Finite Group Databasebeta

← C3×#10590178 #13157105

Subgroups — C3×#10590178 #13157105

Order 1728.

224 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 52 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×#63773C212yesmaximal
2C3×C8×C3⋊Dic3C211yesmaximal, characteristic
3#10590178C311yesmaximal, characteristic, direct product factor
4C2×C33×C8C411yescharacteristic
5C3×C4×C3⋊Dic3C411yescharacteristic
6C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
7C3×#3708—22no—
8#63773C2×C312yes—
9C8×C3⋊Dic3C2×C311yescharacteristic
10C3×C8×Dic3—62no—
11C2×C33×C4C2×C411yescharacteristic
12C2×C3×C3⋊Dic3—21no—
13C33×C8C812yes—
14C3×C3⋊(C3⋊C8)D411yescharacteristic
15C3×C3⋊(C3⋊C8)Q811yescharacteristic
16C3×#364—91nomaximal
17C2×C32×C8—22no—
18C2×C32×C8—42no—
19C2×C32×C8C3×C411yescharacteristic
20C4×C3⋊Dic3C3×C411yescharacteristic
21C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
22C3×C4×Dic3—62no—
23C2×C3×C3⋊C8—62no—
24#3708—22no—
25C22×C33M1611yescharacteristic
26C33×C4C4⋊C411yescharacteristic
27C33×C4C2×C811yescharacteristic
28C3×C3⋊Dic3—22no—
29C3×M32—92no—
30C3×C4×C8—91no—
31C8×Dic3—62no—
32C2×C32×C4—22no—
33C2×C32×C4—42no—
34C2×C32×C4C2×C3×C411yescharacteristic
35C2×C3⋊Dic3—21no—
36C2×C3×Dic3—64no—
37C32×C8—24no—
38C32×C8—44no—
39C32×C8C3×C812yes—
40C3⋊(C3⋊C8)C3×D411yescharacteristic
41C3⋊(C3⋊C8)C3×Q811yescharacteristic
42C3×C3⋊C8—62no—
43C3×C3⋊C8—62no—
44#364—91noSylow
45C2×C33—21no—
46C2×C33C4⋊C811yescharacteristic
47C2×C3×C8—22no—
48C2×C3×C8—42no—
49C2×C3×C8—91no—
50C2×C3×C8#15711yescharacteristic
51C3×C42—91no—
52C4×Dic3—62no—
53C2×C3⋊C8—62no—
54C3×C16—182no—
55C22×C32—22no—
56C22×C32—42no—
57C22×C32C3×M1611yescharacteristic
58C32×C4—22no—
59C32×C4—22no—
60C32×C4—42no—
61C32×C4—42no—
62C32×C4C3×C4⋊C411yescharacteristic
63C32×C4C2×C3×C811yescharacteristic
64C3⋊Dic3—22no—
65C3×Dic3—68no—
66M32—92no—
67C4×C8—91no—
68C33#36411yescharacteristic, complemented, Sylow
69C2×C3×C4—22no—
70C2×C3×C4—42no—
71C2×C3×C4—181no—
72C2×C3×C4C2×#15711yescharacteristic
73C2×Dic3—64no—
74C3×C8—24no—
75C3×C8—44no—
76C3×C8—91no—
77C3×C8—91no—
78C3×C8#62212yes—
79C3⋊C8—62no—
80C3⋊C8—62no—
81C2×C32—21no—
82C2×C32—24no—
83C2×C32—22no—
84C2×C32—44no—
85C2×C32—42no—
86C2×C32C3×C4⋊C811yescharacteristic
87C2×C8—91no—
88C2×C8C3×#15711yescharacteristic
89C42—91no—
90C16—182no—
91C22×C3—22no—
92C22×C3—42no—
93C22×C3#378811yescharacteristic
94C3×C4—22no—
95C3×C4—22no—
96C3×C4—42no—
97C3×C4—42no—
98C3×C4—182no—
99C3×C4C4⋊#15711yescharacteristic
100C3×C4C2×#62211yescharacteristic
101Dic3—68no—
102C32—22no—
103C32—42no—
104C32C3×#36411yescharacteristic, complemented
105C2×C4—181no—
106C2×C4C2×C3×#15711yescharacteristic
107C8—91no—
108C8—91no—
109C8C3×#62212yes—
110C2×C3—24no—
111C2×C3—21no—
112C2×C3—22no—
113C2×C3—44no—
114C2×C3—42no—
115C2×C3C4⋊#62211yescharacteristic
116C22C3×#378811yescharacteristic
117C4—182no—
118C4C3×C4⋊#15711yescharacteristic
119C4C2×C3×#62211yescharacteristic
120C3—22no—
121C3—42no—
122C3#1059017811yescharacteristic, direct product factor
123C2—21no—
124C2C3×C4⋊#62211yescharacteristic
subgroupG/H
centerC3×C4C4⋊#157
commutatorC32×C4C2×C3×C8
FrattiniC2×C4C2×C3×#157
FittingC2×C33×C8C4
socleC2×C33C4⋊C8
prime radicalC22×C33M16
cosocleC2×C3⋊(C3⋊C8)C22×C3
hypercenterC2×C3×C8#157
hypocenterC32C3×#364
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. #10590178C3direct (N × H)
68. C33#364semidirect
104. C32C3×#364semidirect
122. C3#10590178direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637732
C3×C8×C3⋊Dic32
#105901783
C3×#3649