Finite Group Databasebeta

← C3×#87163 #11918973

Subgroups — C3×#87163 #11918973

Order 1152.

2981 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C212yesmaximal, complemented
2C2×C32×#131C211yesmaximal, characteristic, complemented
3C3×#5753C214yesmaximal, complemented
4C3×#5752C214yesmaximal, complemented
5C3×#5751C218yesmaximal, complemented
6C3×#5748C216yesmaximal, complemented
7C2×C3×#904C216yesmaximal, complemented
8C3×#3584—31nomaximal
9#87163C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010C2218yescomplemented
12C3×#1009C2218yescomplemented
13C3×D3×#41—28no—
14C3×#1007—22no—
15C3×#1004—26no—
16C2×C3×#235C22112yescomplemented
17C32×#131C2214yescomplemented
18C32×#131C2214yescomplemented
19C2×C32×M16C2213yescomplemented
20C22×C32×C8C2213yescomplemented
21C3×#909C22124yescomplemented
22C3×#908C22124yescomplemented
23C2×C3×Dic12C2216yescomplemented
24C3×#904C22112yescomplemented
25C3×#904C22112yescomplemented
26C3×#904C22112yescomplemented
27C3×#904C2214yescomplemented
28C2×C3×D24C2216yescomplemented
29C2×C3×#204C22112yescomplemented
30C3×#585—32no—
31#5964C2×C312yescomplemented
32C3×#578—34no—
33C3×#577—38no—
34C3×#576—34no—
35C3×#575—36no—
36C2×C3×#135—36no—
37C2×C3×#131—21no—
38C2×C3×#131D311yescharacteristic, complemented
39C2×C3×#131C2×C311yescharacteristic, complemented
40#5753C2×C314yescomplemented
41#5752C2×C314yescomplemented
42#5751C2×C318yescomplemented
43#5748C2×C316yescomplemented
44C2×#904C2×C316yescomplemented
45C32×#41D414yes—
46C32×#41C2314yes—
47C2×C32×Q8D411yescharacteristic
48C2×C32×D4D413yes—
49C22×C32×C4C2313yes—
50C2×C3×#67—212no—
51C3×#239—28no—
52C3×D3×Q8—28no—
53C3×#237—224no—
54C3×D3×D4—224no—
55C3×#235—212no—
56C3×#235—212no—
57C3×#235C2318yes—
58C3×#235C23124yes—
59C2×C3×D12C23112yes—
60C2×C3×C4×D3—212no—
61C2×C3×Dic6C23112yes—
62C32×M16C23112yes—
63C2×C32×C8C2316yes—
64C2×C32×C8C2316yes—
65C2×C32×C8C2311yescharacteristic
66C2×C32×C8C2313yes—
67C3×Dic12C2314yes—
68C3×Dic12C23112yes—
69C3×D24C2314yes—
70C3×D24C23112yes—
71C3×#204C2318yes—
72C3×#204C23124yes—
73#3584—31noSylow
74C3×2−1+4—38no—
75C3×2−1+4—62no—
76C3×H(5,𝔽2)—38no—
77C3×H(5,𝔽2)—66no—
78C2×C3×#41—21no—
79C2×C3×#41—312no—
80C2×C3×#41—68no—
81C2×C3×#41C2×D311yescharacteristic
82C2×C3×#41C22×C311yescharacteristic
83#1010C22×C318yescomplemented
84#1009C22×C318yescomplemented
85D3×#41—28no—
86#1007—22no—
87#1004—26no—
88C2×#235C22×C3112yescomplemented
89C3×#137—324no—
90C3×AGL(1,ℤ/8)—324no—
91C3×#135—34no—
92C3×#135—312no—
93C3×#135—312no—
94C3×#135—312no—
95C2×C3×Q16—36no—
96C2×C3×SD16—312no—
97C2×C3×D8—36no—
98C3×#131—24no—
99C3×#131—24no—
100C3×#131C2×D314yescomplemented
101C3×#131C2×D314yescomplemented
102C3×#131C22×C314yescomplemented
103C3×#131C22×C314yescomplemented
104C2×C3×M16—23no—
105C2×C3×M16C2×D313yescomplemented
106C2×C3×M16C22×C313yescomplemented
107C22×C3×C8—23no—
108C22×C3×C8C2×D313yescomplemented
109C22×C3×C8C22×C313yescomplemented
110#909C22×C3124yescomplemented
111#908C22×C3124yescomplemented
112C2×Dic12C22×C316yescomplemented
113#904C22×C314yescomplemented
114#904C22×C3112yescomplemented
115#904C22×C3112yescomplemented
116#904C22×C3112yescomplemented
117C2×D24C22×C316yescomplemented
118C2×#204C22×C3112yescomplemented
119C23×C32C2×D413yes—
120C22×C3×D3—212no—
121C32×Q8C2×D414yes—
122C32×D4C2×D4112yes—
123C2×C32×C4C2×D416yes—
124C2×C32×C4C2×D413yes—
125C2×C32×C4C2411yescharacteristic
126C2×C32×C4C2416yes—
127C3×#67—28no—
128C3×#67—224no—
129C2×C3×Dic3—212no—
130C3×D12—212no—
131C3×D12C2418yes—
132C3×C4×D3—28no—
133C3×C4×D3—224no—
134C3×Dic6—212no—
135C3×Dic6C2418yes—
136C32×C8C2411yescharacteristic
137C32×C8C2416yes—
138C32×C8C2411yescharacteristic
139#585—32no—
140#578—34no—
141#577—38no—
142#576—34no—
143#575—36no—
144C2×#135—36no—
145C2×#131C3×D311yescharacteristic, complemented
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—38no—
149C3×#41—324no—
150C3×#41—68no—
151C3×#41—624no—
152C3×#41—612no—
153C3×#41—612no—
154C3×#41D1214yes—
155C3×#41C3×D414yes—
156C3×#41C22×D314yes—
157C3×#41C23×C314yes—
158C2×C3×Q8—21no—
159C2×C3×Q8—312no—
160C2×C3×Q8—68no—
161C2×C3×Q8D1211yescharacteristic
162C2×C3×Q8C3×D411yescharacteristic
163C2×C3×D4—23no—
164C2×C3×D4—312no—
165C2×C3×D4—612no—
166C2×C3×D4—624no—
167C2×C3×D4D1213yes—
168C2×C3×D4C3×D413yes—
169C22×C3×C4—23no—
170C22×C3×C4—612no—
171C22×C3×C4C22×D313yes—
172C22×C3×C4C23×C313yes—
173C2×#67—212no—
174#239—28no—
175D3×Q8—28no—
176#237—224no—
177D3×D4—224no—
178#235—212no—
179#235—212no—
180#235C23×C318yes—
181#235C23×C3124yes—
182C2×D12C23×C3112yes—
183C2×C4×D3—212no—
184C2×Dic6C23×C3112yes—
185C3×Q16—34no—
186C3×Q16—312no—
187C3×SD16—38no—
188C3×SD16—324no—
189C3×D8—34no—
190C3×D8—312no—
191C3×M16—212no—
192C3×M16C22×D3112yes—
193C3×M16C23×C3112yes—
194C2×C3×C8—21no—
195C2×C3×C8—23no—
196C2×C3×C8—26no—
197C2×C3×C8—26no—
198C2×C3×C8C22×D311yescharacteristic
199C2×C3×C8C22×D313yes—
200C2×C3×C8C22×D316yes—
201C2×C3×C8C22×D316yes—
202C2×C3×C8C23×C316yes—
203C2×C3×C8C23×C316yes—
204C2×C3×C8C23×C311yescharacteristic
205C2×C3×C8C23×C313yes—
206Dic12C23×C314yes—
207Dic12C23×C3112yes—
208D24C23×C314yes—
209D24C23×C3112yes—
210#204C23×C318yes—
211#204C23×C3124yes—
212C22×C32—43no—
213C22×C32C22×D416yes—
214C22×C32C22×D411yescharacteristic
215C2×C3×D3—28no—
216C2×C3×D3—812no—
217C32×C4C22×D416yes—
218C32×C4C22×D411yescharacteristic
219C32×C4C2511yescharacteristic
220C3×Dic3—28no—
2212−1+4—38no—
2222−1+4—62no—
223H(5,𝔽2)—38no—
224H(5,𝔽2)—66no—
225C2×#41—312no—
226C2×#41—68no—
227C2×#41C2×C3×D311yescharacteristic
228#137—324no—
229AGL(1,ℤ/8)—324no—
230#135—34no—
231#135—312no—
232#135—312no—
233#135—312no—
234C2×Q16—36no—
235C2×SD16—312no—
236C2×D8—36no—
237#131C2×C3×D314yescomplemented
238#131C2×C3×D314yescomplemented
239C2×M16C2×C3×D313yescomplemented
240C22×C8C2×C3×D313yescomplemented
241C23×C3—23no—
242C23×C3—612no—
243C23×C3C2×D1213yes—
244C23×C3C2×C3×D413yes—
245C22×D3—212no—
246C3×Q8—24no—
247C3×Q8—38no—
248C3×Q8—612no—
249C3×Q8C2×D1214yes—
250C3×Q8C2×C3×D414yes—
251C3×D4—212no—
252C3×D4—38no—
253C3×D4—624no—
254C3×D4—68no—
255C3×D4—612no—
256C3×D4C2×D12112yes—
257C3×D4C2×C3×D4112yes—
258C2×C3×C4—21no—
259C2×C3×C4—26no—
260C2×C3×C4—23no—
261C2×C3×C4—26no—
262C2×C3×C4—612no—
263C2×C3×C4—624no—
264C2×C3×C4—68no—
265C2×C3×C4C2×D1213yes—
266C2×C3×C4C2×D1216yes—
267C2×C3×C4C2×C3×D416yes—
268C2×C3×C4C2×C3×D413yes—
269C2×C3×C4C23×D316yes—
270C2×C3×C4C23×D311yescharacteristic
271C2×C3×C4C24×C311yescharacteristic
272C2×C3×C4C24×C316yes—
273#67—28no—
274#67—224no—
275C2×Dic3—212no—
276D12—212no—
277D12C24×C318yes—
278C4×D3—28no—
279C4×D3—224no—
280Dic6—212no—
281Dic6C24×C318yes—
282C3×C8—21no—
283C3×C8—21no—
284C3×C8—26no—
285C3×C8C23×D311yescharacteristic
286C3×C8C23×D311yescharacteristic
287C3×C8C23×D316yes—
288C3×C8C24×C311yescharacteristic
289C3×C8C24×C316yes—
290C3×C8C24×C311yescharacteristic
291C2×C32—21no—
292C2×C32—26no—
293C2×C32C23×D411yescharacteristic
294C3×D3—48no—
295#41—38no—
296#41—324no—
297#41—68no—
298#41—624no—
299#41—612no—
300#41—612no—
301#41C3×D1214yes—
302#41C22×C3×D314yes—
303C2×Q8—312no—
304C2×Q8—68no—
305C2×Q8C3×D1211yescharacteristic
306C2×D4—312no—
307C2×D4—612no—
308C2×D4—624no—
309C2×D4C3×D1213yes—
310C22×C4—612no—
311C22×C4C22×C3×D313yes—
312Q16—34no—
313Q16—312no—
314SD16—38no—
315SD16—324no—
316D8—34no—
317D8—312no—
318M16C22×C3×D3112yes—
319C2×C8C22×C3×D311yescharacteristic
320C2×C8C22×C3×D313yes—
321C2×C8C22×C3×D316yes—
322C2×C8C22×C3×D316yes—
323C22×C3—21no—
324C22×C3—26no—
325C22×C3—43no—
326C22×C3—43no—
327C22×C3—46no—
328C22×C3—68no—
329C22×C3—2412no—
330C22×C3C22×D1216yes—
331C22×C3C22×D1211yescharacteristic
332C22×C3C22×C3×D416yes—
333C22×C3C22×C3×D411yescharacteristic
334C2×D3—28no—
335C2×D3—812no—
336C3×C4—21no—
337C3×C4—26no—
338C3×C4—21no—
339C3×C4—68no—
340C3×C4C22×D1216yes—
341C3×C4C22×D1211yescharacteristic
342C3×C4C22×C3×D416yes—
343C3×C4C22×C3×D411yescharacteristic
344C3×C4C24×D311yescharacteristic
345C3×C4C25×C311yescharacteristic
346Dic3—28no—
347C32#358411yescharacteristic, complemented, Sylow
348C23—612no—
349C23C2×C3×D1213yes—
350Q8—38no—
351Q8—612no—
352Q8C2×C3×D1214yes—
353D4—38no—
354D4—624no—
355D4—68no—
356D4—612no—
357D4C2×C3×D12112yes—
358C2×C4—612no—
359C2×C4—68no—
360C2×C4—624no—
361C2×C4C2×C3×D1216yes—
362C2×C4C2×C3×D1213yes—
363C2×C4C23×C3×D311yescharacteristic
364C2×C4C23×C3×D316yes—
365C8C23×C3×D311yescharacteristic
366C8C23×C3×D311yescharacteristic
367C8C23×C3×D316yes—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—26no—
371C2×C3—26no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—46no—
375C2×C3—128no—
376C2×C3C23×D1211yescharacteristic
377C2×C3C23×C3×D411yescharacteristic
378D3—48no—
379C22—43no—
380C22—68no—
381C22—2412no—
382C22C22×C3×D1216yes—
383C22C22×C3×D1211yescharacteristic
384C4—68no—
385C4C22×C3×D1211yescharacteristic
386C4C22×C3×D1216yes—
387C4C24×C3×D311yescharacteristic
388C3—21no—
389C3#8716311yescharacteristic, direct product factor
390C3C3×#358411yescharacteristic, complemented
391C2—21no—
392C2—26no—
393C2—128no—
394C2C23×C3×D1211yescharacteristic
subgroupG/H
centerC3×C4C22×D12
commutator, cosocleC3×C4C25×C3
FrattiniC4C24×C3×D3
FittingC2×C32×#131C2
socleC2×C32C23×D4
hypercenterC2×C3×#131D3
hypocenterC3C3×#3584
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C2×C3×#41 ◁ C2×C3×#131

Complements

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

NH ≅ G/Ntype
1. C3×#5964C2semidirect
2. C2×C32×#131C2semidirect
3. C3×#5753C2semidirect
4. C3×#5752C2semidirect
5. C3×#5751C2semidirect
6. C3×#5748C2semidirect
7. C2×C3×#904C2semidirect
9. #87163C3direct (N × H)
11. C3×#1010C22semidirect
12. C3×#1009C22semidirect
16. C2×C3×#235C22semidirect
17. C32×#131C22semidirect
18. C32×#131C22semidirect
19. C2×C32×M16C22semidirect
20. C22×C32×C8C22semidirect
21. C3×#909C22semidirect
22. C3×#908C22semidirect
23. C2×C3×Dic12C22semidirect
24. C3×#904C22semidirect
25. C3×#904C22semidirect
26. C3×#904C22semidirect
27. C3×#904C22semidirect
28. C2×C3×D24C22semidirect
29. C2×C3×#204C22semidirect
31. #5964C2×C3semidirect
38. C2×C3×#131D3semidirect
39. C2×C3×#131C2×C3semidirect
40. #5753C2×C3semidirect
41. #5752C2×C3semidirect
42. #5751C2×C3semidirect
43. #5748C2×C3semidirect
44. C2×#904C2×C3semidirect
83. #1010C22×C3semidirect
84. #1009C22×C3semidirect
88. C2×#235C22×C3semidirect
100. C3×#131C2×D3semidirect
101. C3×#131C2×D3semidirect
102. C3×#131C22×C3semidirect
103. C3×#131C22×C3semidirect
105. C2×C3×M16C2×D3semidirect
106. C2×C3×M16C22×C3semidirect
108. C22×C3×C8C2×D3semidirect
109. C22×C3×C8C22×C3semidirect
110. #909C22×C3semidirect
111. #908C22×C3semidirect
112. C2×Dic12C22×C3semidirect
113. #904C22×C3semidirect
114. #904C22×C3semidirect
115. #904C22×C3semidirect
116. #904C22×C3semidirect
117. C2×D24C22×C3semidirect
118. C2×#204C22×C3semidirect
145. C2×#131C3×D3semidirect
237. #131C2×C3×D3semidirect
238. #131C2×C3×D3semidirect
239. C2×M16C2×C3×D3semidirect
240. C22×C8C2×C3×D3semidirect
347. C32#3584semidirect
389. C3#87163direct (N × H)
390. C3C3×#3584semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C2×C32×#1312
C3×#57532
C3×#57522
C3×#57512
C3×#57482
C2×C3×#9042
C3×#35843
#871633