Finite Group Databasebeta

← C32×#5751 #13179128

Subgroups — C32×#5751 #13179128

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
1C32×#1010C211yesmaximal, characteristic, complemented
2C32×#1009C211yesmaximal, characteristic, complemented
3C33×#131C211yesmaximal, characteristic, complemented
4C32×#909C213yesmaximal, complemented
5C32×#908C213yesmaximal, complemented
6C32×#904C213yesmaximal, complemented
7C2×C32×#204C213yesmaximal, complemented
8C32×#577—31nomaximal
9C3×#5751C314yesmaximal, direct product factor
10C33×#41C2211yescharacteristic
11C32×#239—21no—
12C32×D3×Q8—21no—
13C32×#237—23no—
14C32×D3×D4—23no—
15C32×#235C2213yes—
16C32×#235C2213yescomplemented
17C2×C32×D12C2213yes—
18C2×C32×Dic6C2213yescomplemented
19C33×M16C2213yescomplemented
20C2×C33×C8C2213yescomplemented
21C32×Dic12C2213yescomplemented
22C32×D24C2213yescomplemented
23C32×#204C2216yescomplemented
24C32×#204C2213yescomplemented
25C32×#204C2211yescharacteristic, complemented
26C32×2−1+4—31no—
27C32×H(5,𝔽2)—31no—
28C3×#1010C2×C314yescomplemented
29C3×#1009C2×C314yescomplemented
30C32×#137—33no—
31C32×AGL(1,ℤ/8)—33no—
32C32×#135—33no—
33C2×C32×SD16—33no—
34C32×#131—24no—
35C32×#131D311yescharacteristic, complemented
36C32×#131C2×C314yescomplemented
37C3×#909C2×C3112yescomplemented
38C3×#908C2×C3112yescomplemented
39C3×#904C2×C3112yescomplemented
40C2×C3×#204C2×C3112yescomplemented
41C22×C32×D3—23no—
42C33×Q8D411yescharacteristic
43C33×D4D413yes—
44C2×C33×C4C2313yes—
45C32×#67—23no—
46C32×#67—23no—
47C2×C32×Dic3—23no—
48C32×D12—23no—
49C32×D12C2313yes—
50C32×D12C2311yescharacteristic
51C32×C4×D3—23no—
52C32×C4×D3—23no—
53C32×Dic6—23no—
54C32×Dic6C2313yes—
55C32×Dic6C2311yescharacteristic
56C33×C8C2311yescharacteristic
57C33×C8C2313yes—
58C3×#577—34no—
59#5751C3211yescharacteristic, direct product factor
60C32×#41—24no—
61C32×#41—33no—
62C32×#41—33no—
63C32×#41—61no—
64C32×#41—63no—
65C32×#41C2×D311yescharacteristic
66C32×#41C22×C314yes—
67C2×C32×Q8—33no—
68C2×C32×Q8—61no—
69C2×C32×D4—33no—
70C2×C32×D4—63no—
71C3×#239—24no—
72C3×D3×Q8—24no—
73C3×#237—212no—
74C3×D3×D4—212no—
75C3×#235C22×C3112yes—
76C3×#235C22×C3112yescomplemented
77C2×C3×D12C22×C3112yes—
78C2×C3×Dic6C22×C3112yescomplemented
79C32×Q16—33no—
80C32×SD16—33no—
81C32×SD16—36no—
82C32×SD16—31no—
83C32×D8—33no—
84C32×M16—212no—
85C32×M16C2×D313yescomplemented
86C32×M16C22×C3112yescomplemented
87C2×C32×C8—212no—
88C2×C32×C8C2×D313yescomplemented
89C2×C32×C8C22×C3112yescomplemented
90C3×Dic12C22×C3112yescomplemented
91C3×D24C22×C3112yescomplemented
92C3×#204C22×C3112yescomplemented
93C3×#204C22×C3124yescomplemented
94C3×#204C22×C314yescomplemented
95C22×C33C2×D413yes—
96C2×C32×D3—21no—
97C2×C32×D3—23no—
98C2×C32×D3—83no—
99C33×C4C2×D413yes—
100C33×C4C2411yescharacteristic
101C32×Dic3—23no—
102C32×Dic3—21no—
103C3×2−1+4—34no—
104C3×H(5,𝔽2)—34no—
105#1010C2×C3211yescharacteristic, complemented
106#1009C2×C3211yescharacteristic, complemented
107C3×#137—312no—
108C3×AGL(1,ℤ/8)—312no—
109C3×#135—312no—
110C2×C3×SD16—312no—
111C3×#131—24no—
112C3×#131C3×D314yescomplemented
113C3×#131C2×C3211yescharacteristic, complemented
114#909C2×C3213yescomplemented
115#908C2×C3213yescomplemented
116#904C2×C3213yescomplemented
117C2×#204C2×C3213yescomplemented
118C23×C32—63no—
119C22×C3×D3—212no—
120C32×Q8—24no—
121C32×Q8—33no—
122C32×Q8—31no—
123C32×Q8—63no—
124C32×Q8D1211yescharacteristic
125C32×Q8C3×D414yes—
126C32×D4—212no—
127C32×D4—31no—
128C32×D4—33no—
129C32×D4—63no—
130C32×D4—63no—
131C32×D4—63no—
132C32×D4D1213yes—
133C32×D4C3×D4112yes—
134C2×C32×C4—212no—
135C2×C32×C4—63no—
136C2×C32×C4—63no—
137C2×C32×C4—63no—
138C2×C32×C4C22×D313yes—
139C2×C32×C4C23×C3112yes—
140C3×#67—212no—
141C3×#67—212no—
142C2×C3×Dic3—212no—
143C3×D12—212no—
144C3×D12C23×C3112yes—
145C3×D12C23×C314yes—
146C3×C4×D3—212no—
147C3×C4×D3—212no—
148C3×Dic6—212no—
149C3×Dic6C23×C3112yes—
150C3×Dic6C23×C314yes—
151C32×C8—24no—
152C32×C8—212no—
153C32×C8C22×D311yescharacteristic
154C32×C8C22×D313yes—
155C32×C8C23×C314yes—
156C32×C8C23×C3112yes—
157#577—31noSylow
158C2×C33—23no—
159C2×C33C22×D411yescharacteristic
160C32×D3—41no—
161C32×D3—43no—
162C3×#41—24no—
163C3×#41—312no—
164C3×#41—312no—
165C3×#41—64no—
166C3×#41—612no—
167C3×#41C2×C3×D314yes—
168C3×#41C22×C3211yescharacteristic
169C2×C3×Q8—312no—
170C2×C3×Q8—64no—
171C2×C3×D4—312no—
172C2×C3×D4—612no—
173#239—21no—
174D3×Q8—21no—
175#237—23no—
176D3×D4—23no—
177#235C22×C3213yes—
178#235C22×C3213yescomplemented
179C2×D12C22×C3213yes—
180C2×Dic6C22×C3213yescomplemented
181C3×Q16—312no—
182C3×SD16—312no—
183C3×SD16—324no—
184C3×SD16—34no—
185C3×D8—312no—
186C3×M16—212no—
187C3×M16C2×C3×D3112yescomplemented
188C3×M16C22×C3213yescomplemented
189C2×C3×C8—212no—
190C2×C3×C8C2×C3×D3112yescomplemented
191C2×C3×C8C22×C3213yescomplemented
192Dic12C22×C3213yescomplemented
193D24C22×C3213yescomplemented
194#204C22×C3213yescomplemented
195#204C22×C3216yescomplemented
196#204C22×C3211yescharacteristic, complemented
197C22×C32—212no—
198C22×C32—61no—
199C22×C32—63no—
200C22×C32—243no—
201C22×C32C2×D1213yes—
202C22×C32C2×C3×D4112yes—
203C2×C3×D3—24no—
204C2×C3×D3—212no—
205C2×C3×D3—812no—
206C32×C4—24no—
207C32×C4—212no—
208C32×C4—61no—
209C32×C4—63no—
210C32×C4C2×D1213yes—
211C32×C4C2×C3×D4112yes—
212C32×C4C23×D311yescharacteristic
213C32×C4C24×C314yes—
214C3×Dic3—212no—
215C3×Dic3—24no—
2162−1+4—31no—
217H(5,𝔽2)—31no—
218#137—33no—
219AGL(1,ℤ/8)—33no—
220#135—33no—
221C2×SD16—33no—
222#131C32×D311yescharacteristic, complemented
223C33#57711yescharacteristic, complemented, Sylow
224C23×C3—612no—
225C22×D3—23no—
226C3×Q8—24no—
227C3×Q8—312no—
228C3×Q8—34no—
229C3×Q8—612no—
230C3×Q8C3×D1214yes—
231C3×Q8C32×D411yescharacteristic
232C3×D4—212no—
233C3×D4—34no—
234C3×D4—312no—
235C3×D4—612no—
236C3×D4—612no—
237C3×D4—612no—
238C3×D4C3×D12112yes—
239C3×D4C32×D413yes—
240C2×C3×C4—212no—
241C2×C3×C4—612no—
242C2×C3×C4—612no—
243C2×C3×C4—612no—
244C2×C3×C4C22×C3×D3112yes—
245C2×C3×C4C23×C3213yes—
246#67—23no—
247#67—23no—
248C2×Dic3—23no—
249D12—23no—
250D12C23×C3213yes—
251D12C23×C3211yescharacteristic
252C4×D3—23no—
253C4×D3—23no—
254Dic6—23no—
255Dic6C23×C3213yes—
256Dic6C23×C3211yescharacteristic
257C3×C8—24no—
258C3×C8—212no—
259C3×C8C22×C3×D314yes—
260C3×C8C22×C3×D3112yes—
261C3×C8C23×C3211yescharacteristic
262C3×C8C23×C3213yes—
263C2×C32—23no—
264C2×C32—24no—
265C2×C32—212no—
266C2×C32—412no—
267C2×C32—121no—
268C2×C32—123no—
269C2×C32C22×D1211yescharacteristic
270C2×C32C22×C3×D414yes—
271C3×D3—44no—
272C3×D3—412no—
273#41—33no—
274#41—33no—
275#41—61no—
276#41—63no—
277#41C2×C32×D311yescharacteristic
278C2×Q8—33no—
279C2×Q8—61no—
280C2×D4—33no—
281C2×D4—63no—
282Q16—33no—
283SD16—33no—
284SD16—36no—
285SD16—31no—
286D8—33no—
287M16C2×C32×D313yescomplemented
288C2×C8C2×C32×D313yescomplemented
289C22×C3—212no—
290C22×C3—64no—
291C22×C3—612no—
292C22×C3—2412no—
293C22×C3C2×C3×D12112yes—
294C22×C3C2×C32×D413yes—
295C2×D3—21no—
296C2×D3—23no—
297C2×D3—83no—
298C3×C4—24no—
299C3×C4—212no—
300C3×C4—64no—
301C3×C4—612no—
302C3×C4C2×C3×D12112yes—
303C3×C4C2×C32×D413yes—
304C3×C4C23×C3×D314yes—
305C3×C4C24×C3211yescharacteristic
306Dic3—23no—
307Dic3—21no—
308C32—24no—
309C32#575111yescharacteristic, direct product factor
310C32C3×#57714yescomplemented
311C23—63no—
312Q8—33no—
313Q8—31no—
314Q8—63no—
315Q8C32×D1211yescharacteristic
316D4—31no—
317D4—33no—
318D4—63no—
319D4—63no—
320D4—63no—
321D4C32×D1213yes—
322C2×C4—63no—
323C2×C4—63no—
324C2×C4—63no—
325C2×C4C22×C32×D313yes—
326C8C22×C32×D311yescharacteristic
327C8C22×C32×D313yes—
328C2×C3—212no—
329C2×C3—23no—
330C2×C3—24no—
331C2×C3—412no—
332C2×C3—1212no—
333C2×C3—124no—
334C2×C3C22×C3×D1214yes—
335C2×C3C22×C32×D411yescharacteristic
336D3—43no—
337D3—41no—
338C22—61no—
339C22—63no—
340C22—243no—
341C22C2×C32×D1213yes—
342C4—61no—
343C4—63no—
344C4C2×C32×D1213yes—
345C4C23×C32×D311yescharacteristic
346C3—24no—
347C3C3×#575114yesdirect product factor
348C3C32×#57711yescharacteristic, complemented
349C2—23no—
350C2—123no—
351C2—121no—
352C2C22×C32×D1211yescharacteristic
subgroupG/H
centerC2×C32C22×D12
commutator, cosocleC3×C4C24×C32
FrattiniC4C23×C32×D3
FittingC33×#131C2
socleC2×C33C22×D4
hypercenterC32×#131D3
hypocenterC3C32×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C32×#41 ◁ C32×#131

Complements

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

NH ≅ G/Ntype
1. C32×#1010C2semidirect
2. C32×#1009C2semidirect
3. C33×#131C2semidirect
4. C32×#909C2semidirect
5. C32×#908C2semidirect
6. C32×#904C2semidirect
7. C2×C32×#204C2semidirect
9. C3×#5751C3direct (N × H)
16. C32×#235C22semidirect
18. C2×C32×Dic6C22semidirect
19. C33×M16C22semidirect
20. C2×C33×C8C22semidirect
21. C32×Dic12C22semidirect
22. C32×D24C22semidirect
23. C32×#204C22semidirect
24. C32×#204C22semidirect
25. C32×#204C22semidirect
28. C3×#1010C2×C3semidirect
29. C3×#1009C2×C3semidirect
35. C32×#131D3semidirect
36. C32×#131C2×C3semidirect
37. C3×#909C2×C3semidirect
38. C3×#908C2×C3semidirect
39. C3×#904C2×C3semidirect
40. C2×C3×#204C2×C3semidirect
59. #5751C32direct (N × H)
76. C3×#235C22×C3semidirect
78. C2×C3×Dic6C22×C3semidirect
85. C32×M16C2×D3semidirect
86. C32×M16C22×C3semidirect
88. C2×C32×C8C2×D3semidirect
89. C2×C32×C8C22×C3semidirect
90. C3×Dic12C22×C3semidirect
91. C3×D24C22×C3semidirect
92. C3×#204C22×C3semidirect
93. C3×#204C22×C3semidirect
94. C3×#204C22×C3semidirect
105. #1010C2×C32semidirect
106. #1009C2×C32semidirect
112. C3×#131C3×D3semidirect
113. C3×#131C2×C32semidirect
114. #909C2×C32semidirect
115. #908C2×C32semidirect
116. #904C2×C32semidirect
117. C2×#204C2×C32semidirect
178. #235C22×C32semidirect
180. C2×Dic6C22×C32semidirect
187. C3×M16C2×C3×D3semidirect
188. C3×M16C22×C32semidirect
190. C2×C3×C8C2×C3×D3semidirect
191. C2×C3×C8C22×C32semidirect
192. Dic12C22×C32semidirect
193. D24C22×C32semidirect
194. #204C22×C32semidirect
195. #204C22×C32semidirect
196. #204C22×C32semidirect
222. #131C32×D3semidirect
223. C33#577semidirect
287. M16C2×C32×D3semidirect
288. C2×C8C2×C32×D3semidirect
309. C32#5751direct (N × H)
310. C32C3×#577semidirect
347. C3C3×#5751direct (N × H)
348. C3C32×#577semidirect

Maximal subgroups

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

subgroupindex
C32×#10102
C32×#10092
C33×#1312
C32×#9092
C32×#9082
C32×#9042
C2×C32×#2042
C32×#5773
C3×#57513