Finite Group Databasebeta

← C3×#10590022 #13154872

Subgroups — C3×#10590022 #13154872

Order 1728.

176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 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×C8×#157C211yesmaximal, characteristic
2C3×#63542C212yesmaximal
3#10590022C311yesmaximal, characteristic, direct product factor
4C3×C4×#157C411yescharacteristic
5C3×#3787C411yescharacteristic
6C3×C8×C3⋊D3C2211yescharacteristic
7C32×C3⋊C16—22no—
8C8×#157C2×C311yescharacteristic
9C3×D3×C16—62no—
10#63542C2×C312yes—
11C2×C3×#157C811yescharacteristic
12C2×C3×#157C811yescharacteristic
13C3×C4×C3⋊D3C2×C411yescharacteristic
14C3×#622—21no—
15C33×C8D411yescharacteristic
16C3×C3⋊(C3⋊C8)Q811yescharacteristic
17C3×C4⋊C16—91nomaximal
18C4×#157C3×C411yescharacteristic
19#3787C3×C411yescharacteristic
20C8×C3⋊D3C22×C311yescharacteristic
21C3×D3×C8—62no—
22C32×C16—62no—
23C3×C3⋊C16—22no—
24C3×C3⋊C16—22no—
25C3×C3⋊C16—42no—
26C2×C3×C3⋊D3C2×C811yescharacteristic
27C3×#157—21no—
28C3×#157C1612yescomplemented
29C33×C4C4⋊C411yescharacteristic
30C3×C3⋊Dic3M1611yescharacteristic
31C2×C3×C16—92no—
32C3×C4×C8—91no—
33D3×C16—62no—
34C2×#157C3×C811yescharacteristic
35C2×#157C3×C811yescharacteristic
36C4×C3⋊D3C2×C3×C411yescharacteristic
37C3×C4×D3—62no—
38#622—21no—
39C32×C8—22no—
40C32×C8—42no—
41C32×C8C3×D411yescharacteristic
42C3⋊(C3⋊C8)C3×Q811yescharacteristic
43C3×C3⋊C8—62no—
44C4⋊C16—91noSylow
45C2×C33C4⋊C811yescharacteristic
46C3×C3⋊D3C2×C1611yescharacteristic
47C3×C3⋊D3M3211yescharacteristic
48C2×C3×C8—91no—
49C2×C3×C8—91no—
50C3×C42—91no—
51D3×C8—62no—
52C3×C16—62no—
53C3×C16—62no—
54C3×C16—122no—
55C3⋊C16—22no—
56C2×C3⋊D3C2×C3×C811yescharacteristic
57C2×C3×D3—62no—
58#157—21no—
59#157C3×C1612yescomplemented
60C32×C4—22no—
61C32×C4—42no—
62C32×C4C3×C4⋊C411yescharacteristic
63C3⋊Dic3C3×M1611yescharacteristic
64C3×Dic3—62no—
65C2×C16—92no—
66C4×C8—91no—
67C33C4⋊C1611yescharacteristic, complemented, Sylow
68C2×C3×C4—91no—
69C2×C3×C4—91no—
70C2×C3×C4—91no—
71C4×D3—62no—
72C3×C8—22no—
73C3×C8—42no—
74C3×C8—91no—
75C3×C8—181no—
76C3×C8GO+(4,𝔽2)11yescharacteristic
77C3⋊C8—62no—
78C2×C32—22no—
79C2×C32—42no—
80C2×C32C3×C4⋊C811yescharacteristic
81C3⋊D3C2×C3×C1611yescharacteristic
82C3⋊D3C3×M3211yescharacteristic
83C3×D3—62no—
84C3×D3—62no—
85C2×C8—91no—
86C2×C8—91no—
87C42—91no—
88C16—62no—
89C22×C3—91no—
90C2×D3—62no—
91C3×C4—22no—
92C3×C4—42no—
93C3×C4—92no—
94C3×C4—91no—
95C3×C4—181no—
96C3×C4#377311yescharacteristic
97Dic3—62no—
98C32—22no—
99C32—42no—
100C32C3×C4⋊C1611yescharacteristic, complemented
101C2×C4—91no—
102C2×C4—91no—
103C2×C4—91no—
104C8—91no—
105C8—181no—
106C8C3×GO+(4,𝔽2)11yescharacteristic
107C2×C3—22no—
108C2×C3—42no—
109C2×C3—91no—
110C2×C3—91no—
111C2×C3#6373211yescharacteristic
112D3—62no—
113D3—62no—
114C22—91no—
115C4—92no—
116C4—91no—
117C4—181no—
118C4C3×#377311yescharacteristic
119C3—22no—
120C3—42no—
121C3#1059002211yescharacteristic, direct product factor
122C2—91no—
123C2—91no—
124C2C3×#6373211yescharacteristic
subgroupG/H
center, hypercenterC3×C8GO+(4,𝔽2)
commutatorC3⋊D3C2×C3×C16
FrattiniC8C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33C4⋊C8
prime radicalC2×C3×C3⋊D3C2×C8
cosocleC8×C3⋊D3C22×C3
hypocenterC32C3×C4⋊C16
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
3. #10590022C3direct (N × H)
28. C3×#157C16semidirect
59. #157C3×C16semidirect
67. C33C4⋊C16semidirect
100. C32C3×C4⋊C16semidirect
121. C3#10590022direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C8×#1572
C3×#635422
#105900223
C3×C4⋊C169