Finite Group Databasebeta

← #13157222

Subgroups — #13157222

Order 1728.

360 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 49 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×#3748C211yesmaximal, characteristic
2#11723098C212yesmaximal, complemented
3#10590201—31nomaximal
4C22×C33×C4C411yescharacteristic
5C3×#3748—22no—
6C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
8C3⋊#3708—22no—
9C2×#3748D311yescharacteristic
10C2×C3×#208—62no—
11#63773—32no—
12C23×C33C811yescharacteristic
13C2×C33×C4—21no—
14C2×C33×C4—21no—
15C2×C33×C4C811yescharacteristic
16C2×C33×C4C2×C411yescharacteristic
17C3×C3⋊(C3⋊C8)—21no—
18C3×C3⋊(C3⋊C8)D412yes—
19#4553—91nomaximal
20C22×C32×C4—22no—
21C22×C32×C4—42no—
22C22×C32×C4Dic311yescharacteristic
23#3748—22no—
24C2×C3⋊(C3⋊C8)Dic311yescharacteristic
25C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
26C3×#208—68no—
27C2×C3×C3⋊C8—62no—
28C2×C3×C3⋊C8—62no—
29#3708—62no—
30C22×C33—21no—
31C22×C33—41no—
32C22×C33C2×C811yescharacteristic
33C33×C4—21no—
34C33×C4C22⋊C411yescharacteristic
35C33×C4M1611yescharacteristic
36C2×C3×M16—91no—
37C2×#208—62no—
38#812—92no—
39C23×C32—22no—
40C23×C32—42no—
41C23×C32C3⋊C811yescharacteristic
42C2×C32×C4—21no—
43C2×C32×C4—21no—
44C2×C32×C4—24no—
45C2×C32×C4—24no—
46C2×C32×C4—22no—
47C2×C32×C4—22no—
48C2×C32×C4—44no—
49C2×C32×C4—44no—
50C2×C32×C4—42no—
51C2×C32×C4—42no—
52C2×C32×C4C3⋊C811yescharacteristic
53C2×C32×C4C2×Dic311yescharacteristic
54C3⋊(C3⋊C8)—21no—
55C3⋊(C3⋊C8)#6712yes—
56C3×C3⋊C8—64no—
57C3×C3⋊C8—64no—
58#349—271noSylow
59C2×C33—21no—
60C2×C33—41no—
61C2×C33C22⋊C811yescharacteristic
62C22×C3×C4—22no—
63C22×C3×C4—42no—
64C22×C3×C4#15711yescharacteristic
65C3×M16—182no—
66C2×C3×C8—91no—
67C2×C3×C8—91no—
68#208—68no—
69C2×C3⋊C8—62no—
70C2×C3⋊C8—62no—
71C3⋊C16—182no—
72C22×C32—21no—
73C22×C32—24no—
74C22×C32—22no—
75C22×C32—44no—
76C22×C32—42no—
77C22×C32—41no—
78C22×C32—44no—
79C22×C32—48no—
80C22×C32C2×C3⋊C811yescharacteristic
81C32×C4—21no—
82C32×C4—24no—
83C32×C4—22no—
84C32×C4—22no—
85C32×C4—44no—
86C32×C4—42no—
87C32×C4—42no—
88C32×C4#20811yescharacteristic
89C32×C4#21711yescharacteristic
90C2×M16—91no—
91M32—272no—
92C33#34911yescharacteristic, complemented, Sylow
93C23×C3—22no—
94C23×C3—42no—
95C23×C3#62211yescharacteristic
96C2×C3×C4—24no—
97C2×C3×C4—24no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—44no—
103C2×C3×C4—44no—
104C2×C3×C4—42no—
105C2×C3×C4—42no—
106C2×C3×C4#62211yescharacteristic
107C2×C3×C4C2×#15711yescharacteristic
108C3×C8—92no—
109C3×C8—181no—
110C3⋊C8—64no—
111C3⋊C8—64no—
112C2×C32—21no—
113C2×C32—24no—
114C2×C32—22no—
115C2×C32—44no—
116C2×C32—42no—
117C2×C32—44no—
118C2×C32—48no—
119C2×C32—41no—
120C2×C32#83011yescharacteristic
121C22×C4C3⋊#15711yescharacteristic
122M16—182no—
123C2×C8—91no—
124C2×C8—91no—
125C16—542no—
126C22×C3—24no—
127C22×C3—21no—
128C22×C3—22no—
129C22×C3—44no—
130C22×C3—41no—
131C22×C3—44no—
132C22×C3—48no—
133C22×C3—42no—
134C22×C3C2×#62211yescharacteristic
135C3×C4—24no—
136C3×C4—21no—
137C3×C4—22no—
138C3×C4—22no—
139C3×C4—44no—
140C3×C4—42no—
141C3×C4—42no—
142C3×C4#379211yescharacteristic
143C3×C4C22⋊#15711yescharacteristic
144C32—22no—
145C32—42no—
146C32#455311yescharacteristic, complemented
147C23C3⋊#62211yescharacteristic
148C2×C4—21no—
149C2×C4—21no—
150C2×C4C3⋊#62211yescharacteristic
151C2×C4C2×C3⋊#15711yescharacteristic
152C8—92no—
153C8—181no—
154C2×C3—24no—
155C2×C3—21no—
156C2×C3—22no—
157C2×C3—44no—
158C2×C3—44no—
159C2×C3—48no—
160C2×C3—41no—
161C2×C3—42no—
162C2×C3C22⋊#62211yescharacteristic
163C22—21no—
164C22—41no—
165C22C2×C3⋊#62211yescharacteristic
166C4—21no—
167C4#8899311yescharacteristic
168C4#8899411yescharacteristic
169C3—22no—
170C3—42no—
171C3#1059020111yescharacteristic, complemented
172C2—21no—
173C2—41no—
174C2#1172311211yescharacteristic
subgroupG/H
centerC4#88994
commutatorC22×C33C2×C8
FrattiniC2×C4C2×C3⋊#157
FittingC22×C33×C4C4
socleC2×C33C22⋊C8
prime radicalC23×C33C8
cosocleC2×C3×C3⋊(C3⋊C8)C22
hypercenterC22×C4C3⋊#157
hypocenterC33#349
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11723098C2semidirect
92. C33#349semidirect
146. C32#4553semidirect
171. C3#10590201semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
#117230982
#105902013
#45539