Finite Group Databasebeta

← C3×A4×C3⋊C16 #13157033

Subgroups — C3×A4×C3⋊C16 #13157033

Order 1728.

246 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C32×C8×A4C211yesmaximal, characteristic
2C22×C3×C3⋊C16C311yesmaximal, characteristic, complemented
3C3×A4×C16—31nomaximal
4A4×C3⋊C16C313yesmaximal, direct product factor
5C32×C4×A4C411yescharacteristic
6C32×C3⋊C16—41nomaximal
7C22×C32×C8C2×C311yescharacteristic
8C3×C8×A4—21no—
9C3×C8×A4—23no—
10C3×C8×A4D311yescharacteristic
11C3×C8×A4C2×C313yes—
12C2×C3×C3⋊C16—31no—
13C2×C3×C3⋊C16—31no—
14C2×C32×A4C811yescharacteristic
15C33×C8—41no—
16C22×C3×C16—31no—
17C22×C3⋊C16C3211yescharacteristic, complemented
18A4×C16—33no—
19C22×C32×C4C3×C411yescharacteristic
20C3×C4×A4—21no—
21C3×C4×A4—23no—
22C3×C4×A4Dic311yescharacteristic
23C3×C4×A4C3×C413yes—
24C2×C32×C8—31no—
25C2×C32×C8—31no—
26C32×C16—121no—
27C3×C3⋊C16—31no—
28C3×C3⋊C16—43no—
29C3×C3⋊C16A411yescharacteristic, direct product factor
30C32×A4C1611yescharacteristic, complemented
31C33×C4—41no—
32C22×C3×C8—21no—
33C22×C3×C8C3×D311yescharacteristic
34C22×C3×C8C2×C3211yescharacteristic
35C8×A4—23no—
36C8×A4C3×D313yes—
37C2×C3×C16—91no—
38C2×C3×C16—91no—
39C2×C3⋊C16—31no—
40C2×C3⋊C16—31no—
41C23×C32C3×C811yescharacteristic
42C2×C3×A4—21no—
43C2×C3×A4—23no—
44C2×C3×A4C3⋊C811yescharacteristic
45C2×C3×A4C3×C813yes—
46C2×C32×C4—31no—
47C2×C32×C4—31no—
48C32×C8—31no—
49C32×C8—41no—
50C32×C8—43no—
51C32×C8—83no—
52C32×C8—81no—
53C32×C8C2×A411yescharacteristic
54C22×C16—31noSylow
55C2×C33—41no—
56C22×C3×C4—21no—
57C22×C3×C4C3×Dic311yescharacteristic
58C22×C3×C4C32×C411yescharacteristic
59C4×A4—23no—
60C4×A4C3×Dic313yes—
61C2×C3×C8—31no—
62C2×C3×C8—31no—
63C2×C3×C8—31no—
64C2×C3×C8—31no—
65C2×C3×C8—61no—
66C2×C3×C8—61no—
67C3×C16—31no—
68C3×C16—91no—
69C3×C16—123no—
70C3⋊C16—31no—
71C3⋊C16C3×A411yescharacteristic, direct product factor
72C22×C32—31no—
73C22×C32—31no—
74C22×C32C3×C1611yescharacteristic, complemented
75C3×A4—21no—
76C3×A4—23no—
77C3×A4C3⋊C1611yescharacteristic, direct product factor
78C3×A4C3×C1613yescomplemented
79C32×C4—31no—
80C32×C4—41no—
81C32×C4—43no—
82C32×C4—83no—
83C32×C4—81no—
84C32×C4C4×A411yescharacteristic
85C22×C8C32×D311yescharacteristic
86C2×C16—91no—
87C2×C16—91no—
88C33—41noSylow
89C23×C3—21no—
90C23×C3C3×C3⋊C811yescharacteristic
91C23×C3C32×C811yescharacteristic
92C2×A4—23no—
93C2×A4C3×C3⋊C813yes—
94C2×C3×C4—31no—
95C2×C3×C4—31no—
96C2×C3×C4—31no—
97C2×C3×C4—31no—
98C2×C3×C4—61no—
99C2×C3×C4—61no—
100C3×C8—21no—
101C3×C8—31no—
102C3×C8—31no—
103C3×C8—43no—
104C3×C8—61no—
105C3×C8—83no—
106C3×C8D3×A411yescharacteristic
107C3×C8C2×C3×A411yescharacteristic
108C2×C32—31no—
109C2×C32—31no—
110C2×C32—41no—
111C2×C32—43no—
112C2×C32—83no—
113C2×C32—81no—
114C2×C32C8×A411yescharacteristic
115C22×C4C32×Dic311yescharacteristic
116C2×C8—31no—
117C2×C8—31no—
118C16—31no—
119C16—91no—
120C22×C3—21no—
121C22×C3—31no—
122C22×C3—31no—
123C22×C3—31no—
124C22×C3—31no—
125C22×C3—61no—
126C22×C3—61no—
127C22×C3C3×C3⋊C1611yescharacteristic, complemented
128C22×C3C32×C1611yescharacteristic, complemented
129A4—23no—
130A4C3×C3⋊C1613yesdirect product factor
131C3×C4—21no—
132C3×C4—31no—
133C3×C4—31no—
134C3×C4—43no—
135C3×C4—61no—
136C3×C4—83no—
137C3×C4Dic3×A411yescharacteristic
138C3×C4C3×C4×A411yescharacteristic
139C32—41no—
140C32—43no—
141C32—83no—
142C32—81no—
143C32A4×C1611yescharacteristic, complemented
144C23C32×C3⋊C811yescharacteristic
145C2×C4—31no—
146C2×C4—31no—
147C8—31no—
148C8C3×D3×A411yescharacteristic
149C2×C3—21no—
150C2×C3—31no—
151C2×C3—31no—
152C2×C3—31no—
153C2×C3—31no—
154C2×C3—43no—
155C2×C3—61no—
156C2×C3—61no—
157C2×C3—83no—
158C2×C3A4×C3⋊C811yescharacteristic
159C2×C3C3×C8×A411yescharacteristic
160C22—31no—
161C22—31no—
162C22C32×C3⋊C1611yescharacteristic, complemented
163C4—31no—
164C4C3×Dic3×A411yescharacteristic
165C3—21no—
166C3—43no—
167C3—83no—
168C3A4×C3⋊C1611yescharacteristic, direct product factor
169C3C3×A4×C1611yescharacteristic, complemented
170C2—31no—
171C2—31no—
172C2C3×A4×C3⋊C811yescharacteristic
subgroupG/H
center, hypercenterC3×C8D3×A4
commutator, hypocenterC22×C3C32×C16
FrattiniC8C3×D3×A4
FittingC22×C32×C8C2×C3
socleC23×C32C3×C8
prime radicalC2×C32×A4C8
cosocleC22×C3×C8C2×C32
LCSG ▷ C22×C3
UCSC1 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊C16C3semidirect
4. A4×C3⋊C16C3direct (N × H)
17. C22×C3⋊C16C32semidirect
29. C3×C3⋊C16A4direct (N × H)
30. C32×A4C16semidirect
71. C3⋊C16C3×A4direct (N × H)
74. C22×C32C3×C16semidirect
77. C3×A4C3⋊C16direct (N × H)
78. C3×A4C3×C16semidirect
127. C22×C3C3×C3⋊C16semidirect
128. C22×C3C32×C16semidirect
130. A4C3×C3⋊C16direct (N × H)
143. C32A4×C16semidirect
162. C22C32×C3⋊C16semidirect
168. C3A4×C3⋊C16direct (N × H)
169. C3C3×A4×C16semidirect

Maximal subgroups

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

subgroupindex
C32×C8×A42
C22×C3×C3⋊C163
C3×A4×C163
A4×C3⋊C163
C32×C3⋊C164