Finite Group Databasebeta

← #13173330

Subgroups — #13173330

Order 1728.

632 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 125 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×C4×C3⋊#157C212yesmaximal
2C3×C42×C3⋊D3C211yesmaximal, characteristic
3#11723106C214yesmaximal
4#10593733—31nomaximal
5C22×C3⋊#157C2214yes—
6C2×C3×C4×C3⋊D3C2212yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C4×C3⋊#157C418yescomplemented
9C33×C42C411yescharacteristic, complemented
10C3×C4×C3⋊Dic3C411yescharacteristic, complemented
11C3×C4×C3⋊Dic3C412yescomplemented
12C2×C4×#157—32no—
13C42×C3⋊D3D311yescharacteristic
14C3×C42×D3—62no—
15#63781—34no—
16C22×C3×C3⋊D3C2311yescharacteristic
17C2×C3⋊#157—28no—
18C2×C3⋊#157C2×C418yes—
19C2×C33×C4C2×C412yes—
20C2×C33×C4C2×C411yescharacteristic
21C2×C3×C3⋊Dic3C2×C412yes—
22C2×C3×C3⋊Dic3C2×C411yescharacteristic
23C3×C4×C3⋊D3—24no—
24C3×C4×C3⋊D3C2×C414yes—
25#4932—91nomaximal
26C22×#157—34no—
27C2×C4×C3⋊D3C2×D312yes—
28C2×C4×C3⋊D3C2×D311yescharacteristic
29C2×C3×C4×D3—62no—
30C2×C3×C4×D3—64no—
31C4×#157—38no—
32C32×C42—22no—
33C32×C42—42no—
34C32×C42Dic311yescharacteristic, complemented
35C4×C3⋊Dic3Dic311yescharacteristic, complemented
36C4×C3⋊Dic3Dic312yescomplemented
37C3×C4×Dic3—64no—
38C3×C4×Dic3—62no—
39C22×C33C22×C411yescharacteristic
40C2×C3×C3⋊D3C22×C412yes—
41C2×C3×C3⋊D3#4114yes—
42C3⋊#157—28no—
43C33×C4—22no—
44C33×C4C4212yes—
45C3×C3⋊Dic3—22no—
46C3×C3⋊Dic3C4212yes—
47C2×C3×C42—91no—
48C2×C4×Dic3—92no—
49C42×D3—62no—
50C3⋊#95—94no—
51C22×C3⋊D3C22×D311yescharacteristic
52C22×C3×D3—62no—
53C2×#157—38no—
54C2×#157—68no—
55C2×C32×C4—24no—
56C2×C32×C4—22no—
57C2×C32×C4—44no—
58C2×C32×C4—42no—
59C2×C32×C4C2×Dic312yes—
60C2×C32×C4C2×Dic311yescharacteristic
61C2×C3⋊Dic3C2×Dic312yes—
62C2×C3⋊Dic3C2×Dic311yescharacteristic
63C4×C3⋊D3—24no—
64C4×C3⋊D3C4×D314yes—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—64no—
67C3×C4×D3—68no—
68C3×C4×D3—616no—
69#376—271noSylow
70C2×C33C2×C4211yescharacteristic
71C2×C33#11712yes—
72C3×C3⋊D3#11714yes—
73C22×C3×C4—91no—
74C22×C3×C4—92no—
75C22×Dic3—94no—
76C2×C4×D3—62no—
77C2×C4×D3—64no—
78C3×C42—22no—
79C3×C42—42no—
80C3×C42—92no—
81C3×C42—91no—
82C3×C42#15711yescharacteristic, complemented
83C4×Dic3—64no—
84C4×Dic3—62no—
85C4×Dic3—98no—
86C22×C32—22no—
87C22×C32—42no—
88C22×C32C22×Dic311yescharacteristic
89C2×C3⋊D3C2×C4×D312yes—
90C2×C3⋊D3#23514yes—
91C2×C3×D3—64no—
92C2×C3×D3—68no—
93#157—68no—
94C32×C4—22no—
95C32×C4—28no—
96C32×C4—24no—
97C32×C4—48no—
98C32×C4—44no—
99C32×C4C4×Dic312yes—
100C3⋊Dic3—22no—
101C3⋊Dic3C4×Dic312yes—
102C3×Dic3—64no—
103C3×Dic3—68no—
104C2×C42—91no—
105C2×C42—272no—
106#95—274no—
107C33#37611yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—62no—
110C2×C3×C4—24no—
111C2×C3×C4—22no—
112C2×C3×C4—44no—
113C2×C3×C4—42no—
114C2×C3×C4—91no—
115C2×C3×C4—94no—
116C2×C3×C4—92no—
117C2×C3×C4—184no—
118C2×C3×C4C2×#15712yes—
119C2×C3×C4C2×#15711yescharacteristic
120C2×Dic3—62no—
121C2×Dic3—64no—
122C2×Dic3—98no—
123C2×Dic3—188no—
124C4×D3—68no—
125C4×D3—616no—
126C2×C32—22no—
127C2×C32—24no—
128C2×C32—42no—
129C2×C32—44no—
130C2×C32C2×C4×Dic311yescharacteristic
131C2×C32#92612yes—
132C3⋊D3#87214yes—
133C3×D3—68no—
134C22×C4—91no—
135C22×C4—92no—
136C22×C4—274no—
137C42—92no—
138C42—91no—
139C42—278no—
140C42C3⋊#15711yescharacteristic, complemented
141C22×C3—22no—
142C22×C3—42no—
143C22×C3—92no—
144C22×C3—94no—
145C22×C3C22×#15711yescharacteristic
146C2×D3—64no—
147C2×D3—68no—
148C3×C4—28no—
149C3×C4—22no—
150C3×C4—24no—
151C3×C4—48no—
152C3×C4—44no—
153C3×C4—92no—
154C3×C4—182no—
155C3×C4C4×#15712yes—
156Dic3—64no—
157Dic3—68no—
158Dic3—188no—
159C32—22no—
160C32—42no—
161C32#493211yescharacteristic, complemented
162C23—91no—
163C2×C4—91no—
164C2×C4—94no—
165C2×C4—92no—
166C2×C4—184no—
167C2×C4—278no—
168C2×C4—548no—
169C2×C4C2×C3⋊#15712yes—
170C2×C4C2×C3⋊#15711yescharacteristic
171C2×C3—22no—
172C2×C3—24no—
173C2×C3—42no—
174C2×C3—44no—
175C2×C3—94no—
176C2×C3C2×C4×#15711yescharacteristic
177C2×C3#6428612yes—
178D3—68no—
179C22—92no—
180C22—94no—
181C22C22×C3⋊#15711yescharacteristic
182C4—22no—
183C4—92no—
184C4—182no—
185C4—548no—
186C4C4×C3⋊#15712yes—
187C3—22no—
188C3—42no—
189C3#1059373311yescharacteristic, complemented
190C2—94no—
191C2C2×C4×C3⋊#15711yescharacteristic
192C2#1172451412yes—
subgroupG/H
centerC2×C4C2×C3⋊#157
commutatorC2×C33C2×C42
FrattiniC22C22×C3⋊#157
FittingC33×C42C4
socleC22×C33C22×C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC42C3⋊#157
hypocenterC33#376
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
8. C4×C3⋊#157C4semidirect
9. C33×C42C4semidirect
10. C3×C4×C3⋊Dic3C4semidirect
11. C3×C4×C3⋊Dic3C4semidirect
34. C32×C42Dic3semidirect
35. C4×C3⋊Dic3Dic3semidirect
36. C4×C3⋊Dic3Dic3semidirect
82. C3×C42#157semidirect
107. C33#376semidirect
140. C42C3⋊#157semidirect
161. C32#4932semidirect
189. C3#10593733semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊#1572
C3×C42×C3⋊D32
#117231062
#105937333
#49329