Finite Group Databasebeta

← C3⋊D3×#874 #13175398

Subgroups — C3⋊D3×#874 #13175398

Order 1728.

3420 conjugacy classes of subgroups; only the 364 normal subgroups are shown.

normal subgroups only

This group has 3420 conjugacy classes of subgroups — too many to draw. Only its 364 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⋊D3×D12C216yesmaximal, complemented
2#11723915C211yesmaximal, characteristic, complemented
3C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×#874C211yesmaximal, characteristic, complemented
5#11723524C216yesmaximal, complemented
6C3⋊D3×#127—31nomaximal
7D3×#874—34nomaximal
8C22×D3×C3⋊D3C2214yes—
9C2×C4⋊(C3⋊(C3⋊D3))C2216yescomplemented
10C2×C3×C4×C3⋊D3C2213yes—
11C2×C32×D12C2216yescomplemented
12C2×#6291C22112yescomplemented
13C3⋊D3×D12—224no—
14C33×C42C2211yescharacteristic, complemented
15C3×C4×C3⋊Dic3C2213yescomplemented
16C2×D4×C3⋊D3—36no—
17C2×D3×D12—324no—
18C32×#127—31no—
19#64149—36no—
20#64083—31no—
21#64083—34no—
22#64083—64no—
23C42×C3⋊D3D311yescharacteristic, complemented
24C3×#874D314yescomplemented
25C3×C42×D3—34no—
26#63911—324no—
27C22×C3⋊(C3⋊D3)C2314yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C22×C32×D3C2314yes—
30C2×D3×C3⋊D3—224no—
31C2×C33×C4C2313yes—
32C4⋊(C3⋊(C3⋊D3))—212no—
33C2×C3×C3⋊Dic3C2313yes—
34C3×C4×C3⋊D3D4112yescomplemented
35C32×D12—212no—
36#6291—224no—
37D3×#127—94no—
38C2×#874—91no—
39C23×C3⋊D3—34no—
40C22×D32—316no—
41C2×C32×D4—36no—
42C2×#638—312no—
43D4×C3⋊D3—624no—
44C2×C4⋊(C3⋊D3)—36no—
45C2×C4⋊(C3⋊D3)—324no—
46C2×C4⋊(C3⋊D3)—624no—
47C2×C4×C3⋊D3C2×D313yes—
48C2×C3×D12C2×D3124yescomplemented
49C2×C3×C4×D3—312no—
50C2×#626—348no—
51D3×D12—696no—
52C32×C42—24no—
53C32×C42C2×D314yescomplemented
54C32×C42C2×D311yescharacteristic, complemented
55C4×C3⋊Dic3C2×D313yescomplemented
56C3×C4×Dic3—312no—
57C22×C33C2411yescharacteristic
58C2×C3⋊(C3⋊D3)—212no—
59C2×C3×C3⋊D3C2×D416yes—
60C2×C32×D3—212no—
61D3×C3⋊D3—416no—
62C33×C4C2×D416yescomplemented
63C3×C3⋊Dic3C2×D416yescomplemented
64C2×D3×D4—924no—
65C22×D12—96no—
66C3×#127—34no—
67C2×C3×C42—91no—
68#940—924no—
69#874—96no—
70#874—94no—
71#874—91no—
72#874—184no—
73#874C3⋊D311yescharacteristic, direct product factor
74C42×D3—34no—
75C23×C32—34no—
76C22×C3⋊D3—34no—
77C22×C3⋊D3—316no—
78C22×C3⋊D3—616no—
79C22×C3⋊D3—624no—
80C22×C3⋊D3C22×D311yescharacteristic
81C22×C3×D3—34no—
82C22×C3×D3C22×D3116yes—
83C2×D32—696no—
84C32×D4—612no—
85C2×C32×C4—212no—
86C2×C32×C4C22×D3112yes—
87C2×C32×C4C22×D313yes—
88#638—624no—
89C2×C3⋊Dic3C22×D313yes—
90C4⋊(C3⋊D3)—612no—
91C4⋊(C3⋊D3)—648no—
92C4⋊(C3⋊D3)—1248no—
93C4×C3⋊D3D12112yescomplemented
94C2×C3×Dic3—312no—
95C3×D12—248no—
96C3×C4×D3—348no—
97#626—696no—
98C2×#127—271noSylow
99C2×C33C22×D413yes—
100C3⋊(C3⋊D3)—44no—
101C3×C3⋊D3#12714yescomplemented
102C32×D3—44no—
103C23×D3—916no—
104C23×D3—94no—
105C2×C3×D4—324no—
106C22×C3×C4—93no—
107C2×#67—948no—
108D3×D4—1896no—
109C2×D12—912no—
110C2×D12—924no—
111C2×D12—96no—
112C2×D12—1824no—
113C2×D12—1824no—
114C2×D12C2×C3⋊D316yescomplemented
115C2×C4×D3—312no—
116C3×C42—24no—
117C3×C42—93no—
118C3×C42D3214yescomplemented
119C3×C42C2×C3⋊D311yescharacteristic, complemented
120C4×Dic3—312no—
121C22×C32—24no—
122C22×C32—612no—
123C22×C32C23×D314yes—
124C22×C32C23×D311yescharacteristic
125C2×C3⋊D3—612no—
126C2×C3⋊D3—648no—
127C2×C3⋊D3—1248no—
128C2×C3⋊D3—1216no—
129C2×C3⋊D3C2×D1216yes—
130C2×C3×D3—248no—
131C2×C3×D3—324no—
132D32—1264no—
133C32×C4—224no—
134C32×C4C2×D1216yescomplemented
135C32×C4D3×D4124yescomplemented
136C3⋊Dic3C2×D1216yescomplemented
137C3×Dic3—324no—
138C22×D4—276no—
139#127—31no—
140#127—276no—
141#127—271no—
142C2×C42—91no—
143C33C2×#12711yescharacteristic, complemented, Sylow
144C23×C3—316no—
145C23×C3—91no—
146C22×D3—34no—
147C22×D3—916no—
148C22×D3—94no—
149C22×D3—1816no—
150C22×D3—1824no—
151C22×D3—1896no—
152C22×D3C22×C3⋊D314yes—
153C3×D4—648no—
154C2×C3×C4—212no—
155C2×C3×C4—93no—
156C2×C3×C4—912no—
157C2×C3×C4C2×D32112yes—
158C2×C3×C4C22×C3⋊D313yes—
159#67—1896no—
160C2×Dic3—312no—
161D12—212no—
162D12—1824no—
163D12—1848no—
164D12—1812no—
165D12—3648no—
166C4×D3—348no—
167C2×C32—212no—
168C2×C32—124no—
169C2×C32C22×D1213yes—
170C2×C32C2×D3×D4112yes—
171C3⋊D3—124no—
172C3⋊D3—1216no—
173C3⋊D3—2416no—
174C3⋊D3#87414yesdirect product factor
175C3×D3—316no—
176C3×D3—416no—
177C24—274no—
178C2×D4—36no—
179C2×D4—2712no—
180C2×D4—276no—
181C2×D4—5424no—
182C22×C4—93no—
183C42—93no—
184C42D3×C3⋊D311yescharacteristic, complemented
185C22×C3—24no—
186C22×C3—648no—
187C22×C3—96no—
188C22×C3C22×D3214yes—
189C22×C3C23×C3⋊D311yescharacteristic
190C2×D3—212no—
191C2×D3—324no—
192C2×D3—1848no—
193C2×D3—1812no—
194C2×D3—3648no—
195C2×D3—3664no—
196C2×D3—3616no—
197C3×C4—224no—
198C3×C4—96no—
199C3×C4D3×D12124yescomplemented
200C3×C4D4×C3⋊D316yescomplemented
201Dic3—324no—
202C32—24no—
203C32C2×#87411yescharacteristic, complemented
204C32D3×#12714yescomplemented
205C23—34no—
206C23—91no—
207C23—274no—
208C23—5424no—
209D4—612no—
210D4—5424no—
211D4—5412no—
212C2×C4—93no—
213C2×C4—912no—
214C2×C4C2×D3×C3⋊D313yes—
215C2×C3—212no—
216C2×C3—94no—
217C2×C3—1216no—
218C2×C3C2×D3×D12112yes—
219C2×C3C2×D4×C3⋊D313yes—
220D3—316no—
221D3—44no—
222D3—364no—
223D3—3616no—
224D3—7216no—
225C22—612no—
226C22—96no—
227C22—5412no—
228C22—10816no—
229C22C22×D3×C3⋊D311yescharacteristic
230C4—96no—
231C4C3⋊D3×D1216yescomplemented
232C3—24no—
233C3D3×#87414yescomplemented
234C3C3⋊D3×#12711yescharacteristic, complemented
235C2—94no—
236C2—124no—
237C2—1084no—
238C2C2×C3⋊D3×D1213yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33C2×#127
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×D12C2semidirect
2. #11723915C2semidirect
3. C3×C42×C3⋊D3C2semidirect
4. C32×#874C2semidirect
5. #11723524C2semidirect
9. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
11. C2×C32×D12C22semidirect
12. C2×#6291C22semidirect
14. C33×C42C22semidirect
15. C3×C4×C3⋊Dic3C22semidirect
23. C42×C3⋊D3D3semidirect
24. C3×#874D3semidirect
34. C3×C4×C3⋊D3D4semidirect
48. C2×C3×D12C2×D3semidirect
53. C32×C42C2×D3semidirect
54. C32×C42C2×D3semidirect
55. C4×C3⋊Dic3C2×D3semidirect
62. C33×C4C2×D4semidirect
63. C3×C3⋊Dic3C2×D4semidirect
73. #874C3⋊D3direct (N × H)
93. C4×C3⋊D3D12semidirect
101. C3×C3⋊D3#127semidirect
114. C2×D12C2×C3⋊D3semidirect
118. C3×C42D32semidirect
119. C3×C42C2×C3⋊D3semidirect
134. C32×C4C2×D12semidirect
135. C32×C4D3×D4semidirect
136. C3⋊Dic3C2×D12semidirect
143. C33C2×#127semidirect
174. C3⋊D3#874direct (N × H)
184. C42D3×C3⋊D3semidirect
199. C3×C4D3×D12semidirect
200. C3×C4D4×C3⋊D3semidirect
203. C32C2×#874semidirect
204. C32D3×#127semidirect
231. C4C3⋊D3×D12semidirect
233. C3D3×#874semidirect
234. C3C3⋊D3×#127semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
#117239152
C3×C42×C3⋊D32
C32×#8742
#117235242
C3⋊D3×#1273
D3×#8743