Finite Group Databasebeta

← #13157234

Subgroups — #13157234

Order 1728.

368 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 33 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
2#11723099C212yesmaximal
3#10590217—31nomaximal
4C2×C33×Q8C411yescharacteristic, complemented
5C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
6#88994—22no—
7C3×#3752—22no—
8C3×C4×C3⋊Dic3C411yescharacteristic, complemented
9#64156D311yescharacteristic
10C3×#947—62no—
11#63774—32no—
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—
19#4541—91nomaximal
20C2×C32×Q8—22no—
21C2×C32×Q8—42no—
22C2×C32×Q8Dic311yescharacteristic, complemented
23C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
24C2×C3×D12—62no—
25C22⋊#157—62no—
26#3752—22no—
27C4×C3⋊Dic3Dic311yescharacteristic, 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—
39#834—92no—
40C22×C3⋊D3#6712yes—
41C22×C3×D3—64no—
42C2×#157—122no—
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×Dic311yescharacteristic
52C2×C3⋊Dic3—21no—
53C4⋊(C3⋊D3)—41no—
54C2×C3×Dic3—64no—
55C3×D12—124no—
56#354—271noSylow
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×C42—91no—
67#217—182no—
68(C2×D3)⋊C4—68no—
69C4×Dic3—62no—
70C22×C32—22no—
71C22×C32—42no—
72C22×C32#21711yescharacteristic
73C2×C3⋊D3—22no—
74C2×C3⋊D3—42no—
75C2×C3×D3—124no—
76C2×C3×D3—128no—
77#157—242no—
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—272no—
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×C4C2×#15711yescharacteristic
102C2×Dic3—64no—
103C2×Dic3—362no—
104D12—124no—
105C2×C32—21no—
106C2×C32—24no—
107C2×C32—22no—
108C2×C32—44no—
109C2×C32—42no—
110C2×C32#83411yescharacteristic
111C3⋊D3—42no—
112C3×D3—244no—
113C2×Q8C3⋊#15711yescharacteristic, complemented
114C2×D4—91no—
115C22⋊C4—182no—
116C22⋊C4—542no—
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—
132Dic3—124no—
133Dic3—722no—
134C32—22no—
135C32—42no—
136C32#454111yescharacteristic, complemented
137C23—92no—
138Q8—41no—
139D4—361no—
140C2×C4—21no—
141C2×C4—181no—
142C2×C4—1082no—
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—
154C22#8899411yescharacteristic
155C4—21no—
156C4—41no—
157C4—361no—
158C4—2162no—
159C3—22no—
160C3—42no—
161C3#1059021711yescharacteristic, complemented
162C2—21no—
163C2—362no—
164C2#1172311111yescharacteristic
subgroupG/H
centerC2#11723111
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×Q8C4
socleC2×C33#99
prime radical, cosocleC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8C3⋊#157
hypocenterC33#354
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. C2×C33×Q8C4semidirect
8. C3×C4×C3⋊Dic3C4semidirect
22. C2×C32×Q8Dic3semidirect
27. C4×C3⋊Dic3Dic3semidirect
62. C2×C3×Q8#157semidirect
88. C33#354semidirect
113. C2×Q8C3⋊#157semidirect
136. C32#4541semidirect
161. C3#10590217semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117230992
#105902173
#45419