Finite Group Databasebeta

← #13179894

Subgroups — #13179894

Order 1728.

3752 conjugacy classes of subgroups; only the 619 normal subgroups are shown.

normal subgroups only

This group has 3752 conjugacy classes of subgroups — too many to draw. Only its 619 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
1#11724684C212yesmaximal, complemented
2C33×#135C211yesmaximal, characteristic, complemented
3#11723990C212yesmaximal, complemented
4#11723960C212yesmaximal, complemented
5#11723955C214yesmaximal, complemented
6D8×C3⋊(C3⋊D3)C212yesmaximal, complemented
7C2×#88852C211yesmaximal, characteristic, complemented
8#11723942C211yesmaximal, characteristic, complemented
9#10596240—313nomaximal
10C33×#41C2212yescomplemented
11#89080C2214yescomplemented
12D4×C3⋊(C3⋊D3)—24no—
13D4×C3⋊(C3⋊D3)C2214yescomplemented
14#89076—22no—
15#89076C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))—22no—
17C2×C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
18C33×Q16C2211yescharacteristic, complemented
19C33×SD16C2212yescomplemented
20C33×D8C2211yescharacteristic, complemented
21C2×C33×C8C2211yescharacteristic, complemented
22#88862C2214yescomplemented
23#88860C2214yescomplemented
24#88855C2211yescharacteristic, complemented
25#88852C2212yescomplemented
26#88852C2212yescomplemented
27#88850C2212yescomplemented
28C8×C3⋊(C3⋊D3)C2212yescomplemented
29#64366—326no—
30C32×#135D3113yescomplemented
31#64158—326no—
32#64128—326no—
33#64123—352no—
34D8×C3⋊D3—326no—
35C2×#3745—313no—
36#64110—313no—
37C22×C3⋊(C3⋊D3)—22no—
38C22×C3⋊(C3⋊D3)—24no—
39C33×Q8C2312yes—
40C33×D4—22no—
41C33×D4C2312yes—
42C2×C33×C4—22no—
43C2×C33×C4C2311yescharacteristic
44#6312—24no—
45#6312D412yes—
46C4⋊(C3⋊(C3⋊D3))—24no—
47C4⋊(C3⋊(C3⋊D3))—22no—
48C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
49C4⋊(C3⋊(C3⋊D3))C2314yes—
50C4×C3⋊(C3⋊D3)—24no—
51C4×C3⋊(C3⋊D3)C2312yes—
52#6308D411yescharacteristic
53C33×C8C2311yescharacteristic
54C33×C8C2311yescharacteristic
55C3⋊(C3⋊(C3⋊C8))C2312yes—
56#5769—913no—
57C32×#41C2×D3126yescomplemented
58#3832—352no—
59D4×C3⋊D3—352no—
60D4×C3⋊D3—652no—
61#3828—313no—
62#3828—626no—
63C2×C4⋊(C3⋊D3)—326no—
64C2×C4⋊(C3⋊D3)—626no—
65C32×Q16C2×D3113yescomplemented
66C32×SD16C2×D3126yescomplemented
67C32×D8C2×D3113yescomplemented
68C2×C32×C8C2×D3113yescomplemented
69#3755—352no—
70#3753—352no—
71#3748—313no—
72#3745—326no—
73#3745—326no—
74#3743—326no—
75C8×C3⋊D3—326no—
76C22×C33—22no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—24no—
79C2×C3⋊(C3⋊D3)—84no—
80C2×C3⋊(C3⋊D3)—82no—
81C2×C3⋊(C3⋊D3)C2×D412yes—
82C33×C4—22no—
83C33×C4C2×D411yescharacteristic
84C33×C4C2411yescharacteristic
85C3⋊(C3⋊Dic3)C2×D412yes—
86#1009—926no—
87C3×#135C3⋊D3113yescomplemented
88#949—926no—
89#919—926no—
90#914—952no—
91D3×D8—926no—
92C2×D24—913no—
93#901—913no—
94C22×C3⋊D3—652no—
95C22×C3⋊D3—626no—
96C32×Q8C22×D3126yes—
97C32×D4—226no—
98C32×D4C22×D3126yes—
99C2×C32×C4—226no—
100C2×C32×C4C22×D3113yes—
101#638—326no—
102#638—652no—
103C4⋊(C3⋊D3)—313no—
104C4⋊(C3⋊D3)—352no—
105C4⋊(C3⋊D3)—652no—
106C4⋊(C3⋊D3)—626no—
107C4×C3⋊D3—326no—
108C4×C3⋊D3—652no—
109#634—313no—
110C32×C8C22×D3113yes—
111C32×C8C22×D3113yes—
112C3⋊(C3⋊C8)—326no—
113#576—271noSylow
114C2×C33—21no—
115C2×C33—42no—
116C2×C33C22×D411yescharacteristic
117C3⋊(C3⋊D3)—22no—
118C3⋊(C3⋊D3)—44no—
119C3×#41C2×C3⋊D3126yescomplemented
120#239—952no—
121D3×D4—952no—
122D3×D4—1852no—
123#235—913no—
124#235—1826no—
125C2×D12—926no—
126C2×D12—1826no—
127C3×Q16C2×C3⋊D3113yescomplemented
128C3×SD16C2×C3⋊D3126yescomplemented
129C3×D8C2×C3⋊D3113yescomplemented
130C2×C3×C8C2×C3⋊D3113yescomplemented
131#215—952no—
132#213—952no—
133#208—913no—
134D24—926no—
135D24—926no—
136ΓL(1,𝔽25)—926no—
137D3×C8—926no—
138C22×C32—226no—
139C22×C32D3×D4113yes—
140C2×C3⋊D3—326no—
141C2×C3⋊D3—652no—
142C2×C3⋊D3—2452no—
143C2×C3⋊D3—2426no—
144C32×C4—226no—
145C32×C4D3×D4113yes—
146C32×C4C23×D3113yes—
147C3⋊Dic3—326no—
148H(5,𝔽2)—272no—
149AGL(1,ℤ/8)—272no—
150AGL(1,ℤ/8)—274no—
151#135—272no—
152#135C3⋊(C3⋊D3)11yescharacteristic, complemented
153C2×D8—272no—
154C2×D8—271no—
155#131—271no—
156C33#57611yescharacteristic, complemented, Sylow
157C22×D3—1852no—
158C22×D3—1826no—
159C3×Q8C22×C3⋊D3126yes—
160C3×D4—226no—
161C3×D4C22×C3⋊D3126yes—
162C2×C3×C4—226no—
163C2×C3×C4C22×C3⋊D3113yes—
164#67—926no—
165#67—1852no—
166D12—952no—
167D12—913no—
168D12—1852no—
169D12—1826no—
170C4×D3—926no—
171C4×D3—1852no—
172Dic6—913no—
173C3×C8C22×C3⋊D3113yes—
174C3×C8C22×C3⋊D3113yes—
175C3⋊C8—926no—
176C2×C32—213no—
177C2×C32—426no—
178C2×C32C2×D3×D4113yes—
179C3⋊D3—626no—
180C3⋊D3—1252no—
181#41—271no—
182#41—274no—
183#41—542no—
184#41C2×C3⋊(C3⋊D3)12yescomplemented
185C2×D4—274no—
186C2×D4—272no—
187C2×D4—542no—
188C2×D4—544no—
189Q16C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
190SD16—274no—
191SD16C2×C3⋊(C3⋊D3)12yescomplemented
192D8—274no—
193D8—272no—
194D8—272no—
195D8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
196M16—271no—
197M16—272no—
198C2×C8—272no—
199C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
200C22×C3—226no—
201C22×C3D4×C3⋊D3113yes—
202C2×D3—926no—
203C2×D3—1852no—
204C2×D3—7252no—
205C2×D3—7226no—
206C3×C4—226no—
207C3×C4D4×C3⋊D3113yes—
208C3×C4C23×C3⋊D3113yes—
209Dic3—926no—
210C32#5769113yescomplemented
211C23—544no—
212C23—542no—
213Q8—271no—
214Q8C22×C3⋊(C3⋊D3)12yes—
215D4—22no—
216D4—272no—
217D4—274no—
218D4—271no—
219D4—544no—
220D4—544no—
221D4—542no—
222D4C22×C3⋊(C3⋊D3)12yes—
223C2×C4—22no—
224C2×C4—272no—
225C2×C4—544no—
226C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
227C8—272no—
228C8C22×C3⋊(C3⋊D3)11yescharacteristic
229C8C22×C3⋊(C3⋊D3)11yescharacteristic
230C2×C3—213no—
231C2×C3—426no—
232C2×C3C2×D4×C3⋊D3113yes—
233D3—1826no—
234D3—3652no—
235C22—22no—
236C22—272no—
237C22—544no—
238C22—2162no—
239C22—2164no—
240C22D4×C3⋊(C3⋊D3)11yescharacteristic
241C4—22no—
242C4—272no—
243C4D4×C3⋊(C3⋊D3)11yescharacteristic
244C4C23×C3⋊(C3⋊D3)11yescharacteristic
245C3#10596240113yescomplemented
246C2—21no—
247C2—42no—
248C2—542no—
249C2—1084no—
250C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC33×#135C2
socleC2×C33C22×D4
hypercenter#135C3⋊(C3⋊D3)
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. C33×#135C2semidirect
3. #11723990C2semidirect
4. #11723960C2semidirect
5. #11723955C2semidirect
6. D8×C3⋊(C3⋊D3)C2semidirect
7. C2×#88852C2semidirect
8. #11723942C2semidirect
10. C33×#41C22semidirect
11. #89080C22semidirect
13. D4×C3⋊(C3⋊D3)C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C33×Q16C22semidirect
19. C33×SD16C22semidirect
20. C33×D8C22semidirect
21. C2×C33×C8C22semidirect
22. #88862C22semidirect
23. #88860C22semidirect
24. #88855C22semidirect
25. #88852C22semidirect
26. #88852C22semidirect
27. #88850C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
30. C32×#135D3semidirect
57. C32×#41C2×D3semidirect
65. C32×Q16C2×D3semidirect
66. C32×SD16C2×D3semidirect
67. C32×D8C2×D3semidirect
68. C2×C32×C8C2×D3semidirect
87. C3×#135C3⋊D3semidirect
119. C3×#41C2×C3⋊D3semidirect
127. C3×Q16C2×C3⋊D3semidirect
128. C3×SD16C2×C3⋊D3semidirect
129. C3×D8C2×C3⋊D3semidirect
130. C2×C3×C8C2×C3⋊D3semidirect
152. #135C3⋊(C3⋊D3)semidirect
156. C33#576semidirect
184. #41C2×C3⋊(C3⋊D3)semidirect
189. Q16C2×C3⋊(C3⋊D3)semidirect
191. SD16C2×C3⋊(C3⋊D3)semidirect
195. D8C2×C3⋊(C3⋊D3)semidirect
199. C2×C8C2×C3⋊(C3⋊D3)semidirect
210. C32#5769semidirect
245. C3#10596240semidirect

Maximal subgroups

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

subgroupindex
#117246842
C33×#1352
#117239902
#117239602
#117239552
D8×C3⋊(C3⋊D3)2
C2×#888522
#117239422
#105962403