Finite Group Databasebeta

← C3×#2060 #72775

Subgroups — C3×#2060 #72775

Order 384.

246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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×#544C211yesmaximal, characteristic
2C3×#428C212yesmaximal
3C3×#384C211yesmaximal, characteristic
4C3×#343C211yesmaximal, characteristic
5C3×#339C212yesmaximal
6#2060C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×M16C2212yes—
8C3×C4⋊Q8—22no—
9C3×#125—22no—
10C3×C22⋊Q8—22no—
11C3×#117C2212yes—
12C2×C3×C4⋊C4—22no—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16C3×#107—22no—
17C3×#106—22no—
18#544C2×C311yescharacteristic
19#428C2×C312yes—
20#384C2×C311yescharacteristic
21#343C2×C311yescharacteristic
22#339C2×C312yes—
23C2×C3×Q8—22no—
24C22×C3×C4—21no—
25C22×C3×C4—21no—
26C22×C3×C4D412yes—
27C22×C3×C4C2311yescharacteristic
28C3×M16—22no—
29C3×M16Q814yescomplemented
30C2×C3×C8—22no—
31C3×C4⋊C4—22no—
32C3×C4⋊C4—24no—
33C3×C4⋊C4—22no—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—44no—
36C3×C4⋊C4D412yes—
37C3×C4⋊C4D412yes—
38C3×C22⋊C4—22no—
39C3×C42—22no—
40C3×C42—21no—
41C3×C42Q812yescomplemented
42C2×M16C22×C312yes—
43C4⋊Q8—22no—
44#125—22no—
45C22⋊Q8—22no—
46#117C22×C312yes—
47C2×C4⋊C4—22no—
48C2×C4⋊C4C22×C311yescharacteristic
49C2×C4⋊C4C22×C311yescharacteristic
50C2×C42C22×C311yescharacteristic
51#107—22no—
52#106—22no—
53C23×C3C2×D411yescharacteristic
54C3×Q8—82no—
55C2×C3×C4—22no—
56C2×C3×C4—22no—
57C2×C3×C4—22no—
58C2×C3×C4—22no—
59C2×C3×C4—22no—
60C2×C3×C4—22no—
61C2×C3×C4—42no—
62C2×C3×C4—42no—
63C2×C3×C4—41no—
64C2×C3×C4C2×D412yes—
65C2×C3×C4C2×Q812yes—
66C2×C3×C4C2×Q811yescharacteristic
67C2×C3×C4#4111yescharacteristic
68C3×C8—24no—
69C2×Q8—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4C3×D412yes—
73C22×C4C23×C311yescharacteristic
74M16—22no—
75M16C3×Q814yescomplemented
76C2×C8—22no—
77C4⋊C4—22no—
78C4⋊C4—24no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—44no—
82C4⋊C4C3×D412yes—
83C4⋊C4C3×D412yes—
84C22⋊C4—22no—
85C42—22no—
86C42—21no—
87C42C3×Q812yescomplemented
88C22×C3—21no—
89C22×C3—21no—
90C22×C3#12011yescharacteristic
91C22×C3C22⋊Q811yescharacteristic
92C22×C3C4⋊Q811yescharacteristic
93C3×C4—24no—
94C3×C4—42no—
95C3×C4—42no—
96C3×C4—42no—
97C3×C4—81no—
98C3×C4C22⋊Q812yes—
99C3×C4C4⋊Q812yes—
100C23C2×C3×D411yescharacteristic
101Q8—82no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4C2×C3×D412yes—
112C2×C4C2×C3×Q812yes—
113C2×C4C2×C3×Q811yescharacteristic
114C2×C4C3×#4111yescharacteristic
115C8—24no—
116C2×C3—21no—
117C2×C3—21no—
118C2×C3#39511yescharacteristic
119C2×C3#45311yescharacteristic
120C2×C3#45611yescharacteristic
121C22—21no—
122C22—21no—
123C22C3×#12011yescharacteristic
124C22C3×C22⋊Q811yescharacteristic
125C22C3×C4⋊Q811yescharacteristic
126C4—24no—
127C4—42no—
128C4—42no—
129C4—42no—
130C4—81no—
131C4C3×C22⋊Q812yes—
132C4C3×C4⋊Q812yes—
133C3#206011yescharacteristic, direct product factor, Sylow
134C2—21no—
135C2—21no—
136C2C3×#39511yescharacteristic
137C2C3×#45311yescharacteristic
138C2C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC23×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
6. #2060C3direct (N × H)
29. C3×M16Q8semidirect
41. C3×C42Q8semidirect
75. M16C3×Q8semidirect
87. C42C3×Q8semidirect
133. C3#2060direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5442
C3×#4282
C3×#3842
C3×#3432
C3×#3392
#20603