Finite Group Databasebeta

← C3⋊D3×#817 #13159531

Subgroups — C3⋊D3×#817 #13159531

Order 1728.

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

normal subgroups only

This group has 924 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
2C3⋊D3×C4⋊Dic3C212yesmaximal
3#11722187C211yesmaximal, characteristic, complemented
4C32×#817C211yesmaximal, characteristic, complemented
5#11722054C212yesmaximal, complemented
6C3⋊D3×#106—31nomaximal
7D3×#817—34nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×Dic3×C3⋊D3—22no—
10C2×C33×C8C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C3×C8×C3⋊D3C414yescomplemented
14C32×C4⋊Dic3C2212yes—
15#88807—22no—
16C2×C8×C3⋊D3D311yescharacteristic
17C4⋊C4×C3⋊D3—32no—
18C2×C3×D3×C8—34no—
19D3×C4⋊Dic3—38no—
20C32×#106—31no—
21#63644—31no—
22#63644—34no—
23#63644—64no—
24#63635—32no—
25C3×#817D314yescomplemented
26#63573—38no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C33×C4C2311yescharacteristic
29C2×C3⋊(C3⋊Dic3)—22no—
30C2×C3×C3⋊Dic3D411yescharacteristic
31C3×C4×C3⋊D3C2×C412yes—
32C3×C4×C3⋊D3Q812yes—
33C2×C32×Dic3—22no—
34Dic3×C3⋊D3—44no—
35C33×C8C2×C412yescomplemented
36C3×C3⋊(C3⋊C8)C2×C412yescomplemented
37C2×#817—91no—
38D3×#106—94no—
39C2×C4×C3⋊D3—62no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—34no—
42C2×D3×Dic3—68no—
43C2×C32×C8—24no—
44C2×C32×C8C2×D314yes—
45C2×C32×C8C2×D311yescharacteristic
46C32×C4⋊C4—32no—
47C4⋊(C3⋊Dic3)—32no—
48C4⋊(C3⋊Dic3)—38no—
49C4⋊(C3⋊Dic3)—68no—
50#3750—62no—
51C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
52C8×C3⋊D3Dic314yescomplemented
53C3×C4⋊Dic3C2×D318yes—
54C2×C3×C3⋊C8—34no—
55C3×D3×C8—316no—
56Dic3⋊Dic3—68no—
57C22×C33C2×D411yescharacteristic
58C2×C3×C3⋊D3C4⋊C412yes—
59C2×C3×C3⋊D3SD1614yes—
60C33×C4C22×C411yescharacteristic
61C33×C4C2×Q811yescharacteristic
62C3⋊(C3⋊Dic3)—42no—
63C3×C3⋊Dic3C4⋊C412yes—
64C32×Dic3—42no—
65C22×C3×C8—91no—
66C2×C4⋊Dic3—92no—
67C2×D3×C8—34no—
68D3×C4⋊C4—98no—
69C3×#106—34no—
70#817—94no—
71#817—91no—
72#817—92no—
73#817—184no—
74#817C3⋊D311yescharacteristic, direct product factor
75#808—98no—
76C22×C3⋊D3D1211yescharacteristic
77C22×C3×D3—34no—
78C2×C32×C4—24no—
79C2×C32×C4—62no—
80C2×C32×C4C22×D314yes—
81C2×C32×C4C22×D311yescharacteristic
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—68no—
84C2×C3⋊Dic3—128no—
85C2×C3⋊Dic3D1211yescharacteristic
86C4×C3⋊D3—124no—
87C4×C3⋊D3Dic612yes—
88C4×C3⋊D3C2×Dic312yes—
89C2×C3×Dic3—28no—
90C2×C3×Dic3—34no—
91C3×C4×D3—38no—
92C3×C4×D3—38no—
93D3×Dic3—1216no—
94C32×C8—28no—
95C32×C8C4×D318yescomplemented
96C32×C8C2×Dic312yescomplemented
97C3⋊(C3⋊C8)C2×Dic312yescomplemented
98C3×C3⋊C8—38no—
99C2×#106—271noSylow
100C2×C33C2×C4⋊C411yescharacteristic
101C2×C33C2×SD1612yes—
102C3×C3⋊D3#10614yescomplemented
103C22×C3×C4—91no—
104C22×Dic3—182no—
105C2×C4×D3—34no—
106C2×C4×D3—188no—
107C2×C3×C8—24no—
108C2×C3×C8—91no—
109C2×C3×C8—94no—
110C2×C3×C8D3214yes—
111C2×C3×C8C2×C3⋊D311yescharacteristic
112C3×C4⋊C4—38no—
113C4⋊Dic3—98no—
114C4⋊Dic3—92no—
115C4⋊Dic3—188no—
116C4⋊Dic3—182no—
117C4⋊Dic3C2×C3⋊D312yes—
118Dic3⋊C4—188no—
119C2×C3⋊C8—34no—
120D3×C8—316no—
121C22×C32—24no—
122C22×C32C2×D1211yescharacteristic
123C22×C32D3×D414yes—
124C2×C3⋊D3#20414yes—
125C2×C3⋊D3C4⋊Dic312yes—
126C2×C3×D3—38no—
127C2×C3×D3—316no—
128C32×C4—24no—
129C32×C4—24no—
130C32×C4—122no—
131C32×C4C2×Dic611yescharacteristic
132C32×C4C2×C4×D314yes—
133C32×C4D3×Q814yes—
134C32×C4C22×Dic311yescharacteristic
135C3⋊Dic3—122no—
136C3⋊Dic3—128no—
137C3⋊Dic3—248no—
138C3⋊Dic3C4⋊Dic312yes—
139C3×Dic3—38no—
140C3×Dic3—48no—
141C22×C8—91no—
142C2×C4⋊C4—272no—
143#106—31no—
144#106—272no—
145#106—271no—
146C33C2×#10611yescharacteristic, complemented, Sylow
147C23×C3—91no—
148C22×D3—34no—
149C2×C3×C4—24no—
150C2×C3×C4—68no—
151C2×C3×C4—91no—
152C2×C3×C4—92no—
153C2×C3×C4—92no—
154C2×C3×C4C2×D3214yes—
155C2×C3×C4C22×C3⋊D311yescharacteristic
156C2×Dic3—22no—
157C2×Dic3—34no—
158C2×Dic3—188no—
159C2×Dic3—182no—
160C2×Dic3—368no—
161C2×Dic3—364no—
162C4×D3—38no—
163C4×D3—38no—
164C4×D3—3616no—
165C3×C8—28no—
166C3×C8—92no—
167C3×C8D3×Dic318yescomplemented
168C3×C8C4×C3⋊D312yescomplemented
169C3⋊C8—38no—
170C2×C32—28no—
171C2×C32—24no—
172C2×C32D3×C4⋊C414yes—
173C2×C32C2×#20412yes—
174C2×C32D3×SD1618yes—
175C2×C32C2×C4⋊Dic311yescharacteristic
176C3⋊D3#81714yesdirect product factor
177C3×D3—316no—
178C22×C4—91no—
179C22×C4—542no—
180C2×C8—91no—
181C2×C8—94no—
182C2×C8D3×C3⋊D311yescharacteristic
183C4⋊C4—32no—
184C4⋊C4—272no—
185C4⋊C4—542no—
186C22×C3—24no—
187C22×C3—92no—
188C22×C3—94no—
189C22×C3D3×D1214yes—
190C22×C3D4×C3⋊D311yescharacteristic
191C2×D3—316no—
192C2×D3—38no—
193C3×C4—24no—
194C3×C4—24no—
195C3×C4—92no—
196C3×C4—128no—
197C3×C4D3×Dic614yes—
198C3×C4C2×D3×Dic314yes—
199C3×C4C2×C4×C3⋊D311yescharacteristic
200C3×C4Q8×C3⋊D311yescharacteristic
201Dic3—38no—
202Dic3—42no—
203Dic3—368no—
204Dic3—362no—
205Dic3—728no—
206C32—24no—
207C32D3×#10614yescomplemented
208C32C2×#81711yescharacteristic, complemented
209C23—91no—
210C2×C4—62no—
211C2×C4—91no—
212C2×C4—92no—
213C2×C4—92no—
214C2×C4—542no—
215C2×C4—1084no—
216C2×C4C2×D3×C3⋊D311yescharacteristic
217C8—92no—
218C8Dic3×C3⋊D312yescomplemented
219C2×C3—28no—
220C2×C3—24no—
221C2×C3—94no—
222C2×C3D3×#20418yes—
223C2×C3D3×C4⋊Dic314yes—
224C2×C3C4⋊C4×C3⋊D311yescharacteristic
225C2×C3SD16×C3⋊D312yes—
226D3—316no—
227C22—94no—
228C22—92no—
229C22C3⋊D3×D1211yescharacteristic
230C4—92no—
231C4—122no—
232C4—1082no—
233C4C3⋊D3×Dic611yescharacteristic
234C4C2×Dic3×C3⋊D311yescharacteristic
235C3—24no—
236C3D3×#81714yescomplemented
237C3C3⋊D3×#10611yescharacteristic, complemented
238C2—94no—
239C2C3⋊D3×#20412yes—
240C2C3⋊D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#106
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11722187C2semidirect
4. C32×#817C2semidirect
5. #11722054C2semidirect
13. C3×C8×C3⋊D3C4semidirect
25. C3×#817D3semidirect
35. C33×C8C2×C4semidirect
36. C3×C3⋊(C3⋊C8)C2×C4semidirect
52. C8×C3⋊D3Dic3semidirect
74. #817C3⋊D3direct (N × H)
95. C32×C8C4×D3semidirect
96. C32×C8C2×Dic3semidirect
97. C3⋊(C3⋊C8)C2×Dic3semidirect
102. C3×C3⋊D3#106semidirect
146. C33C2×#106semidirect
167. C3×C8D3×Dic3semidirect
168. C3×C8C4×C3⋊D3semidirect
176. C3⋊D3#817direct (N × H)
207. C32D3×#106semidirect
208. C32C2×#817semidirect
218. C8Dic3×C3⋊D3semidirect
236. C3D3×#817semidirect
237. C3C3⋊D3×#106semidirect

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
C3⋊D3×C4⋊Dic32
#117221872
C32×#8172
#117220542
C3⋊D3×#1063
D3×#8173