Finite Group Databasebeta

← #127×(C3⋊D3)⋊C3 #13153273

Subgroups — #127×(C3⋊D3)⋊C3 #13153273

Order 1728.

1851 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 1851 conjugacy classes of subgroups — too many to draw. Only its 288 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×D4×(C3⋊D3)⋊C3C216yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#64149⋊C3C216yesmaximal, complemented
4#11721385C211yesmaximal, characteristic, complemented
5C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
6C3⋊D3×#127C311yesmaximal, characteristic, complemented
7C3×D3×#127—31nomaximal
8C23×(C3⋊D3)⋊C3C2214yes—
9C2×D4×H(3,𝔽3)C2216yescomplemented
10C2×#638⋊C3C22112yescomplemented
11D4×(C3⋊D3)⋊C3—224no—
12C2×C4⋊((C3⋊D3)⋊C3)C2216yescomplemented
13C2×C4×(C3⋊D3)⋊C3C2213yes—
14C42×H(3,𝔽3)C2211yescharacteristic, complemented
15C4×(C3⋊Dic3)⋊C3C2213yescomplemented
16C2×D4×C3⋊D3C2×C316yescomplemented
17C2×C3×D3×D4—36no—
18C32×#127—21no—
19C32×#127D311yescharacteristic, complemented
20C32×#127C2×C311yescharacteristic, complemented
21#64149C2×C316yescomplemented
22#64083C2×C311yescharacteristic, complemented
23C42×C3⋊D3C2×C311yescharacteristic, complemented
24C3×#940—36no—
25C3×#874—31no—
26C3×C42×D3—31no—
27C23×H(3,𝔽3)C2314yes—
28C22×(C3⋊D3)⋊C3—224no—
29C22×(C3⋊D3)⋊C3C2314yes—
30C22×(C3⋊D3)⋊C3C2311yescharacteristic
31D4×H(3,𝔽3)—212no—
32C2×C4×H(3,𝔽3)C2313yes—
33#638⋊C3—224no—
34C2×(C3⋊Dic3)⋊C3C2313yes—
35C4⋊((C3⋊D3)⋊C3)—212no—
36C4×(C3⋊D3)⋊C3D4112yescomplemented
37C2×C3×#127—91no—
38D3×#127—31no—
39D3×#127—91no—
40C23×C3⋊D3C22×C314yes—
41C23×C3×D3—34no—
42C2×C32×D4—26no—
43C2×C32×D4C2×D316yescomplemented
44C2×C32×D4C22×C316yescomplemented
45C2×#638C22×C3112yescomplemented
46D4×C3⋊D3—224no—
47C2×C4⋊(C3⋊D3)C22×C316yescomplemented
48C2×C4×C3⋊D3C22×C313yes—
49C2×C3×#67—312no—
50C3×D3×D4—624no—
51C2×C3×D12—36no—
52C2×C3×C4×D3—33no—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic, complemented
55C32×C42C22×C311yescharacteristic, complemented
56C4×C3⋊Dic3C22×C313yescomplemented
57C3×C4×Dic3—33no—
58C22×H(3,𝔽3)—212no—
59C22×H(3,𝔽3)C2411yescharacteristic
60C2×(C3⋊D3)⋊C3—212no—
61C2×(C3⋊D3)⋊C3—416no—
62C2×(C3⋊D3)⋊C3C2×D416yes—
63C4×H(3,𝔽3)C2×D416yescomplemented
64(C3⋊Dic3)⋊C3C2×D416yescomplemented
65C22×C3×D4—96no—
66C2×D3×D4—36no—
67C2×D3×D4—96no—
68C3×#127—31no—
69C3×#127—31no—
70C3×#127—61no—
71C3×#127—91no—
72C3×#127—96no—
73C3×#127C3×D311yescharacteristic
74C2×C3×C42—91no—
75#940—36no—
76#940—96no—
77#874—31no—
78#874—91no—
79C42×D3—31no—
80C42×D3—91no—
81C23×C32—24no—
82C23×C32C22×D314yes—
83C23×C32C23×C314yes—
84C22×C3⋊D3—224no—
85C22×C3⋊D3C23×C314yes—
86C22×C3⋊D3C23×C311yescharacteristic
87C22×C3×D3—31no—
88C22×C3×D3—34no—
89C22×C3×D3—624no—
90C32×D4—212no—
91C32×D4—212no—
92C32×D4—412no—
93C2×C32×C4—23no—
94C2×C32×C4C22×D313yes—
95C2×C32×C4C23×C313yes—
96#638—224no—
97C2×C3⋊Dic3C23×C313yes—
98C4⋊(C3⋊D3)—212no—
99C4×C3⋊D3C3×D4112yescomplemented
100C3×#67—624no—
101C2×C3×Dic3—33no—
102C3×D12—612no—
103C3×C4×D3—312no—
104C2×#127—91noSylow
105C2×H(3,𝔽3)—44no—
106C2×H(3,𝔽3)C22×D413yes—
107(C3⋊D3)⋊C3—44no—
108(C3⋊D3)⋊C3#12714yesdirect product factor
109C24×C3—94no—
110C23×D3—34no—
111C23×D3—94no—
112C2×C3×D4—36no—
113C2×C3×D4—36no—
114C2×C3×D4—66no—
115C2×C3×D4—96no—
116C2×C3×D4—912no—
117C2×C3×D4—1824no—
118C2×C3×D4C2×C3×D316yes—
119C22×C3×C4—93no—
120C2×#67—312no—
121C2×#67—912no—
122D3×D4—624no—
123D3×D4—1824no—
124C2×D12—36no—
125C2×D12—96no—
126C2×C4×D3—33no—
127C2×C4×D3—93no—
128C3×C42—31no—
129C3×C42—31no—
130C3×C42—61no—
131C3×C42—93no—
132C3×C42C2×C3×D311yescharacteristic
133C4×Dic3—33no—
134C4×Dic3—93no—
135C22×C32—212no—
136C22×C32—212no—
137C22×C32—21no—
138C22×C32—412no—
139C22×C32C23×D311yescharacteristic
140C22×C32C24×C311yescharacteristic
141C2×C3⋊D3—212no—
142C2×C3⋊D3—416no—
143C2×C3⋊D3C2×C3×D416yes—
144C2×C3×D3—36no—
145C2×C3×D3—612no—
146C2×C3×D3—1216no—
147C32×C4—26no—
148C32×C4D3×D416yescomplemented
149C32×C4C2×C3×D416yescomplemented
150C3⋊Dic3C2×C3×D416yescomplemented
151C3×Dic3—36no—
152C22×D4—96no—
153#127—96no—
154#127—91no—
155#127(C3⋊D3)⋊C311yescharacteristic, direct product factor
156C2×C42—91no—
157H(3,𝔽3)C2×#12711yescharacteristic, complemented, Sylow
158C23×C3—34no—
159C23×C3—34no—
160C23×C3—64no—
161C23×C3—91no—
162C23×C3—94no—
163C23×C3—1824no—
164C23×C3C22×C3×D314yes—
165C22×D3—31no—
166C22×D3—34no—
167C22×D3—624no—
168C22×D3—91no—
169C22×D3—94no—
170C22×D3—1824no—
171C3×D4—212no—
172C3×D4—612no—
173C3×D4—612no—
174C3×D4—1212no—
175C3×D4—1812no—
176C3×D4—1824no—
177C2×C3×C4—33no—
178C2×C3×C4—33no—
179C2×C3×C4—63no—
180C2×C3×C4—912no—
181C2×C3×C4—93no—
182C2×C3×C4C22×C3×D313yes—
183#67—624no—
184#67—1824no—
185C2×Dic3—33no—
186C2×Dic3—93no—
187D12—612no—
188D12—1812no—
189C4×D3—312no—
190C4×D3—912no—
191C2×C32—23no—
192C2×C32—44no—
193C2×C32—44no—
194C2×C32—84no—
195C2×C32C2×D3×D413yes—
196C2×C32C22×C3×D413yes—
197C3⋊D3—44no—
198C3⋊D3C3×#12714yescomplemented
199C3×D3—34no—
200C3×D3—124no—
201C24—94no—
202C2×D4—912no—
203C2×D4—96no—
204C2×D4—1824no—
205C2×D4C2×(C3⋊D3)⋊C316yescomplemented
206C22×C4—93no—
207C42—93no—
208C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
209C22×C3—212no—
210C22×C3—31no—
211C22×C3—31no—
212C22×C3—612no—
213C22×C3—612no—
214C22×C3—61no—
215C22×C3—96no—
216C22×C3—1212no—
217C22×C3—1812no—
218C22×C3—3616no—
219C22×C3C23×C3×D311yescharacteristic
220C2×D3—36no—
221C2×D3—612no—
222C2×D3—96no—
223C2×D3—1216no—
224C2×D3—1812no—
225C2×D3—3616no—
226C3×C4—36no—
227C3×C4—36no—
228C3×C4—66no—
229C3×C4—96no—
230C3×C4C3×D3×D416yes—
231Dic3—36no—
232Dic3—96no—
233C32—21no—
234C32D3×#12711yescharacteristic, complemented
235C32C2×C3×#12711yescharacteristic, complemented
236C23—91no—
237C23—94no—
238C23—1824no—
239C23C22×(C3⋊D3)⋊C314yes—
240D4—212no—
241D4—1824no—
242D4—1812no—
243C2×C4—93no—
244C2×C4—912no—
245C2×C4C22×(C3⋊D3)⋊C313yes—
246C2×C3—33no—
247C2×C3—33no—
248C2×C3—44no—
249C2×C3—63no—
250C2×C3—94no—
251C2×C3—124no—
252C2×C3—124no—
253C2×C3—244no—
254C2×C3—364no—
255C2×C3C2×C3×D3×D413yes—
256D3—34no—
257D3—94no—
258D3—124no—
259D3—364no—
260C22—212no—
261C22—96no—
262C22—1812no—
263C22—3616no—
264C22C23×(C3⋊D3)⋊C311yescharacteristic
265C4—96no—
266C4D4×(C3⋊D3)⋊C316yescomplemented
267C3—31no—
268C3—31no—
269C3—61no—
270C3C3×D3×#12711yescharacteristic
271C2—44no—
272C2—94no—
273C2—364no—
274C2C2×D4×(C3⋊D3)⋊C313yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C2×C3×#127
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×D4×(C3⋊D3)⋊C3C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #64149⋊C3C2semidirect
4. #11721385C2semidirect
5. C42×(C3⋊D3)⋊C3C2semidirect
6. C3⋊D3×#127C3semidirect
9. C2×D4×H(3,𝔽3)C22semidirect
10. C2×#638⋊C3C22semidirect
12. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
14. C42×H(3,𝔽3)C22semidirect
15. C4×(C3⋊Dic3)⋊C3C22semidirect
16. C2×D4×C3⋊D3C2×C3semidirect
19. C32×#127D3semidirect
20. C32×#127C2×C3semidirect
21. #64149C2×C3semidirect
22. #64083C2×C3semidirect
23. C42×C3⋊D3C2×C3semidirect
36. C4×(C3⋊D3)⋊C3D4semidirect
43. C2×C32×D4C2×D3semidirect
44. C2×C32×D4C22×C3semidirect
45. C2×#638C22×C3semidirect
47. C2×C4⋊(C3⋊D3)C22×C3semidirect
54. C32×C42C2×D3semidirect
55. C32×C42C22×C3semidirect
56. C4×C3⋊Dic3C22×C3semidirect
63. C4×H(3,𝔽3)C2×D4semidirect
64. (C3⋊Dic3)⋊C3C2×D4semidirect
99. C4×C3⋊D3C3×D4semidirect
108. (C3⋊D3)⋊C3#127direct (N × H)
148. C32×C4D3×D4semidirect
149. C32×C4C2×C3×D4semidirect
150. C3⋊Dic3C2×C3×D4semidirect
155. #127(C3⋊D3)⋊C3direct (N × H)
157. H(3,𝔽3)C2×#127semidirect
198. C3⋊D3C3×#127semidirect
205. C2×D4C2×(C3⋊D3)⋊C3semidirect
208. C42C2×(C3⋊D3)⋊C3semidirect
234. C32D3×#127semidirect
235. C32C2×C3×#127semidirect
266. C4D4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1272
#64149⋊C32
#117213852
C42×(C3⋊D3)⋊C32
C3⋊D3×#1273
C3×D3×#1273