Finite Group Databasebeta

← C3×#10590217 #13157142

Subgroups — C3×#10590217 #13157142

Order 1728.

368 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 36 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×#64156C211yesmaximal, characteristic
2C3×#63774C212yesmaximal
3#10590217C311yesmaximal, characteristic, direct product factor
4C2×C33×Q8C411yescharacteristic, complemented
5C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
6C3×C22⋊#157—22no—
7C3×#3752—22no—
8C3×C4×C3⋊Dic3C411yescharacteristic, complemented
9#64156C2×C311yescharacteristic
10C3×#947—62no—
11#63774C2×C312yes—
12C22×C3×C3⋊D3D412yes—
13C2×C3×#157—42no—
14C33×Q8—41no—
15C2×C33×C4—21no—
16C2×C33×C4C2×C411yescharacteristic
17C2×C3×C3⋊Dic3—21no—
18C3×C4⋊(C3⋊D3)—41no—
19C3×#354—91nomaximal
20C2×C32×Q8—22no—
21C2×C32×Q8—42no—
22C2×C32×Q8C3×C411yescharacteristic, complemented
23C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
24C2×C3×D12—62no—
25C22⋊#157—22no—
26#3752—22no—
27C4×C3⋊Dic3C3×C411yescharacteristic, complemented
28C3×(C2×D3)⋊C4—68no—
29C3×C4×Dic3—62no—
30C22×C33C22⋊C411yescharacteristic
31C2×C3×C3⋊D3—22no—
32C2×C3×C3⋊D3—42no—
33C3×#157—82no—
34C33×C4—21no—
35C33×C4—41no—
36C3×C3⋊Dic3—41no—
37C3×#124—91no—
38#947—62no—
39C3×#99—92no—
40C22×C3⋊D3C3×D412yes—
41C22×C3×D3—64no—
42C2×#157—42no—
43C32×Q8—41no—
44C32×Q8—44no—
45C32×Q8—48no—
46C2×C32×C4—21no—
47C2×C32×C4—24no—
48C2×C32×C4—22no—
49C2×C32×C4—44no—
50C2×C32×C4—42no—
51C2×C32×C4C2×C3×C411yescharacteristic
52C2×C3⋊Dic3—21no—
53C4⋊(C3⋊D3)—41no—
54C2×C3×Dic3—64no—
55C3×D12—124no—
56#354—91noSylow
57C2×C33—21no—
58C2×C33#9911yescharacteristic
59C3×C3⋊D3—42no—
60C2×C3×Q8—22no—
61C2×C3×Q8—42no—
62C2×C3×Q8#15711yescharacteristic, complemented
63C2×C3×D4—91no—
64C2×D12—62no—
65C3×C22⋊C4—182no—
66C3×C22⋊C4—182no—
67C3×C42—91no—
68(C2×D3)⋊C4—68no—
69C4×Dic3—62no—
70C22×C32—22no—
71C22×C32—42no—
72C22×C32C3×C22⋊C411yescharacteristic
73C2×C3⋊D3—22no—
74C2×C3⋊D3—42no—
75C2×C3×D3—124no—
76C2×C3×D3—128no—
77#157—82no—
78C32×C4—21no—
79C32×C4—24no—
80C32×C4—44no—
81C32×C4—44no—
82C32×C4—48no—
83C32×C4—41no—
84C3⋊Dic3—41no—
85C3×Dic3—124no—
86#124—91no—
87#99—92no—
88C33#35411yescharacteristic, complemented, Sylow
89C23×C3—92no—
90C22×D3—64no—
91C3×Q8—41no—
92C3×Q8—44no—
93C3×Q8—48no—
94C3×D4—361no—
95C2×C3×C4—24no—
96C2×C3×C4—21no—
97C2×C3×C4—22no—
98C2×C3×C4—44no—
99C2×C3×C4—42no—
100C2×C3×C4—181no—
101C2×C3×C4—362no—
102C2×C3×C4C2×#15711yescharacteristic
103C2×Dic3—64no—
104D12—124no—
105C2×C32—21no—
106C2×C32—24no—
107C2×C32—22no—
108C2×C32—44no—
109C2×C32—42no—
110C2×C32C3×#9911yescharacteristic
111C3⋊D3—42no—
112C3×D3—244no—
113C2×Q8C3×#15711yescharacteristic, complemented
114C2×D4—91no—
115C22⋊C4—182no—
116C22⋊C4—182no—
117C42—91no—
118C22×C3—22no—
119C22×C3—42no—
120C22×C3—182no—
121C22×C3—362no—
122C22×C3C22⋊#15711yescharacteristic
123C2×D3—124no—
124C2×D3—128no—
125C3×C4—24no—
126C3×C4—21no—
127C3×C4—44no—
128C3×C4—44no—
129C3×C4—48no—
130C3×C4—41no—
131C3×C4—361no—
132C3×C4—722no—
133Dic3—124no—
134C32—22no—
135C32—42no—
136C32C3×#35411yescharacteristic, complemented
137C23—92no—
138Q8—41no—
139D4—361no—
140C2×C4—21no—
141C2×C4—181no—
142C2×C4—362no—
143C2×C4C2×C3×#15711yescharacteristic
144C2×C3—24no—
145C2×C3—21no—
146C2×C3—22no—
147C2×C3—44no—
148C2×C3—42no—
149C2×C3—362no—
150C2×C3#6378611yescharacteristic
151D3—244no—
152C22—182no—
153C22—362no—
154C22C3×C22⋊#15711yescharacteristic
155C4—21no—
156C4—41no—
157C4—361no—
158C4—722no—
159C3—22no—
160C3—42no—
161C3#1059021711yescharacteristic, direct product factor
162C2—21no—
163C2—362no—
164C2C3×#6378611yescharacteristic
subgroupG/H
centerC2×C3#63786
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×C4C2×C3×#157
FittingC2×C33×Q8C4
socleC2×C33#99
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C4⋊(C3⋊D3)C22×C3
hypercenterC2×C3×Q8#157
hypocenterC32C3×#354
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
3. #10590217C3direct (N × H)
4. C2×C33×Q8C4semidirect
8. C3×C4×C3⋊Dic3C4semidirect
22. C2×C32×Q8C3×C4semidirect
27. C4×C3⋊Dic3C3×C4semidirect
62. C2×C3×Q8#157semidirect
88. C33#354semidirect
113. C2×Q8C3×#157semidirect
136. C32C3×#354semidirect
161. C3#10590217direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#637742
#105902173
C3×#3549