Finite Group Databasebeta

← C3×#10592892 #13163233

Subgroups — C3×#10592892 #13163233

Order 1728.

788 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C33×#135C211yesmaximal, characteristic, complemented
2C2×C3×#3745C211yesmaximal, characteristic, complemented
3C3×#63655C212yesmaximal, complemented
4C3×#63653C212yesmaximal, complemented
5C3×#63639C211yesmaximal, characteristic, complemented
6#10592892C311yesmaximal, characteristic, direct product factor
7C3×#5193—34nomaximal
8C33×#41—21no—
9C2×C3×C4⋊(C3⋊D3)—21no—
10C33×Q16C2211yescharacteristic, complemented
11C33×SD16—21no—
12C33×D8C2211yescharacteristic, complemented
13C2×C33×C8C2211yescharacteristic
14C3×#3745—21no—
15C3×#3745C2212yes—
16C3×C3⋊(C3⋊C16)C2212yescomplemented
17C32×#135—24no—
18C32×#135D314yescomplemented
19C32×#135C2×C311yescharacteristic, complemented
20C2×#3745C2×C311yescharacteristic, complemented
21C2×C3×D24—34no—
22#63655C2×C312yescomplemented
23#63653C2×C312yescomplemented
24#63639C2×C311yescharacteristic, complemented
25C3×#828—38no—
26C3×#826—38no—
27C3×#812—34no—
28C22×C3×C3⋊D3—41no—
29C33×Q8—21no—
30C33×D4—21no—
31C33×D4—41no—
32C2×C33×C4—41no—
33C2×C33×C4D411yescharacteristic
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)—41no—
36C33×C8D411yescharacteristic
37C33×C8C2311yescharacteristic
38C3×#509—91no—
39#5193—34no—
40C32×#41—28no—
41C32×#41—24no—
42C32×#41—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C3×D12—64no—
45C32×Q16—24no—
46C32×Q16C2×D314yescomplemented
47C32×Q16C22×C311yescharacteristic, complemented
48C32×SD16—28no—
49C32×SD16—24no—
50C32×SD16—21no—
51C32×D8—24no—
52C32×D8C2×D314yescomplemented
53C32×D8C22×C311yescharacteristic, complemented
54C2×C32×C8—24no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C22×C311yescharacteristic
57#3745—21no—
58#3745C22×C312yes—
59C3×D24—38no—
60C3×D24—64no—
61C3⋊(C3⋊C16)C22×C312yescomplemented
62C3×C3⋊C16—38no—
63C22×C33—41no—
64C22×C33D811yescharacteristic
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3—82no—
67C33×C4—41no—
68C33×C4D811yescharacteristic
69C33×C4C2×D411yescharacteristic
70C3×#135—24no—
71C3×#135C3×D314yescomplemented
72C3×#135C3⋊D311yescharacteristic, complemented
73C2×C3×D8—91no—
74C2×D24—34no—
75C3×SD32—92no—
76C3×D16—92no—
77C3×M32—91no—
78#828—38no—
79#826—38no—
80#812—34no—
81C22×C3⋊D3—41no—
82C22×C3×D3—124no—
83C32×Q8—28no—
84C32×Q8—24no—
85C32×Q8—21no—
86C32×D4—28no—
87C32×D4—24no—
88C32×D4—21no—
89C32×D4—48no—
90C32×D4—44no—
91C32×D4—41no—
92C2×C32×C4—24no—
93C2×C32×C4—48no—
94C2×C32×C4—44no—
95C2×C32×C4—41no—
96C2×C32×C4#6714yes—
97C2×C32×C4C3×D411yescharacteristic
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)—41no—
100C3×D12—68no—
101C3×D12—124no—
102C32×C8—24no—
103C32×C8—24no—
104C32×C8#6714yes—
105C32×C8C3×D411yescharacteristic
106C32×C8C22×D314yes—
107C32×C8C23×C311yescharacteristic
108#509—91noSylow
109C2×C33—21no—
110C2×C33—81no—
111C2×C33C2×D811yescharacteristic
112C3×C3⋊D3—82no—
113C3×#41—28no—
114C3×#41—24no—
115C3×#41—21no—
116C2×C3×D4—181no—
117C2×D12—64no—
118C3×Q16—24no—
119C3×Q16C2×C3×D314yescomplemented
120C3×Q16C2×C3⋊D311yescharacteristic, complemented
121C3×SD16—28no—
122C3×SD16—24no—
123C3×SD16—21no—
124C3×D8—24no—
125C3×D8—92no—
126C3×D8—181no—
127C3×D8C2×C3×D314yescomplemented
128C3×D8C2×C3⋊D311yescharacteristic, complemented
129C2×C3×C8—24no—
130C2×C3×C8C2×C3×D314yes—
131C2×C3×C8C2×C3⋊D311yescharacteristic
132D24—38no—
133D24—64no—
134C3×C16—92no—
135C3⋊C16—38no—
136C22×C32—24no—
137C22×C32—48no—
138C22×C32—44no—
139C22×C32—41no—
140C22×C32#21314yes—
141C22×C32C3×D811yescharacteristic
142C2×C3⋊D3—42no—
143C2×C3⋊D3—82no—
144C2×C3×D3—128no—
145C2×C3×D3—248no—
146C32×C4—24no—
147C32×C4—24no—
148C32×C4—48no—
149C32×C4—44no—
150C32×C4—41no—
151C32×C4#21314yes—
152C32×C4C3×D811yescharacteristic
153C32×C4C2×#6714yes—
154C32×C4C2×C3×D411yescharacteristic
155#135C3×C3⋊D311yescharacteristic, complemented
156C2×D8—91no—
157SD32—92no—
158D16—92no—
159M32—91no—
160C33#50911yescharacteristic, complemented, Sylow
161C23×C3—361no—
162C22×D3—124no—
163C3×Q8—28no—
164C3×Q8—24no—
165C3×Q8—21no—
166C3×D4—28no—
167C3×D4—24no—
168C3×D4—21no—
169C3×D4—48no—
170C3×D4—44no—
171C3×D4—41no—
172C3×D4—182no—
173C3×D4—361no—
174C2×C3×C4—24no—
175C2×C3×C4—48no—
176C2×C3×C4—44no—
177C2×C3×C4—41no—
178C2×C3×C4C3×#6714yes—
179C2×C3×C4#63811yescharacteristic
180D12—68no—
181D12—124no—
182C3×C8—24no—
183C3×C8—24no—
184C3×C8C3×#6714yes—
185C3×C8#63811yescharacteristic
186C3×C8C22×C3×D314yes—
187C3×C8C22×C3⋊D311yescharacteristic
188C2×C32—21no—
189C2×C32—24no—
190C2×C32—24no—
191C2×C32—44no—
192C2×C32—88no—
193C2×C32—81no—
194C2×C32—84no—
195C2×C32C2×#21314yes—
196C2×C32C2×C3×D811yescharacteristic
197C3⋊D3—82no—
198C3×D3—248no—
199#41—21no—
200C2×D4—181no—
201Q16C2×C3×C3⋊D311yescharacteristic, complemented
202SD16—21no—
203D8—92no—
204D8—181no—
205D8C2×C3×C3⋊D311yescharacteristic, complemented
206C2×C8C2×C3×C3⋊D311yescharacteristic
207C16—92no—
208C22×C3—24no—
209C22×C3—48no—
210C22×C3—44no—
211C22×C3—41no—
212C22×C3—362no—
213C22×C3—722no—
214C22×C3C3×#21314yes—
215C22×C3#375311yescharacteristic
216C2×D3—128no—
217C2×D3—248no—
218C3×C4—24no—
219C3×C4—24no—
220C3×C4—48no—
221C3×C4—44no—
222C3×C4—41no—
223C3×C4C3×#21314yes—
224C3×C4#375311yescharacteristic
225C3×C4C2×C3×#6714yes—
226C3×C4C2×#63811yescharacteristic
227C32—24no—
228C32#519314yescomplemented
229C32C3×#50911yescharacteristic, complemented
230C23—361no—
231Q8—21no—
232D4—21no—
233D4—41no—
234D4—182no—
235D4—361no—
236C2×C4—41no—
237C2×C4C3×#63811yescharacteristic
238C8C3×#63811yescharacteristic
239C8C22×C3×C3⋊D311yescharacteristic
240C2×C3—24no—
241C2×C3—21no—
242C2×C3—24no—
243C2×C3—44no—
244C2×C3—88no—
245C2×C3—84no—
246C2×C3—81no—
247C2×C3—722no—
248C2×C3C2×C3×#21314yes—
249C2×C3C2×#375311yescharacteristic
250D3—248no—
251C22—41no—
252C22—362no—
253C22—722no—
254C22C3×#375311yescharacteristic
255C4—41no—
256C4C3×#375311yescharacteristic
257C4C2×C3×#63811yescharacteristic
258C3—24no—
259C3C3×#519314yescomplemented
260C3#1059289211yescharacteristic, direct product factor
261C2—21no—
262C2—81no—
263C2—722no—
264C2C2×C3×#375311yescharacteristic
subgroupG/H
centerC2×C3C2×#3753
commutator, cosocleC32×C8C23×C3
FrattiniC8C22×C3×C3⋊D3
FittingC33×#135C2
socleC2×C33C2×D8
hypercenterC3×#135C3⋊D3
hypocenterC32C3×#509
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×#135

Complements

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

NH ≅ G/Ntype
1. C33×#135C2semidirect
2. C2×C3×#3745C2semidirect
3. C3×#63655C2semidirect
4. C3×#63653C2semidirect
5. C3×#63639C2semidirect
6. #10592892C3direct (N × H)
10. C33×Q16C22semidirect
12. C33×D8C22semidirect
16. C3×C3⋊(C3⋊C16)C22semidirect
18. C32×#135D3semidirect
19. C32×#135C2×C3semidirect
20. C2×#3745C2×C3semidirect
22. #63655C2×C3semidirect
23. #63653C2×C3semidirect
24. #63639C2×C3semidirect
46. C32×Q16C2×D3semidirect
47. C32×Q16C22×C3semidirect
52. C32×D8C2×D3semidirect
53. C32×D8C22×C3semidirect
61. C3⋊(C3⋊C16)C22×C3semidirect
71. C3×#135C3×D3semidirect
72. C3×#135C3⋊D3semidirect
119. C3×Q16C2×C3×D3semidirect
120. C3×Q16C2×C3⋊D3semidirect
127. C3×D8C2×C3×D3semidirect
128. C3×D8C2×C3⋊D3semidirect
155. #135C3×C3⋊D3semidirect
160. C33#509semidirect
201. Q16C2×C3×C3⋊D3semidirect
205. D8C2×C3×C3⋊D3semidirect
228. C32#5193semidirect
229. C32C3×#509semidirect
259. C3C3×#5193semidirect
260. C3#10592892direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1352
C2×C3×#37452
C3×#636552
C3×#636532
C3×#636392
#105928923
C3×#51933