Finite Group Databasebeta

← C3×#1660 #72375

Subgroups — C3×#1660 #72375

Order 384.

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

normal subgroups only

This group has 196 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×#563C211yesmaximal, characteristic
2C3×#501C212yesmaximal, complemented
3C3×#497C211yesmaximal, characteristic
4C3×#330C212yesmaximal, complemented
5C3×#329C211yesmaximal, characteristic
6#1660C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×Q16—21no—
8C2×C3×SD16—21no—
9C3×C4⋊Q8—21no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×#126—22no—
12C3×#125C2212yes—
13C3×#124C2211yescharacteristic
14C3×#123—21no—
15C3×C22⋊Q8—22no—
16C3×#107—22no—
17C3×#106—22no—
18C3×#97C2212yes—
19C3×#97C2211yescharacteristic
20#563C2×C311yescharacteristic
21#501C2×C312yescomplemented
22#497C2×C311yescharacteristic
23#330C2×C312yescomplemented
24#329C2×C311yescharacteristic
25C2×C3×Q8—21no—
26C2×C3×Q8—21no—
27C2×C3×Q8D411yescharacteristic
28C2×C3×D4D411yescharacteristic
29C22×C3×C4—21no—
30C3×Q16—42no—
31C3×SD16—42no—
32C2×C3×C8—22no—
33C2×C3×C8—21no—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—22no—
36C3×C4⋊C4—22no—
37C3×C4⋊C4—22no—
38C3×C4⋊C4D412yes—
39C3×C4⋊C4D412yes—
40C3×C22⋊C4—22no—
41C3×C22⋊C4—21no—
42C3×C22⋊C4—22no—
43C3×C42—21no—
44C3×C42C2311yescharacteristic
45C2×Q16—21no—
46C2×SD16—21no—
47C4⋊Q8—21no—
48C4⋊Q8C22×C311yescharacteristic
49#126—22no—
50#125C22×C312yes—
51#124C22×C311yescharacteristic
52#123—21no—
53C22⋊Q8—22no—
54#107—22no—
55#106—22no—
56#97C22×C312yes—
57#97C22×C311yescharacteristic
58C23×C3—21no—
59C3×Q8—41no—
60C3×Q8—41no—
61C3×Q8—41no—
62C3×Q8—81no—
63C3×D4—41no—
64C2×C3×C4—22no—
65C2×C3×C4—21no—
66C2×C3×C4—22no—
67C2×C3×C4—41no—
68C2×C3×C4—81no—
69C2×C3×C4C2×D412yes—
70C2×C3×C4C2×D411yescharacteristic
71C3×C8—24no—
72C3×C8—22no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8C3×D411yescharacteristic
76C2×D4C3×D411yescharacteristic
77C22×C4—21no—
78Q16—42no—
79SD16—42no—
80C2×C8—22no—
81C2×C8—21no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4C3×D412yes—
87C4⋊C4C3×D412yes—
88C22⋊C4—22no—
89C22⋊C4—21no—
90C22⋊C4—22no—
91C42—21no—
92C42C23×C311yescharacteristic
93C22×C3—41no—
94C22×C3—42no—
95C22×C3#12011yescharacteristic
96C3×C4—21no—
97C3×C4—22no—
98C3×C4—42no—
99C3×C4—41no—
100C3×C4—42no—
101C3×C4—81no—
102C23—21no—
103Q8—41no—
104Q8—41no—
105Q8—41no—
106Q8—81no—
107D4—41no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—41no—
112C2×C4—81no—
113C2×C4C2×C3×D412yes—
114C2×C4C2×C3×D411yescharacteristic
115C8—22no—
116C8—24no—
117C2×C3—81no—
118C2×C3#45311yescharacteristic
119C2×C3#45612yes—
120C22—41no—
121C22—42no—
122C22C3×#12011yescharacteristic
123C4—21no—
124C4—22no—
125C4—42no—
126C4—41no—
127C4—42no—
128C4—81no—
129C3#166011yescharacteristic, direct product factor, Sylow
130C2—81no—
131C2C3×#45311yescharacteristic
132C2C3×#45612yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC42C23×C3
prime radicalC2×C3×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#501C2semidirect
4. C3×#330C2semidirect
6. #1660C3direct (N × H)
21. #501C2×C3semidirect
23. #330C2×C3semidirect
129. C3#1660direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5632
C3×#5012
C3×#4972
C3×#3302
C3×#3292
#16603