Finite Group Databasebeta

← #13157196

Subgroups — #13157196

Order 1728.

296 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C2×C3×#3743C211yesmaximal, characteristic
2#11723098C212yesmaximal, complemented
3#10590180—31nomaximal
4C2×C3×C4×C3⋊D3C411yescharacteristic
5C2×C33×C8C411yescharacteristic
6C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
7C3×#3743—22no—
8C3⋊#3708—22no—
9C2×#3743D311yescharacteristic
10C2×C3×ΓL(1,𝔽25)—62no—
11#63773—32no—
12C22×C3×C3⋊D3C811yescharacteristic
13C2×C33×C4C2×C411yescharacteristic
14C2×C3×C3⋊Dic3C811yescharacteristic
15C3×C4×C3⋊D3—21no—
16C3×C4×C3⋊D3—21no—
17C33×C8—21no—
18C3×C3⋊(C3⋊C8)D412yes—
19#4553—91nomaximal
20C2×C4×C3⋊D3Dic311yescharacteristic
21C2×C3×C4×D3—62no—
22C2×C32×C8—22no—
23C2×C32×C8—42no—
24C2×C32×C8Dic311yescharacteristic
25C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
26#3743—22no—
27C2×C3×C3⋊C8—62no—
28C3×ΓL(1,𝔽25)—68no—
29#3708—62no—
30C22×C33C2×C811yescharacteristic
31C2×C3×C3⋊D3—21no—
32C2×C3×C3⋊D3—41no—
33C33×C4C22⋊C411yescharacteristic
34C33×C4M1611yescharacteristic
35C3×C3⋊Dic3—21no—
36C2×C3×M16—91no—
37C2×ΓL(1,𝔽25)—62no—
38#812—92no—
39C22×C3⋊D3C3⋊C811yescharacteristic
40C22×C3×D3—62no—
41C2×C32×C4—22no—
42C2×C32×C4—42no—
43C2×C32×C4C2×Dic311yescharacteristic
44C2×C3⋊Dic3C3⋊C811yescharacteristic
45C4×C3⋊D3—21no—
46C4×C3⋊D3—21no—
47C2×C3×Dic3—62no—
48C3×C4×D3—64no—
49C3×C4×D3—64no—
50C32×C8—21no—
51C32×C8—24no—
52C32×C8—44no—
53C3⋊(C3⋊C8)#6712yes—
54C3×C3⋊C8—64no—
55#349—271noSylow
56C2×C33—21no—
57C2×C33C22⋊C811yescharacteristic
58C3×C3⋊D3—41no—
59C22×C3×C4—91no—
60C2×C4×D3—62no—
61C3×M16—182no—
62C2×C3×C8—22no—
63C2×C3×C8—42no—
64C2×C3×C8—91no—
65C2×C3×C8#15711yescharacteristic
66C2×C3⋊C8—62no—
67ΓL(1,𝔽25)—68no—
68C3⋊C16—182no—
69C22×C32—22no—
70C22×C32—42no—
71C22×C32C2×C3⋊C811yescharacteristic
72C2×C3⋊D3—21no—
73C2×C3⋊D3—41no—
74C2×C3×D3—64no—
75C2×C3×D3—124no—
76C32×C4—22no—
77C32×C4—22no—
78C32×C4—42no—
79C32×C4—42no—
80C32×C4#20811yescharacteristic
81C32×C4#21711yescharacteristic
82C3⋊Dic3—21no—
83C3×Dic3—64no—
84C2×M16—91no—
85M32—272no—
86C33#34911yescharacteristic, complemented, Sylow
87C23×C3—91no—
88C22×D3—62no—
89C2×C3×C4—22no—
90C2×C3×C4—42no—
91C2×C3×C4—91no—
92C2×C3×C4—181no—
93C2×C3×C4—181no—
94C2×C3×C4C2×#15711yescharacteristic
95C2×Dic3—62no—
96C4×D3—64no—
97C4×D3—64no—
98C3×C8—24no—
99C3×C8—21no—
100C3×C8—44no—
101C3×C8—92no—
102C3⋊C8—64no—
103C2×C32—21no—
104C2×C32—24no—
105C2×C32—22no—
106C2×C32—44no—
107C2×C32—42no—
108C2×C32#83011yescharacteristic
109C3⋊D3—41no—
110C3×D3—124no—
111C22×C4—91no—
112M16—182no—
113C2×C8—91no—
114C2×C8C3⋊#15711yescharacteristic
115C16—542no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3—181no—
119C22×C3—361no—
120C22×C3#378711yescharacteristic
121C2×D3—64no—
122C2×D3—124no—
123C3×C4—22no—
124C3×C4—22no—
125C3×C4—42no—
126C3×C4—42no—
127C3×C4—181no—
128C3×C4#378811yescharacteristic
129C3×C4C22⋊#15711yescharacteristic
130Dic3—64no—
131C32—22no—
132C32—42no—
133C32#455311yescharacteristic, complemented
134C23—91no—
135C2×C4—91no—
136C2×C4—181no—
137C2×C4—181no—
138C2×C4C2×C3⋊#15711yescharacteristic
139C8—21no—
140C8—92no—
141C2×C3—24no—
142C2×C3—21no—
143C2×C3—22no—
144C2×C3—44no—
145C2×C3—42no—
146C2×C3—361no—
147C2×C3#6377811yescharacteristic
148D3—124no—
149C22—181no—
150C22—361no—
151C22#8898811yescharacteristic
152C4—181no—
153C4#8898911yescharacteristic
154C4#8899411yescharacteristic
155C3—22no—
156C3—42no—
157C3#1059018011yescharacteristic, complemented
158C2—21no—
159C2—361no—
160C2#1172310311yescharacteristic
subgroupG/H
centerC4#88994
commutatorC22×C33C2×C8
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×C8C4
socleC2×C33C22⋊C8
prime radicalC22×C3×C3⋊D3C8
cosocleC2×C3×C3⋊(C3⋊C8)C22
hypercenterC2×C8C3⋊#157
hypocenterC33#349
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723098C2semidirect
86. C33#349semidirect
133. C32#4553semidirect
157. C3#10590180semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117230982
#105901803
#45539