Finite Group Databasebeta

← #102×C3⋊(C3⋊D3) #13163408

Subgroups — #102×C3⋊(C3⋊D3) #13163408

Order 1728.

2828 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 2828 conjugacy classes of subgroups — too many to draw. Only its 427 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⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722191C211yesmaximal, characteristic, complemented
7#11722179C211yesmaximal, characteristic, complemented
8C3⋊D3×#102—313nomaximal
9C23×C3⋊(C3⋊D3)—21no—
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12D4×C3⋊(C3⋊D3)—22no—
13D4×C3⋊(C3⋊D3)C414yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—21no—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic, complemented
18C33×C4⋊C4C2211yescharacteristic, complemented
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20#88857—21no—
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
22C8×C3⋊(C3⋊D3)—22no—
23C2×D4×C3⋊D3—313no—
24C2×C8×C3⋊D3—313no—
25C4⋊C4×C3⋊D3—313no—
26C32×#102D3113yescomplemented
27#63659—313no—
28#63648—313no—
29#63636—313no—
30C23×C33—21no—
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3⋊(C3⋊D3)—24no—
33C22×C3⋊(C3⋊D3)—42no—
34C22×C3⋊(C3⋊D3)—42no—
35C22×C3⋊(C3⋊D3)D411yescharacteristic
36C33×D4—21no—
37C33×D4C2×C412yescomplemented
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40#6312—24no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
45C4×C3⋊(C3⋊D3)—42no—
46C4×C3⋊(C3⋊D3)C2×C412yes—
47C4×C3⋊(C3⋊D3)D412yes—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50D3×#102—913no—
51C23×C3⋊D3—613no—
52C2×C32×D4C2×D3113yes—
53C2×#638—613no—
54D4×C3⋊D3—352no—
55D4×C3⋊D3—626no—
56C2×C4⋊(C3⋊D3)—313no—
57C2×C4×C3⋊D3—313no—
58C2×C4×C3⋊D3—613no—
59C2×C32×C8C2×D3113yescomplemented
60C32×C4⋊C4C2×D3113yescomplemented
61C4⋊(C3⋊Dic3)—313no—
62#3750—613no—
63C2×C3⋊(C3⋊C8)—313no—
64C8×C3⋊D3—626no—
65C22×C33—22no—
66C22×C33—41no—
67C22×C33—41no—
68C22×C33C2×D411yescharacteristic
69C2×C3⋊(C3⋊D3)—22no—
70C2×C3⋊(C3⋊D3)—41no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C3⋊(C3⋊D3)—48no—
73C2×C3⋊(C3⋊D3)C22⋊C412yes—
74C2×C3⋊(C3⋊D3)D812yes—
75C2×C3⋊(C3⋊D3)SD1612yes—
76C33×C4—41no—
77C33×C4C22×C411yescharacteristic
78C33×C4C2×D411yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3⋊(C3⋊Dic3)C22⋊C412yes—
81C2×D3×D4—913no—
82C2×D3×C8—913no—
83D3×C4⋊C4—913no—
84C3×#102C3⋊D3113yescomplemented
85#832—913no—
86#821—913no—
87#809—913no—
88C23×C32—213no—
89C22×C3⋊D3—313no—
90C22×C3⋊D3—652no—
91C22×C3⋊D3—613no—
92C22×C3⋊D3—1226no—
93C22×C3⋊D3—1226no—
94C32×D4—213no—
95C32×D4C4×D3126yescomplemented
96C2×C32×C4—213no—
97C2×C32×C4C22×D3113yes—
98#638—652no—
99C2×C3⋊Dic3—313no—
100C2×C3⋊Dic3—613no—
101C4⋊(C3⋊D3)—326no—
102C4⋊(C3⋊D3)—613no—
103C4×C3⋊D3—326no—
104C4×C3⋊D3—326no—
105C4×C3⋊D3—1226no—
106C32×C8—213no—
107C3⋊(C3⋊C8)—613no—
108C2×#102—271noSylow
109C2×C33—42no—
110C2×C33C2×C22⋊C411yescharacteristic
111C2×C33C2×D811yescharacteristic
112C2×C33C2×SD1611yescharacteristic
113C3⋊(C3⋊D3)—42no—
114C3⋊(C3⋊D3)#10214yesdirect product factor
115C23×D3—1813no—
116C2×C3×D4C2×C3⋊D3113yes—
117C2×#67—1813no—
118D3×D4—952no—
119D3×D4—1826no—
120C2×D12—913no—
121C2×C4×D3—913no—
122C2×C4×D3—1813no—
123C2×C3×C8C2×C3⋊D3113yescomplemented
124C3×C4⋊C4C2×C3⋊D3113yescomplemented
125C4⋊Dic3—913no—
126Dic3⋊C4—1813no—
127C2×C3⋊C8—913no—
128D3×C8—1826no—
129C22×C32—226no—
130C22×C32—413no—
131C22×C32—413no—
132C22×C32D3×D4113yes—
133C2×C3⋊D3—326no—
134C2×C3⋊D3—326no—
135C2×C3⋊D3—326no—
136C2×C3⋊D3—626no—
137C2×C3⋊D3—12104no—
138C2×C3⋊D3—1213no—
139C2×C3⋊D3—1213no—
140C32×C4—413no—
141C32×C4C2×C4×D3113yes—
142C32×C4D3×D4113yes—
143C3⋊Dic3—326no—
144C3⋊Dic3—1213no—
145C22×D4—271no—
146C22×C8—271no—
147C2×C4⋊C4—271no—
148#102—271no—
149#102—271no—
150#102—271no—
151#102C3⋊(C3⋊D3)11yescharacteristic, direct product factor
152C33C2×#10211yescharacteristic, complemented, Sylow
153C23×C3—213no—
154C22×D3—913no—
155C22×D3—1852no—
156C22×D3—1813no—
157C22×D3—3626no—
158C22×D3—3626no—
159C3×D4—213no—
160C3×D4C4×C3⋊D3126yescomplemented
161C2×C3×C4—213no—
162C2×C3×C4C22×C3⋊D3113yes—
163#67—1852no—
164C2×Dic3—913no—
165C2×Dic3—1813no—
166D12—926no—
167D12—1813no—
168C4×D3—926no—
169C4×D3—926no—
170C4×D3—3626no—
171C3×C8—213no—
172C3⋊C8—1813no—
173C2×C32—426no—
174C2×C32D3×C22⋊C4113yes—
175C2×C32D3×D8113yes—
176C2×C32D3×SD16113yes—
177C3⋊D3—352no—
178C3⋊D3—1226no—
179C24—541no—
180C2×D4—274no—
181C2×D4—271no—
182C2×D4—542no—
183C2×D4—541no—
184C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
185C22×C4—271no—
186C22×C4—541no—
187C2×C8—271no—
188C2×C8—542no—
189C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
190C4⋊C4—271no—
191C4⋊C4—541no—
192C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
193C22×C3—226no—
194C22×C3—413no—
195C22×C3—413no—
196C22×C3D4×C3⋊D3113yes—
197C2×D3—926no—
198C2×D3—926no—
199C2×D3—926no—
200C2×D3—1826no—
201C2×D3—36104no—
202C2×D3—3613no—
203C2×D3—3613no—
204C3×C4—413no—
205C3×C4C2×C4×C3⋊D3113yes—
206C3×C4D4×C3⋊D3113yes—
207Dic3—926no—
208Dic3—3613no—
209C32D3×#102113yescomplemented
210C23—21no—
211C23—271no—
212C23—544no—
213C23—541no—
214C23—1082no—
215C23—1082no—
216D4—21no—
217D4—272no—
218D4—544no—
219D4—541no—
220D4C4×C3⋊(C3⋊D3)12yescomplemented
221C2×C4—21no—
222C2×C4—271no—
223C2×C4—272no—
224C2×C4—272no—
225C2×C4—541no—
226C2×C4—1082no—
227C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
228C8—21no—
229C8—541no—
230C2×C3—426no—
231C2×C3C22⋊C4×C3⋊D3113yes—
232C2×C3D8×C3⋊D3113yes—
233C2×C3SD16×C3⋊D3113yes—
234D3—952no—
235D3—3626no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—272no—
240C22—272no—
241C22—272no—
242C22—542no—
243C22—1088no—
244C22—1081no—
245C22—1081no—
246C22D4×C3⋊(C3⋊D3)11yescharacteristic
247C4—41no—
248C4—272no—
249C4—1081no—
250C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
251C4D4×C3⋊(C3⋊D3)11yescharacteristic
252C3C3⋊D3×#102113yescomplemented
253C2—42no—
254C2—274no—
255C2—1082no—
256C2C22⋊C4×C3⋊(C3⋊D3)11yescharacteristic
257C2D8×C3⋊(C3⋊D3)11yescharacteristic
258C2SD16×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33C2×#102
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×C8×C3⋊(C3⋊D3)C2semidirect
3. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
4. C33×#102C2semidirect
5. #11722202C2semidirect
6. #11722191C2semidirect
7. #11722179C2semidirect
13. D4×C3⋊(C3⋊D3)C4semidirect
17. C2×C33×C8C22semidirect
18. C33×C4⋊C4C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
21. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C32×#102D3semidirect
37. C33×D4C2×C4semidirect
44. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
59. C2×C32×C8C2×D3semidirect
60. C32×C4⋊C4C2×D3semidirect
84. C3×#102C3⋊D3semidirect
95. C32×D4C4×D3semidirect
114. C3⋊(C3⋊D3)#102direct (N × H)
123. C2×C3×C8C2×C3⋊D3semidirect
124. C3×C4⋊C4C2×C3⋊D3semidirect
151. #102C3⋊(C3⋊D3)direct (N × H)
152. C33C2×#102semidirect
160. C3×D4C4×C3⋊D3semidirect
189. C2×C8C2×C3⋊(C3⋊D3)semidirect
192. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
209. C32D3×#102semidirect
220. D4C4×C3⋊(C3⋊D3)semidirect
252. C3C3⋊D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×C8×C3⋊(C3⋊D3)2
C4⋊C4×C3⋊(C3⋊D3)2
C33×#1022
#117222022
#117221912
#117221792
C3⋊D3×#1023