Finite Group Databasebeta

← C3×#1662 #72377

Subgroups — C3×#1662 #72377

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×#562C211yesmaximal, characteristic
2C3×#491C211yesmaximal, characteristic, complemented
3C3×#489C212yesmaximal, complemented
4C3×#330C211yesmaximal, characteristic, complemented
5C3×#329C212yesmaximal, complemented
6#1662C311yesmaximal, characteristic, direct product factor, Sylow
7C3×C4⋊Q8C2211yescharacteristic
8C3×#125C2211yescharacteristic
9C3×#124C2212yes—
10C3×#123—22no—
11C3×C22⋊Q8—22no—
12C3×#121—22no—
13C3×#103—22no—
14C3×#103—22no—
15C3×#102—22no—
16C3×#97C2211yescharacteristic
17C3×#97C2212yes—
18#562C2×C311yescharacteristic
19#491C2×C311yescharacteristic, complemented
20#489C2×C312yescomplemented
21#330C2×C311yescharacteristic, complemented
22#329C2×C312yescomplemented
23C2×C3×Q8—21no—
24C2×C3×Q8D412yes—
25C2×C3×D4—22no—
26C2×C3×D4D412yes—
27C22×C3×C4—22no—
28C2×C3×C8—22no—
29C2×C3×C8—21no—
30C3×C4⋊C4—22no—
31C3×C4⋊C4—22no—
32C3×C4⋊C4—22no—
33C3×C4⋊C4D412yes—
34C3×C22⋊C4—22no—
35C3×C22⋊C4—22no—
36C3×C22⋊C4—22no—
37C3×C42C2311yescharacteristic
38C4⋊Q8C22×C311yescharacteristic
39#125C22×C311yescharacteristic
40#124C22×C312yes—
41#123—22no—
42C22⋊Q8—22no—
43#121—22no—
44#103—22no—
45#103—22no—
46#102—22no—
47#97C22×C311yescharacteristic
48#97C22×C312yes—
49C23×C3—22no—
50C3×Q8—42no—
51C3×Q8—42no—
52C3×D4—42no—
53C3×D4—82no—
54C2×C3×C4—22no—
55C2×C3×C4—22no—
56C2×C3×C4—41no—
57C2×C3×C4—82no—
58C2×C3×C4C2×D411yescharacteristic
59C2×C3×C4C2×D412yes—
60C3×C8—41no—
61C3×C8—42no—
62C2×Q8—21no—
63C2×Q8C3×D412yes—
64C2×D4—22no—
65C2×D4C3×D412yes—
66C22×C4—22no—
67C2×C8—22no—
68C2×C8—21no—
69C4⋊C4—22no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4C3×D412yes—
73C22⋊C4—22no—
74C22⋊C4—22no—
75C22⋊C4—22no—
76C42C23×C311yescharacteristic
77C22×C3—42no—
78C22×C3—42no—
79C22×C3—42no—
80C22×C3#12011yescharacteristic
81C3×C4—21no—
82C3×C4—22no—
83C3×C4—42no—
84C3×C4—42no—
85C3×C4—81no—
86C23—22no—
87Q8—42no—
88Q8—42no—
89D4—42no—
90D4—82no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—41no—
94C2×C4—82no—
95C2×C4C2×C3×D411yescharacteristic
96C2×C4C2×C3×D412yes—
97C8—41no—
98C8—42no—
99C2×C3—82no—
100C2×C3#45411yescharacteristic
101C2×C3#45512yes—
102C22—42no—
103C22—42no—
104C22—42no—
105C22C3×#12011yescharacteristic
106C4—21no—
107C4—22no—
108C4—42no—
109C4—42no—
110C4—81no—
111C3#166211yescharacteristic, direct product factor, Sylow
112C2—82no—
113C2C3×#45411yescharacteristic
114C2C3×#45512yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC42C23×C3
prime radicalC3×#562C2
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×#491C2semidirect
3. C3×#489C2semidirect
4. C3×#330C2semidirect
5. C3×#329C2semidirect
6. #1662C3direct (N × H)
19. #491C2×C3semidirect
20. #489C2×C3semidirect
21. #330C2×C3semidirect
22. #329C2×C3semidirect
111. C3#1662direct (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×#4912
C3×#4892
C3×#3302
C3×#3292
#16623