Finite Group Databasebeta

← C3×#10596222 #13179502

Subgroups — C3×#10596222 #13179502

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic, complemented
2C3×#64366C211yesmaximal, characteristic, complemented
3C33×#131C211yesmaximal, characteristic, complemented
4C3×#64118C213yesmaximal, complemented
5C3×#64117C213yesmaximal, complemented
6C3×#64113C213yesmaximal, complemented
7C2×C3×#3744C213yesmaximal, complemented
8#10596222C311yesmaximal, characteristic, direct product factor
9C3×#5751—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832—21no—
12C3×Q8×C3⋊D3—21no—
13C3×#3830—23no—
14C3×D4×C3⋊D3—23no—
15C3×#3828C2213yes—
16C3×#3828C2213yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2213yes—
18C2×C3×#634C2213yescomplemented
19C33×M16C2213yescomplemented
20C2×C33×C8C2213yescomplemented
21C3×#3746C2213yescomplemented
22C3×#3745C2213yescomplemented
23C3×#3744C2213yescomplemented
24C3×#3744C2216yescomplemented
25C3×#3744C2211yescharacteristic, complemented
26#64367C2×C311yescharacteristic, complemented
27#64366C2×C311yescharacteristic, complemented
28C3×#1010—34no—
29C3×#1009—34no—
30C32×#131—24no—
31C32×#131D314yescomplemented
32C32×#131C2×C311yescharacteristic, complemented
33#64118C2×C313yescomplemented
34#64117C2×C313yescomplemented
35#64113C2×C313yescomplemented
36C2×#3744C2×C313yescomplemented
37C3×#909—312no—
38C3×#908—312no—
39C3×#904—312no—
40C2×C3×#204—312no—
41C22×C3×C3⋊D3—23no—
42C33×Q8D411yescharacteristic
43C33×D4D413yes—
44C2×C33×C4C2313yes—
45C3×#638—23no—
46C3×#638—23no—
47C2×C3×C3⋊Dic3—23no—
48C3×C4⋊(C3⋊D3)—23no—
49C3×C4⋊(C3⋊D3)C2313yes—
50C3×C4⋊(C3⋊D3)C2311yescharacteristic
51C3×C4×C3⋊D3—23no—
52C3×C4×C3⋊D3—23no—
53C3×#634—23no—
54C3×#634C2313yes—
55C3×#634C2311yescharacteristic
56C33×C8C2311yescharacteristic
57C33×C8C2313yes—
58C3×#577—91no—
59#5751—34no—
60C32×#41—24no—
61C32×#41C2×D314yes—
62C32×#41C22×C311yescharacteristic
63#3832—21no—
64Q8×C3⋊D3—21no—
65#3830—23no—
66D4×C3⋊D3—23no—
67#3828C22×C313yes—
68#3828C22×C313yescomplemented
69C2×C4⋊(C3⋊D3)C22×C313yes—
70C2×#634C22×C313yescomplemented
71C3×#239—64no—
72C3×D3×Q8—64no—
73C3×#237—612no—
74C3×D3×D4—612no—
75C3×#235—312no—
76C3×#235—312no—
77C2×C3×D12—312no—
78C2×C3×Dic6—312no—
79C32×M16—212no—
80C32×M16C2×D3112yescomplemented
81C32×M16C22×C313yescomplemented
82C2×C32×C8—212no—
83C2×C32×C8C2×D3112yescomplemented
84C2×C32×C8C22×C313yescomplemented
85#3746C22×C313yescomplemented
86#3745C22×C313yescomplemented
87#3744C22×C313yescomplemented
88#3744C22×C316yescomplemented
89#3744C22×C311yescharacteristic, complemented
90C3×Dic12—312no—
91C3×D24—312no—
92C3×#204—312no—
93C3×#204—324no—
94C3×#204—34no—
95C22×C33C2×D413yes—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—23no—
98C2×C3×C3⋊D3—83no—
99C33×C4C2×D413yes—
100C33×C4C2411yescharacteristic
101C3×C3⋊Dic3—23no—
102C3×C3⋊Dic3—21no—
103C3×2−1+4—91no—
104C3×H(5,𝔽2)—91no—
105#1010—34no—
106#1009—34no—
107C3×#137—93no—
108C3×AGL(1,ℤ/8)—93no—
109C3×#135—93no—
110C2×C3×SD16—93no—
111C3×#131—24no—
112C3×#131C3×D314yescomplemented
113C3×#131C3⋊D311yescharacteristic, complemented
114#909—312no—
115#908—312no—
116#904—312no—
117C2×#204—312no—
118C22×C3⋊D3—23no—
119C22×C3×D3—612no—
120C32×Q8—24no—
121C32×Q8D1214yes—
122C32×Q8C3×D411yescharacteristic
123C32×D4—212no—
124C32×D4D12112yes—
125C32×D4C3×D413yes—
126C2×C32×C4—212no—
127C2×C32×C4C22×D3112yes—
128C2×C32×C4C23×C313yes—
129#638—23no—
130#638—23no—
131C2×C3⋊Dic3—23no—
132C4⋊(C3⋊D3)—23no—
133C4⋊(C3⋊D3)C23×C313yes—
134C4⋊(C3⋊D3)C23×C311yescharacteristic
135C4×C3⋊D3—23no—
136C4×C3⋊D3—23no—
137#634—23no—
138#634C23×C313yes—
139#634C23×C311yescharacteristic
140C3×#67—612no—
141C3×#67—612no—
142C2×C3×Dic3—612no—
143C3×D12—312no—
144C3×D12—34no—
145C3×D12—612no—
146C3×C4×D3—612no—
147C3×C4×D3—612no—
148C3×Dic6—312no—
149C3×Dic6—34no—
150C3×Dic6—612no—
151C32×C8—24no—
152C32×C8—212no—
153C32×C8C22×D314yes—
154C32×C8C22×D3112yes—
155C32×C8C23×C311yescharacteristic
156C32×C8C23×C313yes—
157#577—91noSylow
158C2×C33—23no—
159C2×C33C22×D411yescharacteristic
160C3×C3⋊D3—41no—
161C3×C3⋊D3—43no—
162C3×#41—24no—
163C3×#41—93no—
164C3×#41—93no—
165C3×#41—181no—
166C3×#41—183no—
167C3×#41C2×C3×D314yes—
168C3×#41C2×C3⋊D311yescharacteristic
169C2×C3×Q8—93no—
170C2×C3×Q8—181no—
171C2×C3×D4—93no—
172C2×C3×D4—183no—
173#239—64no—
174D3×Q8—64no—
175#237—612no—
176D3×D4—612no—
177#235—312no—
178#235—312no—
179C2×D12—312no—
180C2×Dic6—312no—
181C3×Q16—93no—
182C3×SD16—93no—
183C3×SD16—96no—
184C3×SD16—91no—
185C3×D8—93no—
186C3×M16—212no—
187C3×M16C2×C3×D3112yescomplemented
188C3×M16C2×C3⋊D313yescomplemented
189C2×C3×C8—212no—
190C2×C3×C8C2×C3×D3112yescomplemented
191C2×C3×C8C2×C3⋊D313yescomplemented
192Dic12—312no—
193D24—312no—
194#204—312no—
195#204—324no—
196#204—34no—
197C22×C32—212no—
198C22×C32C2×D12112yes—
199C22×C32C2×C3×D413yes—
200C2×C3⋊D3—21no—
201C2×C3⋊D3—23no—
202C2×C3⋊D3—83no—
203C2×C3×D3—64no—
204C2×C3×D3—612no—
205C2×C3×D3—2412no—
206C32×C4—24no—
207C32×C4—212no—
208C32×C4C2×D12112yes—
209C32×C4C2×C3×D413yes—
210C32×C4C23×D314yes—
211C32×C4C24×C311yescharacteristic
212C3⋊Dic3—23no—
213C3⋊Dic3—21no—
214C3×Dic3—612no—
215C3×Dic3—64no—
2162−1+4—91no—
217H(5,𝔽2)—91no—
218#137—93no—
219AGL(1,ℤ/8)—93no—
220#135—93no—
221C2×SD16—93no—
222#131C3×C3⋊D311yescharacteristic, complemented
223C33#57711yescharacteristic, complemented, Sylow
224C23×C3—183no—
225C22×D3—612no—
226C3×Q8—24no—
227C3×Q8—93no—
228C3×Q8—91no—
229C3×Q8—183no—
230C3×Q8C3×D1214yes—
231C3×Q8C4⋊(C3⋊D3)11yescharacteristic
232C3×D4—212no—
233C3×D4—91no—
234C3×D4—93no—
235C3×D4—183no—
236C3×D4—183no—
237C3×D4—183no—
238C3×D4C3×D12112yes—
239C3×D4C4⋊(C3⋊D3)13yes—
240C2×C3×C4—212no—
241C2×C3×C4—183no—
242C2×C3×C4—183no—
243C2×C3×C4—183no—
244C2×C3×C4C22×C3×D3112yes—
245C2×C3×C4C22×C3⋊D313yes—
246#67—612no—
247#67—612no—
248C2×Dic3—612no—
249D12—312no—
250D12—34no—
251D12—612no—
252C4×D3—612no—
253C4×D3—612no—
254Dic6—312no—
255Dic6—34no—
256Dic6—612no—
257C3×C8—24no—
258C3×C8—212no—
259C3×C8C22×C3×D314yes—
260C3×C8C22×C3×D3112yes—
261C3×C8C22×C3⋊D311yescharacteristic
262C3×C8C22×C3⋊D313yes—
263C2×C32—23no—
264C2×C32—24no—
265C2×C32—212no—
266C2×C32—412no—
267C2×C32C22×D1214yes—
268C2×C32C22×C3×D411yescharacteristic
269C3⋊D3—41no—
270C3⋊D3—43no—
271C3×D3—124no—
272C3×D3—1212no—
273#41—93no—
274#41—93no—
275#41—181no—
276#41—183no—
277#41C2×C3×C3⋊D311yescharacteristic
278C2×Q8—93no—
279C2×Q8—181no—
280C2×D4—93no—
281C2×D4—183no—
282Q16—93no—
283SD16—93no—
284SD16—96no—
285SD16—91no—
286D8—93no—
287M16C2×C3×C3⋊D313yescomplemented
288C2×C8C2×C3×C3⋊D313yescomplemented
289C22×C3—212no—
290C22×C3—181no—
291C22×C3—183no—
292C22×C3—723no—
293C22×C3C2×C3×D12112yes—
294C22×C3C2×C4⋊(C3⋊D3)13yes—
295C2×D3—64no—
296C2×D3—612no—
297C2×D3—2412no—
298C3×C4—24no—
299C3×C4—212no—
300C3×C4—181no—
301C3×C4—183no—
302C3×C4C2×C3×D12112yes—
303C3×C4C2×C4⋊(C3⋊D3)13yes—
304C3×C4C23×C3×D314yes—
305C3×C4C23×C3⋊D311yescharacteristic
306Dic3—612no—
307Dic3—64no—
308C32—24no—
309C32#575114yescomplemented
310C32C3×#57711yescharacteristic, complemented
311C23—183no—
312Q8—93no—
313Q8—91no—
314Q8—183no—
315Q8C3×C4⋊(C3⋊D3)11yescharacteristic
316D4—91no—
317D4—93no—
318D4—183no—
319D4—183no—
320D4—183no—
321D4C3×C4⋊(C3⋊D3)13yes—
322C2×C4—183no—
323C2×C4—183no—
324C2×C4—183no—
325C2×C4C22×C3×C3⋊D313yes—
326C8C22×C3×C3⋊D311yescharacteristic
327C8C22×C3×C3⋊D313yes—
328C2×C3—212no—
329C2×C3—23no—
330C2×C3—24no—
331C2×C3—412no—
332C2×C3—363no—
333C2×C3—361no—
334C2×C3C22×C3×D1214yes—
335C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
336D3—1212no—
337D3—124no—
338C22—181no—
339C22—183no—
340C22—723no—
341C22C2×C3×C4⋊(C3⋊D3)13yes—
342C4—181no—
343C4—183no—
344C4C2×C3×C4⋊(C3⋊D3)13yes—
345C4C23×C3×C3⋊D311yescharacteristic
346C3—24no—
347C3C3×#575114yescomplemented
348C3#1059622211yescharacteristic, direct product factor
349C2—23no—
350C2—363no—
351C2—361no—
352C2C22×C3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2×C3C22×C4⋊(C3⋊D3)
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC33×#131C2
socleC2×C33C22×D4
hypercenterC3×#131C3⋊D3
hypocenterC32C3×#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
1. C3×#64367C2semidirect
2. C3×#64366C2semidirect
3. C33×#131C2semidirect
4. C3×#64118C2semidirect
5. C3×#64117C2semidirect
6. C3×#64113C2semidirect
7. C2×C3×#3744C2semidirect
8. #10596222C3direct (N × H)
16. C3×#3828C22semidirect
18. C2×C3×#634C22semidirect
19. C33×M16C22semidirect
20. C2×C33×C8C22semidirect
21. C3×#3746C22semidirect
22. C3×#3745C22semidirect
23. C3×#3744C22semidirect
24. C3×#3744C22semidirect
25. C3×#3744C22semidirect
26. #64367C2×C3semidirect
27. #64366C2×C3semidirect
31. C32×#131D3semidirect
32. C32×#131C2×C3semidirect
33. #64118C2×C3semidirect
34. #64117C2×C3semidirect
35. #64113C2×C3semidirect
36. C2×#3744C2×C3semidirect
68. #3828C22×C3semidirect
70. C2×#634C22×C3semidirect
80. C32×M16C2×D3semidirect
81. C32×M16C22×C3semidirect
83. C2×C32×C8C2×D3semidirect
84. C2×C32×C8C22×C3semidirect
85. #3746C22×C3semidirect
86. #3745C22×C3semidirect
87. #3744C22×C3semidirect
88. #3744C22×C3semidirect
89. #3744C22×C3semidirect
112. C3×#131C3×D3semidirect
113. C3×#131C3⋊D3semidirect
187. C3×M16C2×C3×D3semidirect
188. C3×M16C2×C3⋊D3semidirect
190. C2×C3×C8C2×C3×D3semidirect
191. C2×C3×C8C2×C3⋊D3semidirect
222. #131C3×C3⋊D3semidirect
223. C33#577semidirect
287. M16C2×C3×C3⋊D3semidirect
288. C2×C8C2×C3×C3⋊D3semidirect
309. C32#5751semidirect
310. C32C3×#577semidirect
347. C3C3×#5751semidirect
348. C3#10596222direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643672
C3×#643662
C33×#1312
C3×#641182
C3×#641172
C3×#641132
C2×C3×#37442
#105962223
C3×#57513