Finite Group Databasebeta

← #13175417

Subgroups — #13175417

Order 1728.

3420 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

normal subgroups only

This group has 3420 conjugacy classes of subgroups — too many to draw. Only its 336 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×#89026C216yesmaximal, complemented
2C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64083C211yesmaximal, characteristic, complemented
4C32×#874C211yesmaximal, characteristic, complemented
5#11723529C216yesmaximal, complemented
6C3⋊D3×#127—31nomaximal
7#10594008—34nomaximal
8C22×D3×C3⋊D3C2214yes—
9C2×C4×C3⋊(C3⋊D3)C2213yes—
10C2×C3×C4⋊(C3⋊D3)C2216yescomplemented
11C2×C32×D12C2216yescomplemented
12C2×#6290C22112yescomplemented
13#89026—224no—
14C33×C42C2211yescharacteristic, complemented
15C4×C3⋊(C3⋊Dic3)C2213yescomplemented
16C2×D4×C3⋊D3—36no—
17C2×#3802—324no—
18C32×#127—31no—
19#64149—36no—
20#64083D311yescharacteristic, complemented
21C42×C3⋊D3—31no—
22C42×C3⋊D3—34no—
23C42×C3⋊D3—64no—
24C3×#874—34no—
25C3×#874D314yescomplemented
26#63914—324no—
27C22×C3⋊(C3⋊D3)C2311yescharacteristic
28C22×C3×C3⋊D3C2314yes—
29C22×C32×D3C2314yes—
30C2×D3×C3⋊D3—224no—
31C2×C33×C4C2313yes—
32C2×C3⋊(C3⋊Dic3)C2313yes—
33C4×C3⋊(C3⋊D3)D4112yescomplemented
34C3×C4⋊(C3⋊D3)—212no—
35C32×D12—212no—
36#6290—224no—
37D3×#127—94no—
38D3×#127—91no—
39C23×C3⋊D3—34no—
40C22×D32—316no—
41C2×C32×D4—36no—
42C2×#638—312no—
43D4×C3⋊D3—624no—
44C2×C4⋊(C3⋊D3)C2×D316yescomplemented
45C2×C4×C3⋊D3—33no—
46C2×C4×C3⋊D3—312no—
47C2×C4×C3⋊D3—612no—
48C2×C3×D12—324no—
49C2×C3×D12C2×D3124yescomplemented
50C2×#625—348no—
51#3802—696no—
52C32×C42—24no—
53C32×C42C2×D314yescomplemented
54C32×C42C2×D311yescharacteristic, complemented
55C4×C3⋊Dic3—33no—
56C4×C3⋊Dic3—312no—
57C4×C3⋊Dic3—612no—
58C22×C33C2411yescharacteristic
59C2×C3⋊(C3⋊D3)C2×D416yes—
60C2×C3×C3⋊D3—212no—
61C2×C32×D3—212no—
62D3×C3⋊D3—416no—
63C33×C4C2×D416yescomplemented
64C3⋊(C3⋊Dic3)C2×D416yescomplemented
65C2×D3×D4—924no—
66C2×D3×D4—96no—
67C3×#127—34no—
68C3×#127—91no—
69#940—924no—
70#940—96no—
71#874—34no—
72#874C3⋊D311yescharacteristic, complemented
73C42×D3—94no—
74C42×D3—91no—
75C42×D3—184no—
76C23×C32—34no—
77C22×C3⋊D3—31no—
78C22×C3⋊D3—34no—
79C22×C3⋊D3—64no—
80C22×C3⋊D3—624no—
81C22×C3⋊D3C22×D314yes—
82C22×C3×D3—316no—
83C22×C3×D3C22×D3116yes—
84C2×D32—696no—
85C32×D4—612no—
86C2×C32×C4—212no—
87C2×C32×C4C22×D3112yes—
88C2×C32×C4C22×D313yes—
89#638—624no—
90C2×C3⋊Dic3—33no—
91C2×C3⋊Dic3—312no—
92C2×C3⋊Dic3—612no—
93C4⋊(C3⋊D3)—212no—
94C4×C3⋊D3—312no—
95C4×C3⋊D3—348no—
96C4×C3⋊D3—648no—
97C3×D12—248no—
98C3×D12—648no—
99#625—696no—
100C2×#127—271noSylow
101C2×C33C22×D413yes—
102C3⋊(C3⋊D3)#12714yescomplemented
103C3×C3⋊D3—44no—
104C32×D3—44no—
105C23×D3—916no—
106C23×D3—94no—
107C2×C3×D4—324no—
108C2×C3×D4—96no—
109C2×#67—948no—
110C2×#67—912no—
111D3×D4—1824no—
112D3×D4—1896no—
113C2×D12—324no—
114C2×D12C2×C3⋊D316yescomplemented
115C2×C4×D3—912no—
116C2×C4×D3—93no—
117C2×C4×D3—1812no—
118C3×C42—24no—
119C3×C42D3214yescomplemented
120C3×C42C2×C3⋊D311yescharacteristic, complemented
121C4×Dic3—912no—
122C4×Dic3—93no—
123C4×Dic3—1812no—
124C22×C32—24no—
125C22×C32—612no—
126C22×C32C23×D314yes—
127C22×C32C23×D311yescharacteristic
128C2×C3⋊D3—212no—
129C2×C3⋊D3—36no—
130C2×C3⋊D3—324no—
131C2×C3⋊D3—624no—
132C2×C3⋊D3—1216no—
133C2×C3×D3—248no—
134C2×C3×D3—648no—
135D32—1264no—
136C32×C4—224no—
137C32×C4D3×D4124yescomplemented
138C32×C4D3×D416yescomplemented
139C3⋊Dic3—36no—
140C3⋊Dic3—324no—
141C3⋊Dic3—624no—
142C22×D4—276no—
143#127—31no—
144#127—91no—
145#127—276no—
146C2×C42—271no—
147C33C2×#12711yescharacteristic, complemented, Sylow
148C23×C3—316no—
149C23×C3—94no—
150C22×D3—316no—
151C22×D3—94no—
152C22×D3—91no—
153C22×D3—184no—
154C22×D3—1824no—
155C22×D3—1896no—
156C22×D3C22×C3⋊D314yes—
157C3×D4—648no—
158C3×D4—1812no—
159C2×C3×C4—212no—
160C2×C3×C4C2×D32112yes—
161C2×C3×C4C22×C3⋊D313yes—
162#67—1824no—
163#67—1896no—
164C2×Dic3—912no—
165C2×Dic3—93no—
166C2×Dic3—1812no—
167D12—212no—
168D12—648no—
169C4×D3—948no—
170C4×D3—912no—
171C4×D3—1848no—
172C2×C32—212no—
173C2×C32—124no—
174C2×C32C2×D3×D4112yes—
175C2×C32C2×D3×D413yes—
176C3⋊D3—34no—
177C3⋊D3—316no—
178C3⋊D3—44no—
179C3⋊D3—616no—
180C3×D3—416no—
181C3×D3—1216no—
182C24—274no—
183C2×D4—36no—
184C2×D4—96no—
185C2×D4—2712no—
186C2×D4—5424no—
187C22×C4—273no—
188C42—273no—
189C42D3×C3⋊D311yescharacteristic, complemented
190C22×C3—24no—
191C22×C3—648no—
192C22×C3—1812no—
193C22×C3C22×D3214yes—
194C22×C3C23×C3⋊D311yescharacteristic
195C2×D3—212no—
196C2×D3—648no—
197C2×D3—924no—
198C2×D3—96no—
199C2×D3—1824no—
200C2×D3—3616no—
201C2×D3—3664no—
202C3×C4—224no—
203C3×C4#3802124yescomplemented
204C3×C4D4×C3⋊D316yescomplemented
205Dic3—924no—
206Dic3—96no—
207Dic3—1824no—
208C32—24no—
209C32D3×#12714yescomplemented
210C32D3×#12711yescharacteristic, complemented
211C23—34no—
212C23—94no—
213C23—271no—
214C23—5424no—
215D4—612no—
216D4—1812no—
217D4—5424no—
218C2×C4—273no—
219C2×C4—2712no—
220C2×C4C2×D3×C3⋊D313yes—
221C2×C3—212no—
222C2×C3—1216no—
223C2×C3—364no—
224C2×C3C2×#3802112yes—
225C2×C3C2×D4×C3⋊D313yes—
226D3—44no—
227D3—94no—
228D3—916no—
229D3—1216no—
230D3—1816no—
231C22—612no—
232C22—1812no—
233C22—276no—
234C22—10816no—
235C22C22×D3×C3⋊D311yescharacteristic
236C4—276no—
237C4#8902616yescomplemented
238C3—24no—
239C3#1059400814yescomplemented
240C3C3⋊D3×#12711yescharacteristic, complemented
241C2—124no—
242C2—274no—
243C2—364no—
244C2C2×#8902613yes—
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×#89026C2semidirect
2. C42×C3⋊(C3⋊D3)C2semidirect
3. C3×#64083C2semidirect
4. C32×#874C2semidirect
5. #11723529C2semidirect
10. C2×C3×C4⋊(C3⋊D3)C22semidirect
11. C2×C32×D12C22semidirect
12. C2×#6290C22semidirect
14. C33×C42C22semidirect
15. C4×C3⋊(C3⋊Dic3)C22semidirect
20. #64083D3semidirect
25. C3×#874D3semidirect
33. C4×C3⋊(C3⋊D3)D4semidirect
44. C2×C4⋊(C3⋊D3)C2×D3semidirect
49. C2×C3×D12C2×D3semidirect
53. C32×C42C2×D3semidirect
54. C32×C42C2×D3semidirect
63. C33×C4C2×D4semidirect
64. C3⋊(C3⋊Dic3)C2×D4semidirect
72. #874C3⋊D3semidirect
102. C3⋊(C3⋊D3)#127semidirect
114. C2×D12C2×C3⋊D3semidirect
119. C3×C42D32semidirect
120. C3×C42C2×C3⋊D3semidirect
137. C32×C4D3×D4semidirect
138. C32×C4D3×D4semidirect
147. C33C2×#127semidirect
189. C42D3×C3⋊D3semidirect
203. C3×C4#3802semidirect
204. C3×C4D4×C3⋊D3semidirect
209. C32D3×#127semidirect
210. C32D3×#127semidirect
237. C4#89026semidirect
239. C3#10594008semidirect
240. C3C3⋊D3×#127semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C42×C3⋊(C3⋊D3)2
C3×#640832
C32×#8742
#117235292
C3⋊D3×#1273
#105940083