Finite Group Databasebeta

← C3×#87167 #11919037

Subgroups — C3×#87167 #11919037

Order 1152.

3057 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic, complemented
2C2×C3×#1009C211yesmaximal, characteristic, complemented
3C32×#568C211yesmaximal, characteristic, complemented
4C3×#5752C218yesmaximal, complemented
5C3×#5751C218yesmaximal, complemented
6C3×#5748C216yesmaximal, complemented
7C2×C3×#908C216yesmaximal, complemented
8C3×#3585—31nomaximal
9#87167C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010C2214yescomplemented
12C3×#1009—24no—
13C3×#1009C2218yescomplemented
14C3×#1009C2214yescomplemented
15C3×D3×#41—24no—
16C3×#1007—21no—
17C2×C3×#239—21no—
18C3×#1004—23no—
19C2×C3×D3×D4—23no—
20C2×C3×#235C2213yescomplemented
21C2×C3×#235C2216yescomplemented
22C22×C3×D12C2213yescomplemented
23C32×#131C2218yescomplemented
24C2×C32×M16C2216yescomplemented
25C3×#909C2214yescomplemented
26C3×#909C22112yescomplemented
27C3×#908C22112yescomplemented
28C3×#908C22124yescomplemented
29C3×#908C22112yescomplemented
30C3×#904C22124yescomplemented
31C2×C3×D24C22112yescomplemented
32C2×C3×#204C22112yescomplemented
33C3×#585—31no—
34C2×C3×H(5,𝔽2)—31no—
35#5964C2×C311yescharacteristic, complemented
36C2×#1009C2×C311yescharacteristic, complemented
37C3×#577—38no—
38C3×#576—38no—
39C3×#575—36no—
40C2×C3×AGL(1,ℤ/8)—36no—
41C3×#568—21no—
42C3×#568D311yescharacteristic, complemented
43C3×#568C2×C311yescharacteristic, complemented
44#5752C2×C318yescomplemented
45#5751C2×C318yescomplemented
46#5748C2×C316yescomplemented
47C2×#908C2×C316yescomplemented
48C23×C3×D3—23no—
49C32×#41D414yes—
50C32×#41C2314yes—
51C2×C32×Q8D411yescharacteristic
52C2×C32×D4D413yes—
53C22×C32×C4C2313yes—
54C2×C3×#67—23no—
55C2×C3×#67—26no—
56C3×#239—28no—
57C3×#239—24no—
58C3×D3×Q8—24no—
59C3×#237—212no—
60C3×D3×D4—224no—
61C3×D3×D4—212no—
62C3×#235—26no—
63C3×#235—26no—
64C3×#235—26no—
65C3×#235C23112yes—
66C3×#235C2314yes—
67C3×#235C23112yescomplemented
68C2×C3×D12—23no—
69C2×C3×D12—26no—
70C2×C3×D12C23112yes—
71C2×C3×D12C2313yes—
72C2×C3×D12C2316yescomplemented
73C2×C3×C4×D3—23no—
74C2×C3×C4×D3—26no—
75C2×C3×Dic6C2311yescharacteristic
76C2×C3×Dic6C2316yescomplemented
77C32×M16C2314yes—
78C32×M16C23112yescomplemented
79C2×C32×C8C23112yescomplemented
80C3×Dic12C2318yescomplemented
81C3×D24C23124yescomplemented
82C3×#204C23124yescomplemented
83C3×#204C2318yescomplemented
84#3585—31noSylow
85C3×2−1+4—34no—
86C3×2−1+4—61no—
87C3×H(5,𝔽2)—34no—
88C3×H(5,𝔽2)—38no—
89C3×H(5,𝔽2)—64no—
90C3×H(5,𝔽2)—63no—
91C2×C3×#41—21no—
92C2×C3×#41—36no—
93C2×C3×#41—33no—
94C2×C3×#41—64no—
95C2×C3×#41—61no—
96C2×C3×#41C2×D311yescharacteristic
97C2×C3×#41C22×C311yescharacteristic
98C22×C3×D4—33no—
99C22×C3×D4—63no—
100#1010C22×C314yescomplemented
101#1009—24no—
102#1009C22×C318yescomplemented
103#1009C22×C314yescomplemented
104D3×#41—24no—
105#1007—21no—
106C2×#239—21no—
107#1004—23no—
108C2×D3×D4—23no—
109C2×#235C22×C313yescomplemented
110C2×#235C22×C316yescomplemented
111C22×D12C22×C313yescomplemented
112C3×#137—34no—
113C3×#137—312no—
114C3×AGL(1,ℤ/8)—312no—
115C3×AGL(1,ℤ/8)—324no—
116C3×AGL(1,ℤ/8)—312no—
117C3×#135—324no—
118C2×C3×SD16—312no—
119C2×C3×D8—312no—
120C3×#131—28no—
121C3×#131C2×D318yescomplemented
122C3×#131C22×C318yescomplemented
123C2×C3×M16—26no—
124C2×C3×M16C2×D316yescomplemented
125C2×C3×M16C22×C316yescomplemented
126#909C22×C314yescomplemented
127#909C22×C3112yescomplemented
128#908C22×C3112yescomplemented
129#908C22×C3124yescomplemented
130#908C22×C3112yescomplemented
131#904C22×C3124yescomplemented
132C2×D24C22×C3112yescomplemented
133C2×#204C22×C3112yescomplemented
134C23×C32C2×D413yes—
135C22×C3×D3—23no—
136C22×C3×D3—212no—
137C22×C3×D3—26no—
138C22×C3×D3—86no—
139C32×Q8C2×D414yes—
140C32×D4C2×D4112yes—
141C2×C32×C4C2×D413yes—
142C2×C32×C4C2×D416yes—
143C2×C32×C4C2416yes—
144C2×C32×C4C2411yescharacteristic
145C3×#67—24no—
146C3×#67—212no—
147C3×#67—212no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—26no—
150C3×D12—23no—
151C3×D12—212no—
152C3×D12—26no—
153C3×D12C2416yes—
154C3×D12C2414yes—
155C3×C4×D3—24no—
156C3×C4×D3—212no—
157C3×C4×D3—212no—
158C3×Dic6—21no—
159C3×Dic6—26no—
160C3×Dic6C2414yes—
161C3×Dic6C2412yes—
162C32×C8C2418yes—
163#585—31no—
164C2×H(5,𝔽2)—31no—
165#577—38no—
166#576—38no—
167#575—36no—
168C2×AGL(1,ℤ/8)—36no—
169#568C3×D311yescharacteristic, complemented
170C24×C3—63no—
171C23×D3—23no—
172C3×#41—24no—
173C3×#41—24no—
174C3×#41—312no—
175C3×#41—312no—
176C3×#41—34no—
177C3×#41—68no—
178C3×#41—64no—
179C3×#41—612no—
180C3×#41—66no—
181C3×#41—66no—
182C3×#41—66no—
183C3×#41D1214yes—
184C3×#41C3×D414yes—
185C3×#41C22×D314yes—
186C3×#41C23×C314yes—
187C2×C3×Q8—21no—
188C2×C3×Q8—36no—
189C2×C3×Q8—31no—
190C2×C3×Q8—64no—
191C2×C3×Q8D1211yescharacteristic
192C2×C3×Q8C3×D411yescharacteristic
193C2×C3×D4—23no—
194C2×C3×D4—36no—
195C2×C3×D4—312no—
196C2×C3×D4—33no—
197C2×C3×D4—63no—
198C2×C3×D4—66no—
199C2×C3×D4—624no—
200C2×C3×D4—612no—
201C2×C3×D4—63no—
202C2×C3×D4—66no—
203C2×C3×D4D1213yes—
204C2×C3×D4C3×D413yes—
205C22×C3×C4—23no—
206C22×C3×C4—63no—
207C22×C3×C4—66no—
208C22×C3×C4C22×D313yes—
209C22×C3×C4C23×C313yes—
210C2×#67—23no—
211C2×#67—26no—
212#239—28no—
213#239—24no—
214D3×Q8—24no—
215#237—212no—
216D3×D4—224no—
217D3×D4—212no—
218#235—26no—
219#235—26no—
220#235—26no—
221#235C23×C3112yes—
222#235C23×C314yes—
223#235C23×C3112yescomplemented
224C2×D12—23no—
225C2×D12—26no—
226C2×D12C23×C3112yes—
227C2×D12C23×C313yes—
228C2×D12C23×C316yescomplemented
229C2×C4×D3—23no—
230C2×C4×D3—26no—
231C2×Dic6C23×C311yescharacteristic
232C2×Dic6C23×C316yescomplemented
233C3×Q16—38no—
234C3×SD16—324no—
235C3×SD16—38no—
236C3×D8—324no—
237C3×M16—24no—
238C3×M16—212no—
239C3×M16C22×D314yes—
240C3×M16C22×D3112yescomplemented
241C3×M16C23×C314yes—
242C3×M16C23×C3112yescomplemented
243C2×C3×C8—212no—
244C2×C3×C8C22×D3112yescomplemented
245C2×C3×C8C23×C3112yescomplemented
246Dic12C23×C318yescomplemented
247D24C23×C3124yescomplemented
248#204C23×C3124yescomplemented
249#204C23×C318yescomplemented
250C22×C32—43no—
251C22×C32C22×D411yescharacteristic
252C22×C32C22×D416yes—
253C2×C3×D3—24no—
254C2×C3×D3—26no—
255C2×C3×D3—46no—
256C2×C3×D3—86no—
257C2×C3×D3—812no—
258C32×C4C22×D411yescharacteristic
259C32×C4C22×D416yes—
260C32×C4C2511yescharacteristic
261C3×Dic3—24no—
262C3×Dic3—22no—
2632−1+4—34no—
2642−1+4—61no—
265H(5,𝔽2)—34no—
266H(5,𝔽2)—38no—
267H(5,𝔽2)—64no—
268H(5,𝔽2)—63no—
269C2×#41—36no—
270C2×#41—33no—
271C2×#41—64no—
272C2×#41—61no—
273C2×#41C2×C3×D311yescharacteristic
274C22×D4—33no—
275C22×D4—63no—
276#137—34no—
277#137—312no—
278AGL(1,ℤ/8)—312no—
279AGL(1,ℤ/8)—324no—
280AGL(1,ℤ/8)—312no—
281#135—324no—
282C2×SD16—312no—
283C2×D8—312no—
284#131C2×C3×D318yescomplemented
285C2×M16C2×C3×D316yescomplemented
286C23×C3—23no—
287C23×C3—63no—
288C23×C3—66no—
289C23×C3—612no—
290C23×C3—246no—
291C23×C3C2×D1213yes—
292C23×C3C2×C3×D413yes—
293C22×D3—23no—
294C22×D3—26no—
295C22×D3—212no—
296C22×D3—86no—
297C3×Q8—24no—
298C3×Q8—34no—
299C3×Q8—32no—
300C3×Q8—66no—
301C3×Q8—61no—
302C3×Q8C2×D1214yes—
303C3×Q8C2×C3×D414yes—
304C3×D4—212no—
305C3×D4—34no—
306C3×D4—36no—
307C3×D4—612no—
308C3×D4—64no—
309C3×D4—612no—
310C3×D4—63no—
311C3×D4—612no—
312C3×D4—66no—
313C3×D4C2×D12112yes—
314C3×D4C2×C3×D4112yes—
315C2×C3×C4—26no—
316C2×C3×C4—21no—
317C2×C3×C4—23no—
318C2×C3×C4—26no—
319C2×C3×C4—61no—
320C2×C3×C4—66no—
321C2×C3×C4—64no—
322C2×C3×C4—612no—
323C2×C3×C4—612no—
324C2×C3×C4C2×D1213yes—
325C2×C3×C4C2×D1216yes—
326C2×C3×C4C2×C3×D413yes—
327C2×C3×C4C2×C3×D416yes—
328C2×C3×C4C23×D316yes—
329C2×C3×C4C23×D311yescharacteristic
330C2×C3×C4C24×C316yes—
331C2×C3×C4C24×C311yescharacteristic
332#67—24no—
333#67—212no—
334#67—212no—
335C2×Dic3—21no—
336C2×Dic3—26no—
337D12—23no—
338D12—212no—
339D12—26no—
340D12C24×C316yes—
341D12C24×C314yes—
342C4×D3—24no—
343C4×D3—212no—
344C4×D3—212no—
345Dic6—21no—
346Dic6—26no—
347Dic6C24×C314yes—
348Dic6C24×C312yes—
349C3×C8—28no—
350C3×C8C23×D318yes—
351C3×C8C24×C318yes—
352C2×C32—21no—
353C2×C32—26no—
354C2×C32C23×D411yescharacteristic
355C3×D3—46no—
356C3×D3—44no—
357C24—63no—
358#41—312no—
359#41—312no—
360#41—34no—
361#41—68no—
362#41—64no—
363#41—612no—
364#41—66no—
365#41—66no—
366#41—66no—
367#41C3×D1214yes—
368#41C22×C3×D314yes—
369C2×Q8—36no—
370C2×Q8—31no—
371C2×Q8—64no—
372C2×Q8C3×D1211yescharacteristic
373C2×D4—36no—
374C2×D4—312no—
375C2×D4—33no—
376C2×D4—63no—
377C2×D4—66no—
378C2×D4—624no—
379C2×D4—612no—
380C2×D4—63no—
381C2×D4—66no—
382C2×D4C3×D1213yes—
383C22×C4—63no—
384C22×C4—66no—
385C22×C4C22×C3×D313yes—
386Q16—38no—
387SD16—324no—
388SD16—38no—
389D8—324no—
390M16C22×C3×D314yes—
391M16C22×C3×D3112yescomplemented
392C2×C8C22×C3×D3112yescomplemented
393C22×C3—21no—
394C22×C3—26no—
395C22×C3—43no—
396C22×C3—43no—
397C22×C3—64no—
398C22×C3—66no—
399C22×C3—83no—
400C22×C3—126no—
401C22×C3—2412no—
402C22×C3—246no—
403C22×C3C22×D1211yescharacteristic
404C22×C3C22×D1216yes—
405C22×C3C22×C3×D411yescharacteristic
406C22×C3C22×C3×D416yes—
407C2×D3—24no—
408C2×D3—26no—
409C2×D3—46no—
410C2×D3—812no—
411C2×D3—86no—
412C3×C4—21no—
413C3×C4—26no—
414C3×C4—21no—
415C3×C4—62no—
416C3×C4—64no—
417C3×C4C22×D1211yescharacteristic
418C3×C4C22×D1216yes—
419C3×C4C22×C3×D411yescharacteristic
420C3×C4C22×C3×D416yes—
421C3×C4C24×D311yescharacteristic
422C3×C4C25×C311yescharacteristic
423Dic3—24no—
424Dic3—22no—
425C32#358511yescharacteristic, complemented, Sylow
426C23—63no—
427C23—66no—
428C23—612no—
429C23—246no—
430C23C2×C3×D1213yes—
431Q8—34no—
432Q8—32no—
433Q8—66no—
434Q8—61no—
435Q8C2×C3×D1214yes—
436D4—34no—
437D4—36no—
438D4—612no—
439D4—64no—
440D4—612no—
441D4—63no—
442D4—612no—
443D4—66no—
444D4C2×C3×D12112yes—
445C2×C4—61no—
446C2×C4—66no—
447C2×C4—64no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4C2×C3×D1213yes—
451C2×C4C2×C3×D1216yes—
452C2×C4C23×C3×D316yes—
453C2×C4C23×C3×D311yescharacteristic
454C8C23×C3×D318yes—
455C2×C3—21no—
456C2×C3—26no—
457C2×C3—21no—
458C2×C3—26no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—46no—
462C2×C3—126no—
463C2×C3—124no—
464C2×C3C23×D1211yescharacteristic
465C2×C3C23×C3×D411yescharacteristic
466D3—46no—
467D3—44no—
468C22—43no—
469C22—64no—
470C22—66no—
471C22—126no—
472C22—2412no—
473C22—246no—
474C22C22×C3×D1211yescharacteristic
475C22C22×C3×D1216yes—
476C4—62no—
477C4—64no—
478C4C22×C3×D1211yescharacteristic
479C4C22×C3×D1216yes—
480C4C24×C3×D311yescharacteristic
481C3—21no—
482C3#8716711yescharacteristic, direct product factor
483C3C3×#358511yescharacteristic, complemented
484C2—21no—
485C2—26no—
486C2—126no—
487C2—124no—
488C2C23×C3×D1211yescharacteristic
subgroupG/H
centerC2×C3C23×D12
commutator, cosocleC3×C4C25×C3
FrattiniC4C24×C3×D3
FittingC32×#568C2
socleC2×C32C23×D4
hypercenterC3×#568D3
hypocenterC3C3×#3585
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×#41 ◁ C3×#568

Complements

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

NH ≅ G/Ntype
1. C3×#5964C2semidirect
2. C2×C3×#1009C2semidirect
3. C32×#568C2semidirect
4. C3×#5752C2semidirect
5. C3×#5751C2semidirect
6. C3×#5748C2semidirect
7. C2×C3×#908C2semidirect
9. #87167C3direct (N × H)
11. C3×#1010C22semidirect
13. C3×#1009C22semidirect
14. C3×#1009C22semidirect
20. C2×C3×#235C22semidirect
21. C2×C3×#235C22semidirect
22. C22×C3×D12C22semidirect
23. C32×#131C22semidirect
24. C2×C32×M16C22semidirect
25. C3×#909C22semidirect
26. C3×#909C22semidirect
27. C3×#908C22semidirect
28. C3×#908C22semidirect
29. C3×#908C22semidirect
30. C3×#904C22semidirect
31. C2×C3×D24C22semidirect
32. C2×C3×#204C22semidirect
35. #5964C2×C3semidirect
36. C2×#1009C2×C3semidirect
42. C3×#568D3semidirect
43. C3×#568C2×C3semidirect
44. #5752C2×C3semidirect
45. #5751C2×C3semidirect
46. #5748C2×C3semidirect
47. C2×#908C2×C3semidirect
67. C3×#235C23semidirect
72. C2×C3×D12C23semidirect
76. C2×C3×Dic6C23semidirect
78. C32×M16C23semidirect
79. C2×C32×C8C23semidirect
80. C3×Dic12C23semidirect
81. C3×D24C23semidirect
82. C3×#204C23semidirect
83. C3×#204C23semidirect
100. #1010C22×C3semidirect
102. #1009C22×C3semidirect
103. #1009C22×C3semidirect
109. C2×#235C22×C3semidirect
110. C2×#235C22×C3semidirect
111. C22×D12C22×C3semidirect
121. C3×#131C2×D3semidirect
122. C3×#131C22×C3semidirect
124. C2×C3×M16C2×D3semidirect
125. C2×C3×M16C22×C3semidirect
126. #909C22×C3semidirect
127. #909C22×C3semidirect
128. #908C22×C3semidirect
129. #908C22×C3semidirect
130. #908C22×C3semidirect
131. #904C22×C3semidirect
132. C2×D24C22×C3semidirect
133. C2×#204C22×C3semidirect
169. #568C3×D3semidirect
223. #235C23×C3semidirect
228. C2×D12C23×C3semidirect
232. C2×Dic6C23×C3semidirect
240. C3×M16C22×D3semidirect
242. C3×M16C23×C3semidirect
244. C2×C3×C8C22×D3semidirect
245. C2×C3×C8C23×C3semidirect
246. Dic12C23×C3semidirect
247. D24C23×C3semidirect
248. #204C23×C3semidirect
249. #204C23×C3semidirect
284. #131C2×C3×D3semidirect
285. C2×M16C2×C3×D3semidirect
391. M16C22×C3×D3semidirect
392. C2×C8C22×C3×D3semidirect
425. C32#3585semidirect
482. C3#87167direct (N × H)
483. C3C3×#3585semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C2×C3×#10092
C32×#5682
C3×#57522
C3×#57512
C3×#57482
C2×C3×#9082
C3×#35853
#871673