Finite Group Databasebeta

← #13157193

Subgroups — #13157193

Order 1728.

328 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 57 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×C8×C3⋊D3C211yesmaximal, characteristic
2C2×C3⋊#3708C212yesmaximal, complemented
3#10590176—31nomaximal
4C2×C3×C4×C3⋊D3C411yescharacteristic
5C2×C33×C8C411yescharacteristic
6C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
7C3×C8×C3⋊D3—22no—
8C3⋊#3708—22no—
9C2×C8×C3⋊D3D311yescharacteristic
10C2×C3×D3×C8—62no—
11C2×#3708—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#4549—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
26C8×C3⋊D3—22no—
27C2×C3×C3⋊C8—62no—
28C3×D3×C8—68no—
29#3708—62no—
30C22×C33C2×C811yescharacteristic
31C2×C3×C3⋊D3—21no—
32C2×C3×C3⋊D3—21no—
33C2×C3×C3⋊D3C1612yescomplemented
34C33×C4C22⋊C411yescharacteristic
35C33×C4M1611yescharacteristic
36C3×C3⋊Dic3—21no—
37C22×C3×C8—91no—
38C2×D3×C8—62no—
39C2×C3⋊C16—92no—
40C22×C3⋊D3C3⋊C811yescharacteristic
41C22×C3×D3—62no—
42C2×C32×C4—22no—
43C2×C32×C4—42no—
44C2×C32×C4C2×Dic311yescharacteristic
45C2×C3⋊Dic3C3⋊C811yescharacteristic
46C4×C3⋊D3—21no—
47C4×C3⋊D3—21no—
48C2×C3×Dic3—62no—
49C3×C4×D3—64no—
50C3×C4×D3—64no—
51C32×C8—21no—
52C32×C8—24no—
53C32×C8—44no—
54C3⋊(C3⋊C8)#6712yes—
55C3×C3⋊C8—64no—
56C22⋊C16—271noSylow
57C2×C33C22⋊C811yescharacteristic
58C2×C33C2×C1611yescharacteristic
59C2×C33M3211yescharacteristic
60C3×C3⋊D3—22no—
61C22×C3×C4—91no—
62C2×C4×D3—62no—
63C2×C3×C8—22no—
64C2×C3×C8—42no—
65C2×C3×C8—91no—
66C2×C3×C8—182no—
67C2×C3×C8#15711yescharacteristic
68C2×C3⋊C8—62no—
69D3×C8—68no—
70C3⋊C16—182no—
71C22×C32—22no—
72C22×C32—42no—
73C22×C32C2×C3⋊C811yescharacteristic
74C2×C3⋊D3—21no—
75C2×C3⋊D3—21no—
76C2×C3⋊D3C3⋊C1612yescomplemented
77C2×C3×D3—64no—
78C2×C3×D3—64no—
79C2×C3×D3—64no—
80C32×C4—22no—
81C32×C4—22no—
82C32×C4—42no—
83C32×C4—42no—
84C32×C4#20811yescharacteristic
85C32×C4#21711yescharacteristic
86C3⋊Dic3—21no—
87C3×Dic3—64no—
88C22×C8—91no—
89C2×C16—272no—
90C33C22⋊C1611yescharacteristic, complemented, Sylow
91C23×C3—91no—
92C22×D3—62no—
93C2×C3×C4—22no—
94C2×C3×C4—42no—
95C2×C3×C4—91no—
96C2×C3×C4—181no—
97C2×C3×C4—181no—
98C2×C3×C4C2×#15711yescharacteristic
99C2×Dic3—62no—
100C4×D3—64no—
101C4×D3—64no—
102C3×C8—24no—
103C3×C8—21no—
104C3×C8—44no—
105C3×C8—92no—
106C3⋊C8—64no—
107C2×C32—22no—
108C2×C32—22no—
109C2×C32—22no—
110C2×C32—42no—
111C2×C32—42no—
112C2×C32—42no—
113C2×C32C2×C3⋊C1611yescharacteristic
114C2×C32#81211yescharacteristic
115C2×C32#83011yescharacteristic
116C3⋊D3—22no—
117C3×D3—68no—
118C22×C4—91no—
119C2×C8—91no—
120C2×C8—182no—
121C2×C8C3⋊#15711yescharacteristic
122C16—542no—
123C22×C3—22no—
124C22×C3—42no—
125C22×C3—92no—
126C22×C3—181no—
127C22×C3—181no—
128C22×C3#378711yescharacteristic
129C2×D3—64no—
130C2×D3—64no—
131C2×D3—64no—
132C3×C4—22no—
133C3×C4—22no—
134C3×C4—42no—
135C3×C4—42no—
136C3×C4—181no—
137C3×C4#378811yescharacteristic
138C3×C4C22⋊#15711yescharacteristic
139Dic3—64no—
140C32—22no—
141C32—42no—
142C32#454911yescharacteristic, complemented
143C23—91no—
144C2×C4—91no—
145C2×C4—181no—
146C2×C4—181no—
147C2×C4C2×C3⋊#15711yescharacteristic
148C8—21no—
149C8—92no—
150C2×C3—22no—
151C2×C3—22no—
152C2×C3—22no—
153C2×C3—42no—
154C2×C3—42no—
155C2×C3—42no—
156C2×C3—182no—
157C2×C3#6376411yescharacteristic
158C2×C3#6376511yescharacteristic
159C2×C3#6377811yescharacteristic
160D3—68no—
161C22—92no—
162C22—181no—
163C22—181no—
164C22#8898811yescharacteristic
165C4—181no—
166C4#8898911yescharacteristic
167C4#8899411yescharacteristic
168C3—22no—
169C3—42no—
170C3#1059017611yescharacteristic, complemented
171C2—182no—
172C2#1172308911yescharacteristic
173C2#1172309011yescharacteristic
174C2#1172310311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊#157
commutatorC2×C33C2×C16
FittingC2×C33×C8C4
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C8
cosocleC2×C3×C3⋊(C3⋊C8)C22
hypercenterC2×C8C3⋊#157
hypocenterC33C22⋊C16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#3708C2semidirect
33. C2×C3×C3⋊D3C16semidirect
76. C2×C3⋊D3C3⋊C16semidirect
90. C33C22⋊C16semidirect
142. C32#4549semidirect
170. C3#10590176semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C2×C3⋊#37082
#105901763
#45499