Finite Group Databasebeta

← C3×#1656 #72371

Subgroups — C3×#1656 #72371

Order 384.

230 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 60 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×#562C211yesmaximal, characteristic, complemented
2C3×#496C212yesmaximal, complemented
3C3×#490C211yesmaximal, characteristic, complemented
4C3×#330C211yesmaximal, characteristic, complemented
5C3×#329C212yesmaximal, complemented
6#1656C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×SD16—22no—
8C2×C3×D8—22no—
9C3×#127C2211yescharacteristic
10C3×#125C2211yescharacteristic
11C3×#124C2212yes—
12C3×#123—22no—
13C3×C22⋊Q8—22no—
14C3×#121—22no—
15C3×#102—22no—
16C3×#97C2211yescharacteristic
17C3×#97C2212yes—
18#562C2×C311yescharacteristic, complemented
19#496C2×C312yescomplemented
20#490C2×C311yescharacteristic, complemented
21#330C2×C311yescharacteristic, complemented
22#329C2×C312yescomplemented
23C2×C3×Q8D412yes—
24C2×C3×D4—22no—
25C2×C3×D4—21no—
26C2×C3×D4—22no—
27C2×C3×D4D412yes—
28C22×C3×C4—22no—
29C3×SD16—44no—
30C3×D8—44no—
31C2×C3×C8—22no—
32C2×C3×C8—21no—
33C3×C4⋊C4—22no—
34C3×C4⋊C4—22no—
35C3×C4⋊C4D412yes—
36C3×C22⋊C4—22no—
37C3×C22⋊C4—22no—
38C3×C22⋊C4—22no—
39C3×C42C2311yescharacteristic
40C2×SD16—22no—
41C2×D8—22no—
42#127C22×C311yescharacteristic
43#125C22×C311yescharacteristic
44#124C22×C312yes—
45#123—22no—
46C22⋊Q8—22no—
47#121—22no—
48#102—22no—
49#97C22×C311yescharacteristic
50#97C22×C312yes—
51C23×C3—22no—
52C23×C3—41no—
53C3×Q8—42no—
54C3×D4—42no—
55C3×D4—42no—
56C3×D4—42no—
57C3×D4—42no—
58C3×D4—82no—
59C2×C3×C4—22no—
60C2×C3×C4—22no—
61C2×C3×C4—82no—
62C2×C3×C4C2×D411yescharacteristic
63C2×C3×C4C2×D412yes—
64C3×C8—24no—
65C3×C8—41no—
66C2×Q8C3×D412yes—
67C2×D4—22no—
68C2×D4—21no—
69C2×D4—22no—
70C2×D4C3×D412yes—
71C22×C4—22no—
72SD16—44no—
73D8—44no—
74C2×C8—22no—
75C2×C8—21no—
76C4⋊C4—22no—
77C4⋊C4—22no—
78C4⋊C4C3×D412yes—
79C22⋊C4—22no—
80C22⋊C4—22no—
81C22⋊C4—22no—
82C42C23×C311yescharacteristic
83C22×C3—42no—
84C22×C3—42no—
85C22×C3—42no—
86C22×C3—82no—
87C22×C3—81no—
88C22×C3#12011yescharacteristic
89C3×C4—22no—
90C3×C4—21no—
91C3×C4—42no—
92C3×C4—42no—
93C23—22no—
94C23—41no—
95Q8—42no—
96D4—42no—
97D4—42no—
98D4—42no—
99D4—42no—
100D4—82no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—82no—
104C2×C4C2×C3×D411yescharacteristic
105C2×C4C2×C3×D412yes—
106C8—24no—
107C8—41no—
108C2×C3—82no—
109C2×C3—161no—
110C2×C3#45312yes—
111C2×C3#45411yescharacteristic
112C22—42no—
113C22—42no—
114C22—42no—
115C22—82no—
116C22—81no—
117C22C3×#12011yescharacteristic
118C4—22no—
119C4—21no—
120C4—42no—
121C4—42no—
122C3#165611yescharacteristic, direct product factor, Sylow
123C2—82no—
124C2—161no—
125C2C3×#45312yes—
126C2C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC42C23×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#562C2semidirect
2. C3×#496C2semidirect
3. C3×#490C2semidirect
4. C3×#330C2semidirect
5. C3×#329C2semidirect
6. #1656C3direct (N × H)
18. #562C2×C3semidirect
19. #496C2×C3semidirect
20. #490C2×C3semidirect
21. #330C2×C3semidirect
22. #329C2×C3semidirect
122. C3#1656direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5622
C3×#4962
C3×#4902
C3×#3302
C3×#3292
#16563