Finite Group Databasebeta

← #13179882

Subgroups — #13179882

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#11724679C212yesmaximal, complemented
2C2×C33×D8C211yesmaximal, characteristic, complemented
3#11723973C212yesmaximal, complemented
4#11723953C212yesmaximal, complemented
5#11723952C214yesmaximal, complemented
6D8×C3⋊(C3⋊D3)C212yesmaximal, complemented
7#11723945C211yesmaximal, characteristic, complemented
8#11723942C211yesmaximal, characteristic, complemented
9#10596228—313nomaximal
10C2×C33×D4C2212yescomplemented
11C2×#6312—24no—
12#89078—22no—
13#89078C2214yescomplemented
14D4×C3⋊(C3⋊D3)—22no—
15D4×C3⋊(C3⋊D3)C2214yescomplemented
16#89076C2211yescharacteristic
17#89076C2212yescomplemented
18C33×D8C2212yescomplemented
19C33×D8C2212yescomplemented
20C2×C33×C8C2211yescharacteristic, complemented
21#88861C2214yescomplemented
22#88860C2214yescomplemented
23#88855C2211yescharacteristic, complemented
24#88853C2211yescharacteristic, complemented
25#88852C2211yescharacteristic, complemented
26#88851C2212yescomplemented
27#88850C2212yescomplemented
28C8×C3⋊(C3⋊D3)C2212yescomplemented
29#64361—326no—
30C2×C32×D8D3113yescomplemented
31#64141—326no—
32#64121—326no—
33#64120—352no—
34D8×C3⋊D3—326no—
35#64113—313no—
36#64110—313no—
37C23×C33—22no—
38C22×C3⋊(C3⋊D3)—24no—
39C33×D4—22no—
40C33×D4C2314yes—
41C2×C33×C4C2311yescharacteristic
42#6312—22no—
43#6312—24no—
44#6312—24no—
45#6312D412yes—
46C2×C3⋊(C3⋊Dic3)—24no—
47C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
48C4⋊(C3⋊(C3⋊D3))C2312yes—
49C4×C3⋊(C3⋊D3)—22no—
50C4×C3⋊(C3⋊D3)C2312yes—
51#6308D411yescharacteristic
52#6308C2312yes—
53C33×C8C2311yescharacteristic
54C33×C8C2311yescharacteristic
55C3⋊(C3⋊(C3⋊C8))C2312yes—
56#5757—913no—
57C2×C32×D4C2×D3126yescomplemented
58C2×#638—652no—
59#3830—352no—
60#3830—626no—
61D4×C3⋊D3—352no—
62D4×C3⋊D3—626no—
63#3828—313no—
64#3828—326no—
65C32×D8C2×D3126yescomplemented
66C32×D8C2×D3126yescomplemented
67C2×C32×C8C2×D3113yescomplemented
68#3754—352no—
69#3753—352no—
70#3748—313no—
71#3746—313no—
72#3745—313no—
73#3744—326no—
74#3743—326no—
75C8×C3⋊D3—326no—
76C22×C33—24no—
77C22×C33—82no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—22no—
80C2×C3⋊(C3⋊D3)—84no—
81C2×C3⋊(C3⋊D3)C2×D412yes—
82C33×C4C2×D411yescharacteristic
83C33×C4C2411yescharacteristic
84C3⋊(C3⋊Dic3)—22no—
85C3⋊(C3⋊Dic3)C2×D412yes—
86#1004—926no—
87C2×C3×D8C3⋊D3113yescomplemented
88#932—926no—
89#912—926no—
90#911—952no—
91D3×D8—926no—
92#904—913no—
93#901—913no—
94C23×C32—226no—
95C22×C3⋊D3—652no—
96C32×D4—226no—
97C32×D4C22×D3152yes—
98C2×C32×C4C22×D3113yes—
99#638—326no—
100#638—652no—
101#638—652no—
102#638—626no—
103C2×C3⋊Dic3—652no—
104C4⋊(C3⋊D3)—313no—
105C4⋊(C3⋊D3)—326no—
106C4×C3⋊D3—326no—
107C4×C3⋊D3—626no—
108#634—313no—
109#634—326no—
110C32×C8C22×D3113yes—
111C32×C8C22×D3113yes—
112C3⋊(C3⋊C8)—326no—
113#576—271noSylow
114C2×C33—21no—
115C2×C33—44no—
116C2×C33C22×D411yescharacteristic
117C3⋊(C3⋊D3)—22no—
118C3⋊(C3⋊D3)—42no—
119C2×C3×D4C2×C3⋊D3126yescomplemented
120C2×#67—1852no—
121#237—952no—
122#237—1826no—
123D3×D4—952no—
124D3×D4—1826no—
125#235—913no—
126#235—926no—
127C3×D8C2×C3⋊D3126yescomplemented
128C3×D8C2×C3⋊D3126yescomplemented
129C2×C3×C8C2×C3⋊D3113yescomplemented
130#214—952no—
131#213—952no—
132#208—913no—
133Dic12—913no—
134D24—913no—
135#204—926no—
136ΓL(1,𝔽25)—926no—
137D3×C8—926no—
138C22×C32—252no—
139C22×C32—826no—
140C22×C32D3×D4113yes—
141C2×C3⋊D3—326no—
142C2×C3⋊D3—626no—
143C2×C3⋊D3—2452no—
144C32×C4D3×D4113yes—
145C32×C4C23×D3113yes—
146C3⋊Dic3—326no—
147C3⋊Dic3—626no—
148H(5,𝔽2)—272no—
149AGL(1,ℤ/8)—272no—
150AGL(1,ℤ/8)—274no—
151#135—272no—
152#135—271no—
153C2×D8—272no—
154C2×D8C3⋊(C3⋊D3)11yescharacteristic, complemented
155#131—271no—
156C33#57611yescharacteristic, complemented, Sylow
157C23×C3—226no—
158C22×D3—1852no—
159C3×D4—226no—
160C3×D4C22×C3⋊D3152yes—
161C2×C3×C4C22×C3⋊D3113yes—
162#67—926no—
163#67—1852no—
164#67—1852no—
165#67—1826no—
166C2×Dic3—1852no—
167D12—926no—
168D12—913no—
169C4×D3—926no—
170C4×D3—1826no—
171Dic6—913no—
172Dic6—926no—
173C3×C8C22×C3⋊D3113yes—
174C3×C8C22×C3⋊D3113yes—
175C3⋊C8—926no—
176C2×C32—213no—
177C2×C32—452no—
178C2×C32C2×D3×D4113yes—
179C3⋊D3—626no—
180C3⋊D3—1226no—
181#41—271no—
182#41—274no—
183#41—272no—
184#41—542no—
185C2×D4—274no—
186C2×D4—542no—
187C2×D4—544no—
188C2×D4C2×C3⋊(C3⋊D3)12yescomplemented
189Q16—271no—
190SD16—274no—
191SD16—272no—
192D8—274no—
193D8—271no—
194D8C2×C3⋊(C3⋊D3)12yescomplemented
195D8C2×C3⋊(C3⋊D3)12yescomplemented
196M16—271no—
197M16—272no—
198C2×C8—272no—
199C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
200C22×C3—252no—
201C22×C3—826no—
202C22×C3D4×C3⋊D3113yes—
203C2×D3—926no—
204C2×D3—1826no—
205C2×D3—7252no—
206C3×C4D4×C3⋊D3113yes—
207C3×C4C23×C3⋊D3113yes—
208Dic3—926no—
209Dic3—1826no—
210C32#5757113yescomplemented
211C23—22no—
212C23—544no—
213Q8—271no—
214Q8—272no—
215D4—22no—
216D4—272no—
217D4—272no—
218D4—271no—
219D4—544no—
220D4—544no—
221D4—542no—
222D4C22×C3⋊(C3⋊D3)14yes—
223C2×C4—272no—
224C2×C4—544no—
225C2×C4—542no—
226C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
227C8—272no—
228C8C22×C3⋊(C3⋊D3)11yescharacteristic
229C8C22×C3⋊(C3⋊D3)11yescharacteristic
230C2×C3—213no—
231C2×C3—452no—
232C2×C3C2×D4×C3⋊D3113yes—
233D3—1826no—
234D3—3626no—
235C22—24no—
236C22—82no—
237C22—272no—
238C22—542no—
239C22—2164no—
240C22D4×C3⋊(C3⋊D3)11yescharacteristic
241C4—272no—
242C4—542no—
243C4D4×C3⋊(C3⋊D3)11yescharacteristic
244C4C23×C3⋊(C3⋊D3)11yescharacteristic
245C3#10596228113yescomplemented
246C2—21no—
247C2—44no—
248C2—542no—
249C2—1082no—
250C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC2×C33×D8C2
socleC2×C33C22×D4
hypercenterC2×D8C3⋊(C3⋊D3)
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. #11724679C2semidirect
2. C2×C33×D8C2semidirect
3. #11723973C2semidirect
4. #11723953C2semidirect
5. #11723952C2semidirect
6. D8×C3⋊(C3⋊D3)C2semidirect
7. #11723945C2semidirect
8. #11723942C2semidirect
10. C2×C33×D4C22semidirect
13. #89078C22semidirect
15. D4×C3⋊(C3⋊D3)C22semidirect
17. #89076C22semidirect
18. C33×D8C22semidirect
19. C33×D8C22semidirect
20. C2×C33×C8C22semidirect
21. #88861C22semidirect
22. #88860C22semidirect
23. #88855C22semidirect
24. #88853C22semidirect
25. #88852C22semidirect
26. #88851C22semidirect
27. #88850C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
30. C2×C32×D8D3semidirect
57. C2×C32×D4C2×D3semidirect
65. C32×D8C2×D3semidirect
66. C32×D8C2×D3semidirect
67. C2×C32×C8C2×D3semidirect
87. C2×C3×D8C3⋊D3semidirect
119. C2×C3×D4C2×C3⋊D3semidirect
127. C3×D8C2×C3⋊D3semidirect
128. C3×D8C2×C3⋊D3semidirect
129. C2×C3×C8C2×C3⋊D3semidirect
154. C2×D8C3⋊(C3⋊D3)semidirect
156. C33#576semidirect
188. C2×D4C2×C3⋊(C3⋊D3)semidirect
194. D8C2×C3⋊(C3⋊D3)semidirect
195. D8C2×C3⋊(C3⋊D3)semidirect
199. C2×C8C2×C3⋊(C3⋊D3)semidirect
210. C32#5757semidirect
245. C3#10596228semidirect

Maximal subgroups

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

subgroupindex
#117246792
C2×C33×D82
#117239732
#117239532
#117239522
D8×C3⋊(C3⋊D3)2
#117239452
#117239422
#105962283