Finite Group Databasebeta

← C3×#2135 #72850

Subgroups — C3×#2135 #72850

Order 384.

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

normal subgroups only

This group has 230 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×#413C212yesmaximal
3C3×#356C214yesmaximal, complemented
4#2135C311yesmaximal, characteristic, direct product factor, Sylow
5C2×C3×#41C2211yescharacteristic
6C22×C3×Q8C411yescharacteristic
7C2×C3×M16—22no—
8C3×C4⋊Q8C2212yes—
9C3×#124C412yes—
10C3×#123—22no—
11C3×C22⋊Q8—22no—
12C3×#117C411yescharacteristic
13C3×#101C2214yes—
14C3×#100—22no—
15#536C2×C311yescharacteristic
16#413C2×C312yes—
17#356C2×C314yescomplemented
18C3×#41—42no—
19C2×C3×Q8—21no—
20C2×C3×Q8—42no—
21C2×C3×Q8C2×C412yes—
22C2×C3×Q8C2311yescharacteristic
23C2×C3×D4C2×C411yescharacteristic
24C2×C3×D4D412yes—
25C22×C3×C4—21no—
26C22×C3×C4C2×C411yescharacteristic
27C22×C3×C4D412yes—
28C3×M16—24no—
29C3×M16—42no—
30C2×C3×C8—42no—
31C3×C4⋊C4—21no—
32C3×C4⋊C4—24no—
33C3×C22⋊C4—21no—
34C3×C22⋊C4—24no—
35C3×C42C2×C412yes—
36C2×#41C22×C311yescharacteristic
37C22×Q8C3×C411yescharacteristic
38C2×M16—22no—
39C4⋊Q8C22×C312yes—
40#124C3×C412yes—
41#123—22no—
42C22⋊Q8—22no—
43#117C3×C411yescharacteristic
44#101C22×C314yes—
45#100—22no—
46C23×C3C22⋊C412yes—
47C23×C3C22⋊C411yescharacteristic
48C3×Q8—41no—
49C3×Q8—42no—
50C3×Q8—42no—
51C3×D4—41no—
52C3×D4—42no—
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×C4C22⋊C411yescharacteristic
60C2×C3×C4C22×C411yescharacteristic
61C2×C3×C4C2×D412yes—
62C3×C8—44no—
63#41—42no—
64C2×Q8—21no—
65C2×Q8—42no—
66C2×Q8C2×C3×C412yes—
67C2×Q8C23×C311yescharacteristic
68C2×D4C2×C3×C411yescharacteristic
69C2×D4C3×D412yes—
70C22×C4—21no—
71C22×C4C2×C3×C411yescharacteristic
72C22×C4C3×D412yes—
73M16—24no—
74M16—42no—
75C2×C8—42no—
76C4⋊C4—21no—
77C4⋊C4—24no—
78C22⋊C4—21no—
79C22⋊C4—24no—
80C42C2×C3×C412yes—
81C22×C3—21no—
82C22×C3—22no—
83C22×C3—42no—
84C22×C3—41no—
85C22×C3C2×C22⋊C411yescharacteristic
86C3×C4—22no—
87C3×C4—21no—
88C3×C4—42no—
89C3×C4—42no—
90C3×C4#9912yes—
91C23C3×C22⋊C412yes—
92C23C3×C22⋊C411yescharacteristic
93Q8—41no—
94Q8—42no—
95Q8—42no—
96D4—41no—
97D4—42no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4C3×C22⋊C411yescharacteristic
105C2×C4C22×C3×C411yescharacteristic
106C2×C4C2×C3×D412yes—
107C8—44no—
108C2×C3—21no—
109C2×C3—41no—
110C2×C3—42no—
111C2×C3C2×#9911yescharacteristic
112C22—21no—
113C22—22no—
114C22—42no—
115C22—41no—
116C22C2×C3×C22⋊C411yescharacteristic
117C4—22no—
118C4—21no—
119C4—42no—
120C4—42no—
121C4C3×#9912yes—
122C3#213511yescharacteristic, direct product factor, Sylow
123C2—21no—
124C2—41no—
125C2—42no—
126C2C2×C3×#9911yescharacteristic
subgroupG/H
center, socleC2×C3C2×#99
commutatorC2×C4C22×C3×C4
Frattini, cosocleC2×Q8C23×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×#356C2semidirect
4. #2135C3direct (N × H)
17. #356C2×C3semidirect
122. C3#2135direct (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×#4132
C3×#3562
#21353