Finite Group Databasebeta

← #13157201

Subgroups — #13157201

Order 1728.

368 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#11724514C212yesmaximal
2C2×C3×#3743C211yesmaximal, characteristic
3#10590184—31nomaximal
4C2×C3×C4×C3⋊D3C2211yescharacteristic
5#88994—22no—
6C4⋊(C3⋊#157)—22no—
7C4×C3⋊#157C414yescomplemented
8C2×C33×C8C411yescharacteristic, complemented
9C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
10C3×#3743—22no—
11#64286—32no—
12C2×#3743D311yescharacteristic
13C2×C3×ΓL(1,𝔽25)—62no—
14C22×C3×C3⋊D3D411yescharacteristic
15C2×C3⋊#157—24no—
16C2×C33×C4C2×C411yescharacteristic
17C2×C3×C3⋊Dic3Q811yescharacteristic
18C3×C4×C3⋊D3C2×C412yes—
19C3×C4×C3⋊D3D412yes—
20C33×C8—21no—
21C3×C3⋊(C3⋊C8)—21no—
22#4557—91nomaximal
23C2×C4×C3⋊D3C2×D311yescharacteristic
24C2×C3×C4×D3—62no—
25C22⋊#157—62no—
26C4⋊#157—62no—
27C4×#157—34no—
28C2×C32×C8—22no—
29C2×C32×C8—42no—
30C2×C32×C8Dic311yescharacteristic, complemented
31C2×C3⋊(C3⋊C8)Dic311yescharacteristic, complemented
32#3743—22no—
33C2×C3×C3⋊C8—62no—
34C3×ΓL(1,𝔽25)—68no—
35C22×C33C4⋊C411yescharacteristic
36C2×C3×C3⋊D3—41no—
37C2×C3×C3⋊D3C22⋊C412yes—
38C3⋊#157—44no—
39C33×C4C4211yescharacteristic
40C33×C4C22⋊C411yescharacteristic
41C3×C3⋊Dic3C4⋊C412yes—
42C2×C3×M16—91no—
43#926—92no—
44C2×ΓL(1,𝔽25)—62no—
45C22×C3⋊D3D1211yescharacteristic
46C22×C3×D3—62no—
47C2×#157—64no—
48C2×C32×C4—22no—
49C2×C32×C4—42no—
50C2×C32×C4C2×Dic311yescharacteristic
51C2×C3⋊Dic3Dic611yescharacteristic
52C4×C3⋊D3C4×D312yes—
53C4×C3⋊D3#6712yes—
54C2×C3×Dic3—62no—
55C3×C4×D3—64no—
56C3×C4×D3—64no—
57C32×C8—21no—
58C32×C8—24no—
59C32×C8—44no—
60C3⋊(C3⋊C8)—21no—
61C3×C3⋊C8—64no—
62H(3,ℤ/4)—271noSylow
63C2×C33—21no—
64C2×C33#9511yescharacteristic
65C3×C3⋊D3—22no—
66C22×C3×C4—91no—
67C2×C4×D3—62no—
68C3×M16—182no—
69C2×C3×C8—22no—
70C2×C3×C8—42no—
71C2×C3×C8—91no—
72C2×C3×C8#15711yescharacteristic, complemented
73#217—182no—
74C4⋊Dic3—182no—
75C4×Dic3—94no—
76C2×C3⋊C8—62no—
77ΓL(1,𝔽25)—68no—
78C22×C32—22no—
79C22×C32—42no—
80C22×C32C4⋊Dic311yescharacteristic
81C2×C3⋊D3—41no—
82C2×C3⋊D3(C2×D3)⋊C412yes—
83C2×C3×D3—64no—
84C2×C3×D3—124no—
85#157—124no—
86C32×C4—22no—
87C32×C4—22no—
88C32×C4—42no—
89C32×C4—42no—
90C32×C4C4×Dic311yescharacteristic
91C32×C4#21711yescharacteristic
92C3⋊Dic3Dic3⋊C412yes—
93C3×Dic3—64no—
94C2×M16—91no—
95#117—272no—
96C33H(3,ℤ/4)11yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C22×D3—62no—
99C2×C3×C4—22no—
100C2×C3×C4—42no—
101C2×C3×C4—91no—
102C2×C3×C4—92no—
103C2×C3×C4—92no—
104C2×C3×C4C2×#15711yescharacteristic
105C2×Dic3—62no—
106C2×Dic3—184no—
107C4×D3—64no—
108C4×D3—64no—
109C3×C8—24no—
110C3×C8—21no—
111C3×C8—44no—
112C3×C8—181no—
113C3⋊C8—64no—
114C2×C32—21no—
115C2×C32—24no—
116C2×C32—22no—
117C2×C32—44no—
118C2×C32—42no—
119C2×C32C3⋊#9511yescharacteristic
120C3⋊D3—22no—
121C3×D3—124no—
122C22×C4—91no—
123M16—182no—
124C2×C8—91no—
125C2×C8C3⋊#15711yescharacteristic, complemented
126C4⋊C4—542no—
127C22⋊C4—542no—
128C42—274no—
129C22×C3—22no—
130C22×C3—42no—
131C22×C3—92no—
132C22×C3—361no—
133C22×C3C4⋊#15711yescharacteristic
134C2×D3—64no—
135C2×D3—124no—
136C3×C4—22no—
137C3×C4—22no—
138C3×C4—42no—
139C3×C4—42no—
140C3×C4—92no—
141C3×C4C4×#15711yescharacteristic
142C3×C4C22⋊#15711yescharacteristic
143Dic3—64no—
144Dic3—364no—
145C32—22no—
146C32—42no—
147C32#455711yescharacteristic, complemented
148C23—91no—
149C2×C4—91no—
150C2×C4—92no—
151C2×C4—92no—
152C2×C4—544no—
153C2×C4C2×C3⋊#15711yescharacteristic
154C8—21no—
155C8—181no—
156C2×C3—24no—
157C2×C3—21no—
158C2×C3—22no—
159C2×C3—44no—
160C2×C3—42no—
161C2×C3—182no—
162C2×C3#6378111yescharacteristic
163D3—124no—
164C22—92no—
165C22—361no—
166C22C4⋊(C3⋊#157)11yescharacteristic
167C4—92no—
168C4—1084no—
169C4C4×C3⋊#15711yescharacteristic
170C4#8899411yescharacteristic
171C3—22no—
172C3—42no—
173C3#1059018411yescharacteristic, complemented
174C2—21no—
175C2—182no—
176C2#1172310611yescharacteristic
subgroupG/H
centerC4C4×C3⋊#157
commutatorC33×C4C42
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×C8C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypercenterC2×C8C3⋊#157
hypocenterC33H(3,ℤ/4)
LCSG ▷ C33×C4 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
7. C4×C3⋊#157C4semidirect
8. C2×C33×C8C4semidirect
9. C2×C3×C3⋊(C3⋊C8)C4semidirect
30. C2×C32×C8Dic3semidirect
31. C2×C3⋊(C3⋊C8)Dic3semidirect
72. C2×C3×C8#157semidirect
96. C33H(3,ℤ/4)semidirect
125. C2×C8C3⋊#157semidirect
147. C32#4557semidirect
173. C3#10590184semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×C3×#37432
#105901843
#45579