2×C3×Q8C2×C411yescharacteristic
23C2×C3×D4D412yes—
24C2×C3×D4C2311yescharacteristic
25C22×C3×C4—21no—
26C22×C3×C4C2×C411yescharacteristic
27C22×C3×C4D412yes—
28C3×C4⋊C4—24no—
29C3×C4⋊C4—21no—
30C3×C4⋊C4—42no—
31C3×C22⋊C4—21no—
32C3×C22⋊C4—24no—
33C3×C22⋊C4—24no—
34C3×C42—42no—
35C3×C42C2×C412yes—
36C2×#41C22×C311yescharacteristic
37C22×Q8C3×C411yescharacteristic, complemented
38C4⋊Q8C3×C412yescomplemented
39#124C22×C312yes—
40#123—22no—
41C22⋊Q8—22no—
42#117—22no—
43#117C3×C411yescharacteristic, complemented
44#99—22no—
45#99C22×C314yes—
46C23×C3C22⋊C411yescharacteristic
47C23×C3C2×D412yes—
48C3×Q8—41no—
49C3×Q8—42no—
50C3×Q8—42no—
51C3×D4—42no—
52C3×D4—41no—
53C2×C3×C4—22no—
54C2×C3×C4—22no—
55C2×C3×C4—22no—
56C2×C3×C4—22no—
57C2×C3×C4—22no—
58C2×C3×C4—42no—
59C2×C3×C4—44no—
60C2×C3×C4C22⋊C412yes—
61C2×C3×C4C22⋊C411yescharacteristic
62C2×C3×C4C22×C411yescharacteristic
63#41—42no—
64C2×Q8—21no—
65C2×Q8—42no—
66C2×Q8C2×C3×C412yes—
67C2×Q8C2×C3×C411yescharacteristic
68C2×D4C3×D412yes—
69C2×D4C23×C311yescharacteristic
70C22×C4—21no—
71C22×C4C2×C3×C411yescharacteristic
72C22×C4C3×D412yes—
73C4⋊C4—21no—
74C4⋊C4—24no—
75C4⋊C4—42no—
76C22⋊C4—21no—
77C22⋊C4—24no—
78C22⋊C4—24no—
79C42—42no—
80C42C2×C3×C412yes—
81C22×C3—21no—
82C22×C3—22no—
83C22×C3—41no—
84C22×C3—42no—
85C22×C3C2×C22⋊C411yescharacteristic
86C3×C4—22no—
87C3×C4—21no—
88C3×C4—42no—
89C3×C4—42no—
90C3×C4—84no—
91C3×C4#9912yes—
92C23C3×C22⋊C411yescharacteristic
93C23C2×C3×D412yes—
94Q8—41no—
95Q8—42no—
96Q8—42no—
97D4—42no—
98D4—41no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—42no—
105C2×C4—44no—
106C2×C4C3×C22⋊C412yes—
107C2×C4C3×C22⋊C411yescharacteristic
108C2×C4C22×C3×C411yescharacteristic
109C2×C3—21no—
110C2×C3—42no—
111C2×C3—41no—
112C2×C3C2×#9911yescharacteristic
113C22—21no—
114C22—22no—
115C22—41no—
116C22—42no—
117C22C2×C3×C22⋊C411yescharacteristic
118C4—22no—
119C4—21no—
120C4—42no—
121C4—42no—
122C4—84no—
123C4C3×#9912yes—
124C3#212911yescharacteristic, direct product factor, Sylow
125C2—21no—
126C2—42no—
127C2—41no—
128C2C2×C3×#9911yescharacteristic
subgroupG/H
center, socleC2×C3C2×#99
commutatorC2×C4C22×C3×C4
Frattini, cosocleC2×D4C23×C3
prime radicalC2×C3×#41C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#354C2semidirect
4. #2129C3direct (N × H)
6. C22×C3×Q8C4semidirect
7. C3×C4⋊Q8C4semidirect
12. C3×#117C4semidirect
17. #354C2×C3semidirect
37. C22×Q8C3×C4semidirect
38. C4⋊Q8C3×C4semidirect
43. #117C3×C4semidirect
124. C3#2129direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5362
C3×#4102
C3×#3542
#21293