Finite Group Databasebeta

← C3×#1352 #67912

Subgroups — C3×#1352 #67912

Order 384.

154 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 40 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×#399C211yesmaximal, characteristic
2C3×#342C211yesmaximal, characteristic
3C3×#323C211yesmaximal, characteristic
4#1352C311yesmaximal, characteristic, direct product factor, Sylow
5C2×C3×C4⋊C4—21no—
6C2×C3×C4⋊C4C411yescharacteristic, complemented
7C2×C3×C22⋊C4—21no—
8C2×C3×C22⋊C4—21no—
9C2×C3×C22⋊C4C2211yescharacteristic
10C3×C22⋊C8—21no—
11C3×#95C411yescharacteristic, complemented
12#399C2×C311yescharacteristic
13#342C2×C311yescharacteristic
14#323C2×C311yescharacteristic
15C24×C3D411yescharacteristic
16C22×C3×C4—21no—
17C22×C3×C4—21no—
18C22×C3×C4—41no—
19C22×C3×C4C2×C411yescharacteristic
20C22×C3×C4D411yescharacteristic
21C2×C3×C8—41no—
22C3×C4⋊C4—41no—
23C3×C4⋊C4—42no—
24C3×C22⋊C4—22no—
25C3×C22⋊C4—41no—
26C3×C22⋊C4—42no—
27C3×C22⋊C4—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4C3×C411yescharacteristic, complemented
30C2×C22⋊C4—21no—
31C2×C22⋊C4—21no—
32C2×C22⋊C4C22×C311yescharacteristic
33C22⋊C8—21no—
34#95C3×C411yescharacteristic, complemented
35C23×C3—21no—
36C23×C3—21no—
37C23×C3—41no—
38C23×C3—41no—
39C23×C3C22⋊C411yescharacteristic
40C23×C3SD1611yescharacteristic
41C23×C3Q1611yescharacteristic
42C2×C3×C4—22no—
43C2×C3×C4—21no—
44C2×C3×C4—21no—
45C2×C3×C4—41no—
46C2×C3×C4—41no—
47C2×C3×C4—41no—
48C2×C3×C4—41no—
49C2×C3×C4—42no—
50C2×C3×C4—41no—
51C2×C3×C4—41no—
52C2×C3×C4—41no—
53C2×C3×C4—81no—
54C2×C3×C4—81no—
55C3×C8—81no—
56C24C3×D411yescharacteristic
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—41no—
60C22×C4C2×C3×C411yescharacteristic
61C22×C4C3×D411yescharacteristic
62C2×C8—41no—
63C4⋊C4—41no—
64C4⋊C4—42no—
65C22⋊C4—22no—
66C22⋊C4—41no—
67C22⋊C4—42no—
68C22⋊C4—41no—
69C22×C3—21no—
70C22×C3—21no—
71C22×C3—21no—
72C22×C3—21no—
73C22×C3—41no—
74C22×C3—41no—
75C22×C3—41no—
76C22×C3—41no—
77C22×C3—81no—
78C22×C3#9911yescharacteristic
79C22×C3#10311yescharacteristic
80C22×C3#10411yescharacteristic
81C3×C4—42no—
82C3×C4—41no—
83C3×C4—41no—
84C3×C4—82no—
85C3×C4—81no—
86C23—21no—
87C23—21no—
88C23—41no—
89C23—41no—
90C23C3×C22⋊C411yescharacteristic
91C23C3×SD1611yescharacteristic
92C23C3×Q1611yescharacteristic
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—81no—
105C2×C4—81no—
106C8—81no—
107C2×C3—21no—
108C2×C3—21no—
109C2×C3—41no—
110C2×C3—41no—
111C2×C3#32811yescharacteristic
112C2×C3#35211yescharacteristic
113C2×C3#35311yescharacteristic
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—81no—
123C22C3×#9911yescharacteristic
124C22C3×#10311yescharacteristic
125C22C3×#10411yescharacteristic
126C4—42no—
127C4—41no—
128C4—41no—
129C4—82no—
130C4—81no—
131C3#135211yescharacteristic, direct product factor, Sylow
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2C3×#32811yescharacteristic
137C2C3×#35211yescharacteristic
138C2C3×#35311yescharacteristic
subgroupG/H
center, socleC22×C3#99
commutatorC22×C4C2×C3×C4
Frattini, cosocleC2×C22⋊C4C22×C3
prime radicalC24×C3D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
4. #1352C3direct (N × H)
6. C2×C3×C4⋊C4C4semidirect
11. C3×#95C4semidirect
29. C2×C4⋊C4C3×C4semidirect
34. #95C3×C4semidirect
131. C3#1352direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#3992
C3×#3422
C3×#3232
#13523