Finite Group Databasebeta

← C3×#1316 #67876

Subgroups — C3×#1316 #67876

Order 384.

294 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 44 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×#379C211yesmaximal, characteristic
2C3×#323C212yesmaximal, complemented
3#1316C311yesmaximal, characteristic, direct product factor, Sylow
4C25×C3C411yescharacteristic
5C2×C3×C22⋊C4—22no—
6C2×C3×C22⋊C4C411yescharacteristic
7C2×C3×C22⋊C4C2211yescharacteristic
8C3×C22⋊C8—22no—
9#379C2×C311yescharacteristic
10#323C2×C312yescomplemented
11C24×C3—21no—
12C24×C3—21no—
13C24×C3—44no—
14C24×C3—41no—
15C24×C3—41no—
16C24×C3C812yescomplemented
17C24×C3C2×C411yescharacteristic
18C22×C3×C4—21no—
19C22×C3×C4D412yes—
20C2×C3×C8—42no—
21C3×C22⋊C4—41no—
22C3×C22⋊C4—44no—
23C3×C22⋊C4—41no—
24C25C3×C411yescharacteristic
25C2×C22⋊C4—22no—
26C2×C22⋊C4C3×C411yescharacteristic
27C2×C22⋊C4C22×C311yescharacteristic
28C22⋊C8—22no—
29C23×C3—24no—
30C23×C3—22no—
31C23×C3—21no—
32C23×C3—21no—
33C23×C3—44no—
34C23×C3—42no—
35C23×C3—42no—
36C23×C3—42no—
37C23×C3—44no—
38C23×C3—42no—
39C23×C3—44no—
40C23×C3—48no—
41C23×C3—44no—
42C23×C3—41no—
43C23×C3—41no—
44C23×C3C22⋊C411yescharacteristic
45C23×C3C2×C811yescharacteristic
46C23×C3M1611yescharacteristic
47C2×C3×C4—22no—
48C2×C3×C4—42no—
49C2×C3×C4—42no—
50C2×C3×C4—41no—
51C3×C8—82no—
52C24—21no—
53C24—21no—
54C24—44no—
55C24—41no—
56C24—41no—
57C24C3×C812yescomplemented
58C24C2×C3×C411yescharacteristic
59C22×C4—21no—
60C22×C4C3×D412yes—
61C2×C8—42no—
62C22⋊C4—41no—
63C22⋊C4—44no—
64C22⋊C4—41no—
65C22×C3—22no—
66C22×C3—24no—
67C22×C3—21no—
68C22×C3—21no—
69C22×C3—41no—
70C22×C3—41no—
71C22×C3—44no—
72C22×C3—42no—
73C22×C3—42no—
74C22×C3—44no—
75C22×C3—44no—
76C22×C3—48no—
77C22×C3—42no—
78C22×C3—44no—
79C22×C3—42no—
80C22×C3C22⋊C811yescharacteristic
81C22×C3#9911yescharacteristic
82C22×C3#10011yescharacteristic
83C3×C4—42no—
84C3×C4—81no—
85C23—24no—
86C23—22no—
87C23—21no—
88C23—21no—
89C23—44no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23—44no—
94C23—42no—
95C23—44no—
96C23—48no—
97C23—44no—
98C23—41no—
99C23—41no—
100C23C3×C22⋊C411yescharacteristic
101C23C2×C3×C811yescharacteristic
102C23C3×M1611yescharacteristic
103C2×C4—22no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C8—82no—
108C2×C3—21no—
109C2×C3—21no—
110C2×C3—44no—
111C2×C3—41no—
112C2×C3—41no—
113C2×C3#32311yescharacteristic
114C2×C3AΣL(1,𝔽16)12yes—
115C22—22no—
116C22—24no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—44no—
122C22—42no—
123C22—42no—
124C22—44no—
125C22—44no—
126C22—48no—
127C22—42no—
128C22—44no—
129C22—42no—
130C22C3×C22⋊C811yescharacteristic
131C22C3×#9911yescharacteristic
132C22C3×#10011yescharacteristic
133C4—42no—
134C4—81no—
135C3#131611yescharacteristic, direct product factor, Sylow
136C2—21no—
137C2—21no—
138C2—44no—
139C2—41no—
140C2—41no—
141C2C3×#32311yescharacteristic
142C2C3×AΣL(1,𝔽16)12yes—
subgroupG/H
center, socleC22×C3#99
commutatorC23C2×C3×C8
Frattini, cosocleC2×C22⋊C4C22×C3
prime radicalC25×C3C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#323C2semidirect
3. #1316C3direct (N × H)
10. #323C2×C3semidirect
16. C24×C3C8semidirect
57. C24C3×C8semidirect
135. C3#1316direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#3792
C3×#3232
#13163