Finite Group Databasebeta

← C3×#1659 #72374

Subgroups — C3×#1659 #72374

Order 384.

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

normal subgroups only

This group has 228 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×H(3,𝔽4)C211yesmaximal, characteristic
2C3×#497C211yesmaximal, characteristic, complemented
3C3×#489C212yesmaximal, complemented
4C3×#329C211yesmaximal, characteristic, complemented
5C3×#329C212yesmaximal, complemented
6#1659C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×Q16—21no—
8C2×C3×SD16—21no—
9C3×C4⋊Q8C2211yescharacteristic, complemented
10C3×#124—22no—
11C3×#124—21no—
12C3×#124C2212yes—
13C3×#124C2211yescharacteristic
14C3×#120—22no—
15C3×#120—21no—
16C3×#103—22no—
17C3×#102—22no—
18C3×#97C2212yescomplemented
19C3×#97C2211yescharacteristic, complemented
20H(3,𝔽4)C2×C311yescharacteristic
21#497C2×C311yescharacteristic, complemented
22#489C2×C312yescomplemented
23#329C2×C311yescharacteristic, complemented
24#329C2×C312yescomplemented
25C24×C3—21no—
26C2×C3×Q8—21no—
27C2×C3×Q8D412yes—
28C2×C3×Q8D411yescharacteristic
29C2×C3×D4—21no—
30C2×C3×D4—22no—
31C2×C3×D4D412yes—
32C2×C3×D4D411yescharacteristic
33C3×Q16—42no—
34C3×SD16—42no—
35C2×C3×C8—21no—
36C2×C3×C8—22no—
37C3×C4⋊C4—22no—
38C3×C22⋊C4—22no—
39C3×C22⋊C4—21no—
40C3×C22⋊C4—22no—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—22no—
43C3×C42—21no—
44C3×C42C2311yescharacteristic
45C2×Q16—21no—
46C2×SD16—21no—
47C4⋊Q8C22×C311yescharacteristic, complemented
48#124—22no—
49#124—21no—
50#124C22×C312yes—
51#124C22×C311yescharacteristic
52#120—22no—
53#120—21no—
54#103—22no—
55#102—22no—
56#97C22×C312yescomplemented
57#97C22×C311yescharacteristic, complemented
58C23×C3—21no—
59C23×C3—22no—
60C23×C3—81no—
61C23×C3—82no—
62C3×Q8—41no—
63C3×Q8—42no—
64C3×Q8—41no—
65C3×Q8—41no—
66C3×D4—41no—
67C3×D4—42no—
68C3×D4—81no—
69C3×D4—82no—
70C2×C3×C4—21no—
71C2×C3×C4—22no—
72C2×C3×C4—41no—
73C2×C3×C4C2×D412yes—
74C2×C3×C4C2×D411yescharacteristic
75C3×C8—22no—
76C3×C8—42no—
77C24—21no—
78C2×Q8—21no—
79C2×Q8C3×D412yes—
80C2×Q8C3×D411yescharacteristic
81C2×D4—21no—
82C2×D4—22no—
83C2×D4C3×D412yes—
84C2×D4C3×D411yescharacteristic
85Q16—42no—
86SD16—42no—
87C2×C8—21no—
88C2×C8—22no—
89C4⋊C4—22no—
90C22⋊C4—22no—
91C22⋊C4—21no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C22⋊C4—22no—
95C42—21no—
96C42C23×C311yescharacteristic
97C22×C3—41no—
98C22×C3—42no—
99C22×C3—42no—
100C22×C3—42no—
101C22×C3—42no—
102C22×C3—84no—
103C22×C3#12011yescharacteristic
104C3×C4—21no—
105C3×C4—22no—
106C3×C4—41no—
107C3×C4—42no—
108C3×C4—81no—
109C23—21no—
110C23—22no—
111C23—81no—
112C23—82no—
113Q8—41no—
114Q8—42no—
115Q8—41no—
116Q8—41no—
117D4—41no—
118D4—42no—
119D4—81no—
120D4—82no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4C2×C3×D412yes—
125C2×C4C2×C3×D411yescharacteristic
126C8—22no—
127C8—42no—
128C2×C3—81no—
129C2×C3—82no—
130C2×C3#45311yescharacteristic
131C2×C3#45512yes—
132C22—41no—
133C22—42no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—84no—
138C22C3×#12011yescharacteristic
139C4—21no—
140C4—22no—
141C4—41no—
142C4—42no—
143C4—81no—
144C3#165911yescharacteristic, direct product factor, Sylow
145C2—81no—
146C2—82no—
147C2C3×#45311yescharacteristic
148C2C3×#45512yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC42C23×C3
prime radicalC3×H(3,𝔽4)C2
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×#497C2semidirect
3. C3×#489C2semidirect
4. C3×#329C2semidirect
5. C3×#329C2semidirect
6. #1659C3direct (N × H)
9. C3×C4⋊Q8C22semidirect
18. C3×#97C22semidirect
19. C3×#97C22semidirect
21. #497C2×C3semidirect
22. #489C2×C3semidirect
23. #329C2×C3semidirect
24. #329C2×C3semidirect
47. C4⋊Q8C22×C3semidirect
56. #97C22×C3semidirect
57. #97C22×C3semidirect
144. C3#1659direct (N × H)

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽4)2
C3×#4972
C3×#4892
C3×#3292
#16593