Finite Group Databasebeta

← C3×#73203 #11919476

Subgroups — C3×#73203 #11919476

Order 1152.

240 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 46 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×#395⋊C3C213yesmaximal
2C3×#2811C311yesmaximal, characteristic, complemented
3#73203C313yesmaximal, direct product factor
4C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
5C3×Q8×A4—41nomaximal
6C3×#395—31no—
7C3×#395C2×C313yes—
8C3×#391—33no—
9#395⋊C3C2×C319yes—
10C3×C4×A4—43no—
11#2811C3211yescharacteristic, complemented, Sylow
12C22×C3×Q8—31no—
13C22×C3×Q8A411yescharacteristic
14C22×C3×Q8C22×C311yescharacteristic
15C22⋊SL(2,𝔽3)C22×C313yes—
16Q8×A4—43no—
17C2×C3×C4⋊C4—33no—
18C2×C3×C42—31no—
19C3×#95—33no—
20C3×#95—33no—
21C2×C3×A4—41no—
22C32×Q8—161no—
23C3×SL(2,𝔽3)—161no—
24#395—31no—
25#395C2×C3213yes—
26#391—33no—
27C2×C3×Q8—34no—
28C2×C3×Q8—121no—
29C2×C3×Q8—121no—
30C2×C3×Q8—121no—
31C2×C3×Q8—121no—
32C22×C3×C4—33no—
33C22×C3×C4—31no—
34C22×C3×C4C2×A413yes—
35C4×A4—49no—
36C3×C4⋊C4—123no—
37C3×C42—121no—
38C3×A4—41no—
39C32×C4—163no—
40C22×Q8—31no—
41C22×Q8C3×A411yescharacteristic
42C22×Q8C22×C3211yescharacteristic
43C2×C4⋊C4—33no—
44C2×C42—31no—
45#95—33no—
46#95—33no—
47C23×C3C22×A411yescharacteristic
48C2×A4—43no—
49C3×Q8—41no—
50C3×Q8—41no—
51C3×Q8—121no—
52C3×Q8—124no—
53C3×Q8—121no—
54C3×Q8—163no—
55C2×C3×C4—36no—
56C2×C3×C4—63no—
57C2×C3×C4—63no—
58C2×C3×C4—63no—
59C2×C3×C4—63no—
60SL(2,𝔽3)—163no—
61C2×C32—161no—
62C2×Q8—34no—
63C2×Q8—121no—
64C2×Q8—121no—
65C2×Q8—121no—
66C2×Q8—121no—
67C22×C4—33no—
68C22×C4—31no—
69C22×C4C2×C3×A413yes—
70C4⋊C4—123no—
71C42—121no—
72C22×C3—31no—
73C22×C3—31no—
74C22×C3#99411yescharacteristic
75A4—43no—
76C3×C4—43no—
77C3×C4—66no—
78C3×C4—121no—
79C3×C4—169no—
80C32—161noSylow
81C23C22×C3×A411yescharacteristic
82Q8—41no—
83Q8—41no—
84Q8—121no—
85Q8—124no—
86Q8—121no—
87C2×C4—36no—
88C2×C4—63no—
89C2×C4—63no—
90C2×C4—63no—
91C2×C4—63no—
92C2×C3—31no—
93C2×C3—31no—
94C2×C3—163no—
95C2×C3#545011yescharacteristic
96C22—31no—
97C22—31no—
98C22C3×#99411yescharacteristic
99C4—43no—
100C4—66no—
101C4—121no—
102C3—163no—
103C3#7320311yescharacteristic, direct product factor
104C2—31no—
105C2—31no—
106C2C3×#545011yescharacteristic
subgroupG/H
center, hypercenterC2×C3#5450
commutator, cosocle, hypocenterC22×Q8C22×C32
FrattiniC23C22×C3×A4
FittingC3×#2811C3
socleC23×C3C22×A4
prime radicalC3×C22⋊SL(2,𝔽3)C22
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3×#2811C3semidirect
3. #73203C3direct (N × H)
11. #2811C32semidirect
103. C3#73203direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#395⋊C32
C3×#28113
#732033
C3×Q8×A44