Finite Group Databasebeta

← C2×C3⋊D3×C3⋊C16 #13159459

Subgroups — C2×C3⋊D3×C3⋊C16 #13159459

Order 1728.

700 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 212 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⋊(C3⋊(C3⋊C16))C211yesmaximal, characteristic, complemented
3C2×C32×C3⋊C16C211yesmaximal, characteristic, complemented
4C3⋊D3×C3⋊C16C214yesmaximal, direct product factor
5C2×C16×C3⋊D3—31nomaximal
6C2×D3×C3⋊C16—34nomaximal
7C2×C3×C4×C3⋊D3C411yescharacteristic
8C2×C33×C8C2211yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
10C3×C8×C3⋊D3C412yes—
11C3×C8×C3⋊D3C2212yes—
12C3⋊(C3⋊(C3⋊C16))C2212yescomplemented
13C32×C3⋊C16C2212yescomplemented
14C2×C8×C3⋊D3D311yescharacteristic
15C2×C3×D3×C8—34no—
16C2×C32×C16—31no—
17C2×C3⋊(C3⋊C16)—31no—
18C2×C3⋊(C3⋊C16)—34no—
19C2×C3⋊(C3⋊C16)—64no—
20C16×C3⋊D3—34no—
21C2×C3×C3⋊C16D314yescomplemented
22D3×C3⋊C16—316no—
23C22×C3×C3⋊D3C811yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C811yescharacteristic
26C3×C4×C3⋊D3C812yes—
27C3×C4×C3⋊D3C2×C412yes—
28C33×C8C2×C411yescharacteristic
29C33×C8C2311yescharacteristic
30C3×C3⋊(C3⋊C8)C2×C412yes—
31C22×C3⋊C16—91no—
32C2×D3×C16—94no—
33C2×C4×C3⋊D3Dic311yescharacteristic
34C2×C3×C4×D3—34no—
35C2×C32×C8—24no—
36C2×C32×C8C2×D314yes—
37C2×C32×C8C2×D311yescharacteristic
38C2×C3⋊(C3⋊C8)Dic311yescharacteristic
39C8×C3⋊D3Dic312yes—
40C8×C3⋊D3C2×D312yes—
41C2×C3×C3⋊C8—34no—
42C3×D3×C8—38no—
43C3×D3×C8—38no—
44C32×C16—32no—
45C3⋊(C3⋊C16)—32no—
46C3⋊(C3⋊C16)—38no—
47C3⋊(C3⋊C16)—68no—
48C3×C3⋊C16C2×D318yescomplemented
49C22×C33C2×C811yescharacteristic
50C2×C3×C3⋊D3C1614yescomplemented
51C2×C3×C3⋊D3C2×C812yes—
52C33×C4C2×C811yescharacteristic
53C33×C4C22×C411yescharacteristic
54C3×C3⋊Dic3C2×C812yes—
55C22×C3×C8—91no—
56C2×D3×C8—34no—
57C2×C3×C16—34no—
58C2×C3⋊C16—94no—
59C2×C3⋊C16—91no—
60C2×C3⋊C16—94no—
61C2×C3⋊C16—184no—
62C2×C3⋊C16C3⋊D311yescharacteristic, direct product factor
63D3×C16—916no—
64C22×C3⋊D3C3⋊C811yescharacteristic
65C22×C3×D3—34no—
66C2×C32×C4—24no—
67C2×C32×C4C4×D314yes—
68C2×C32×C4C2×Dic311yescharacteristic
69C2×C3⋊Dic3C3⋊C811yescharacteristic
70C4×C3⋊D3C3⋊C812yes—
71C4×C3⋊D3C2×Dic312yes—
72C2×C3×Dic3—34no—
73C3×C4×D3—38no—
74C3×C4×D3—38no—
75C32×C8—24no—
76C32×C8—24no—
77C32×C8C4×D314yes—
78C32×C8C2×Dic311yescharacteristic
79C32×C8C22×D314yes—
80C32×C8C22×D311yescharacteristic
81C3⋊(C3⋊C8)C2×Dic312yes—
82C3×C3⋊C8—38no—
83C22×C16—271noSylow
84C2×C33C2×C1612yescomplemented
85C2×C33C22×C811yescharacteristic
86C3×C3⋊D3C2×C1614yescomplemented
87C22×C3×C4—91no—
88C2×C4×D3—34no—
89C2×C3×C8—24no—
90C2×C3×C8—91no—
91C2×C3×C8—92no—
92C2×C3×C8—92no—
93C2×C3×C8D3214yes—
94C2×C3×C8C2×C3⋊D311yescharacteristic
95C2×C3⋊C8—34no—
96D3×C8—38no—
97D3×C8—38no—
98C3×C16—38no—
99C3⋊C16—98no—
100C3⋊C16—92no—
101C3⋊C16—188no—
102C3⋊C16C2×C3⋊D312yesdirect product factor
103C22×C32—24no—
104C22×C32D3×C814yes—
105C22×C32C2×C3⋊C811yescharacteristic
106C2×C3⋊D3C3⋊C1614yesdirect product factor
107C2×C3⋊D3C2×C3⋊C812yes—
108C2×C3×D3—316no—
109C2×C3×D3—38no—
110C32×C4—24no—
111C32×C4—24no—
112C32×C4D3×C814yes—
113C32×C4C2×C3⋊C811yescharacteristic
114C32×C4C2×C4×D314yes—
115C32×C4C22×Dic311yescharacteristic
116C3⋊Dic3C2×C3⋊C812yes—
117C3×Dic3—38no—
118C22×C8—91no—
119C2×C16—31no—
120C2×C16—274no—
121C2×C16—271no—
122C33C22×C1611yescharacteristic, complemented, Sylow
123C23×C3—91no—
124C22×D3—34no—
125C2×C3×C4—24no—
126C2×C3×C4—91no—
127C2×C3×C4—92no—
128C2×C3×C4—92no—
129C2×C3×C4D3×Dic314yes—
130C2×C3×C4C4×C3⋊D311yescharacteristic
131C2×Dic3—34no—
132C4×D3—38no—
133C4×D3—38no—
134C3×C8—24no—
135C3×C8—24no—
136C3×C8—92no—
137C3×C8D3×Dic314yes—
138C3×C8C4×C3⋊D311yescharacteristic
139C3×C8C2×D3214yes—
140C3×C8C22×C3⋊D311yescharacteristic
141C3⋊C8—38no—
142C2×C32—28no—
143C2×C32—24no—
144C2×C32D3×C1618yescomplemented
145C2×C32C2×C3⋊C1612yescomplemented
146C2×C32C2×D3×C814yes—
147C2×C32C22×C3⋊C811yescharacteristic
148C3⋊D3C2×C3⋊C1614yesdirect product factor
149C3×D3—316no—
150C22×C4—91no—
151C2×C8—91no—
152C2×C8—92no—
153C2×C8—92no—
154C2×C8D3×C3⋊D311yescharacteristic
155C16—32no—
156C16—272no—
157C22×C3—24no—
158C22×C3—92no—
159C22×C3—94no—
160C22×C3D3×C3⋊C814yes—
161C22×C3C8×C3⋊D311yescharacteristic
162C2×D3—316no—
163C2×D3—38no—
164C3×C4—24no—
165C3×C4—24no—
166C3×C4—92no—
167C3×C4D3×C3⋊C814yes—
168C3×C4C8×C3⋊D311yescharacteristic
169C3×C4C2×D3×Dic314yes—
170C3×C4C2×C4×C3⋊D311yescharacteristic
171Dic3—38no—
172C32—24no—
173C32C2×D3×C1614yescomplemented
174C32C22×C3⋊C1611yescharacteristic, complemented
175C23—91no—
176C2×C4—91no—
177C2×C4—92no—
178C2×C4—92no—
179C2×C4Dic3×C3⋊D311yescharacteristic
180C8—92no—
181C8Dic3×C3⋊D311yescharacteristic
182C8C2×D3×C3⋊D311yescharacteristic
183C2×C3—28no—
184C2×C3—24no—
185C2×C3—94no—
186C2×C3D3×C3⋊C1618yescomplemented
187C2×C3C16×C3⋊D312yescomplemented
188C2×C3C2×D3×C3⋊C814yes—
189C2×C3C2×C8×C3⋊D311yescharacteristic
190D3—316no—
191C22—94no—
192C22—92no—
193C22C3⋊D3×C3⋊C811yescharacteristic
194C4—92no—
195C4C3⋊D3×C3⋊C811yescharacteristic
196C4C2×Dic3×C3⋊D311yescharacteristic
197C3—24no—
198C3C2×D3×C3⋊C1614yescomplemented
199C3C2×C16×C3⋊D311yescharacteristic, complemented
200C2—94no—
201C2C3⋊D3×C3⋊C1612yesdirect product factor
202C2C2×C3⋊D3×C3⋊C811yescharacteristic
subgroupG/H
center, hypercenterC2×C8D3×C3⋊D3
commutator, hypocenterC33C22×C16
FrattiniC8C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C8
cosocleC33×C8C23
LCSG ▷ C33
UCSC1 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊(C3⋊C16))C2semidirect
3. C2×C32×C3⋊C16C2semidirect
4. C3⋊D3×C3⋊C16C2direct (N × H)
12. C3⋊(C3⋊(C3⋊C16))C22semidirect
13. C32×C3⋊C16C22semidirect
21. C2×C3×C3⋊C16D3semidirect
48. C3×C3⋊C16C2×D3semidirect
50. C2×C3×C3⋊D3C16semidirect
62. C2×C3⋊C16C3⋊D3direct (N × H)
84. C2×C33C2×C16semidirect
86. C3×C3⋊D3C2×C16semidirect
102. C3⋊C16C2×C3⋊D3direct (N × H)
106. C2×C3⋊D3C3⋊C16direct (N × H)
122. C33C22×C16semidirect
144. C2×C32D3×C16semidirect
145. C2×C32C2×C3⋊C16semidirect
148. C3⋊D3C2×C3⋊C16direct (N × H)
173. C32C2×D3×C16semidirect
174. C32C22×C3⋊C16semidirect
186. C2×C3D3×C3⋊C16semidirect
187. C2×C3C16×C3⋊D3semidirect
198. C3C2×D3×C3⋊C16semidirect
199. C3C2×C16×C3⋊D3semidirect
201. C2C3⋊D3×C3⋊C16direct (N × H)

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⋊(C3⋊(C3⋊C16))2
C2×C32×C3⋊C162
C3⋊D3×C3⋊C162
C2×C16×C3⋊D33
C2×D3×C3⋊C163