Finite Group Databasebeta

← C3×#86572 #11901982

Subgroups — C3×#86572 #11901982

Order 1152.

2441 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#543C211yesmaximal, characteristic, complemented
2C3×#5645C216yesmaximal, complemented
3C3×#5644C216yesmaximal, complemented
4C3×#5627C216yesmaximal, complemented
5C3×#5624C216yesmaximal, complemented
6C3×#5493C213yesmaximal, complemented
7C2×C3×#67⋊C4C213yesmaximal, complemented
8C3×#3459—31nomaximal
9#86572C311yesmaximal, characteristic, direct product factor
10C22×C3×#67C2213yes—
11C23×C3×Dic3C2211yescharacteristic
12C2×C3×#239C2212yescomplemented
13C22×C3×C4×D3C2213yescomplemented
14C32×C22⋊Q8C2216yescomplemented
15C32×C22⋊Q8C2216yescomplemented
16C2×C32×C22⋊C4C2213yes—
17C3×#947C2216yescomplemented
18C3×#946C2216yescomplemented
19C3×Q8×Dic3C2212yescomplemented
20C3×#930C2216yes—
21C3×#930C2216yes—
22C2×C3×(C2×D3)⋊C4C2213yescomplemented
23C2×C3×Dic3⋊C4C2213yescomplemented
24C2×C3×C4×Dic3C2213yescomplemented
25C3×#898C22112yescomplemented
26C3×#895C2216yescomplemented
27C3×#894C22112yescomplemented
28C3×#893C2216yescomplemented
29C3×#892C2216yescomplemented
30C3×#890C2216yescomplemented
31C3×#886C22112yescomplemented
32C3×#882C22112yescomplemented
33C3×#67⋊C4C22112yescomplemented
34C3×#67⋊C4C22112yescomplemented
35C3×#543—21no—
36C3×#543D311yescharacteristic, complemented
37C3×#543C2×C311yescharacteristic, complemented
38C3×#539—36no—
39C3×#538—36no—
40C3×#534—33no—
41C3×#529—36no—
42C3×#525—36no—
43C2×C3×C4×D4—33no—
44#5645C2×C316yescomplemented
45#5644C2×C316yescomplemented
46#5627C2×C316yescomplemented
47#5624C2×C316yescomplemented
48#5493C2×C313yescomplemented
49C2×#67⋊C4C2×C313yescomplemented
50C24×C32C2311yescharacteristic
51C23×C3×D3C2313yes—
52C2×C32×Q8C2312yes—
53C2×C32×Q8C2312yes—
54C22×C32×C4C2313yes—
55C22×C32×C4C2313yes—
56C2×C3×#67—212no—
57C2×C3×#67C23112yes—
58C22×C3×Dic3—42no—
59C22×C3×Dic3C2313yes—
60C22×C3×Dic3C2313yes—
61C22×C3×Dic3C2313yes—
62C3×#239—44no—
63C2×C3×D12C2316yes—
64C2×C3×D12C2316yes—
65C2×C3×C4×D3—46no—
66C2×C3×C4×D3C2316yes—
67C2×C3×C4×D3C2316yes—
68C32×C4⋊C4C2316yes—
69C32×C4⋊C4C2316yes—
70C32×C22⋊C4C23112yes—
71C3×(C2×D3)⋊C4C23112yes—
72C3×(C2×D3)⋊C4C23112yes—
73C3×(C2×D3)⋊C4C23112yes—
74C3×C4⋊Dic3C2316yes—
75C3×C4⋊Dic3C2316yes—
76C3×Dic3⋊C4C23112yes—
77C3×C4×Dic3C2316yes—
78C3×C4×Dic3C2316yes—
79#3459—31noSylow
80C2×C3×#41—32no—
81C22×C3×D4—33no—
82C23×C3×C4—31no—
83C23×C3×C4—33no—
84C22×#67C22×C313yes—
85C23×Dic3C22×C311yescharacteristic
86C2×#239C22×C312yescomplemented
87C22×C4×D3C22×C313yescomplemented
88C3×#126—312no—
89C3×#125—36no—
90C3×#124—36no—
91C3×#123—312no—
92C3×#123—312no—
93C3×C22⋊Q8—26no—
94C3×C22⋊Q8—26no—
95C3×C22⋊Q8—36no—
96C3×C22⋊Q8C2×D316yescomplemented
97C3×C22⋊Q8C2×D316yescomplemented
98C3×C22⋊Q8C22×C316yescomplemented
99C3×C22⋊Q8C22×C316yescomplemented
100C3×#121—36no—
101C3×#121—36no—
102C3×#121—36no—
103C3×#120—312no—
104C3×C4×Q8—32no—
105C3×C4×D4—36no—
106C3×C4×D4—312no—
107C3×C4×D4—312no—
108C3×#117—36no—
109C2×C3×C4⋊C4—33no—
110C2×C3×C22⋊C4—23no—
111C2×C3×C22⋊C4—33no—
112C2×C3×C22⋊C4C2×D313yes—
113C2×C3×C22⋊C4C22×C313yes—
114C2×C3×C42—33no—
115#947C22×C316yescomplemented
116#946C22×C316yescomplemented
117Q8×Dic3C22×C312yescomplemented
118#930C22×C316yes—
119#930C22×C316yes—
120C2×(C2×D3)⋊C4C22×C313yescomplemented
121C2×Dic3⋊C4C22×C313yescomplemented
122C2×C4×Dic3C22×C313yescomplemented
123#898C22×C3112yescomplemented
124#895C22×C316yescomplemented
125#894C22×C3112yescomplemented
126#893C22×C316yescomplemented
127#892C22×C316yescomplemented
128#890C22×C316yescomplemented
129#886C22×C3112yescomplemented
130#882C22×C3112yescomplemented
131#67⋊C4C22×C3112yescomplemented
132#67⋊C4C22×C3112yescomplemented
133C23×C32—41no—
134C23×C32—41no—
135C23×C32#4114yescomplemented
136C23×C32C2413yes—
137C22×C3×D3—26no—
138C22×C3×D3—43no—
139C22×C3×D3—43no—
140C22×C3×D3C2416yes—
141C32×Q8—42no—
142C32×Q8—42no—
143C2×C32×C4—43no—
144C2×C32×C4—43no—
145C2×C32×C4C2416yes—
146C2×C32×C4C2416yes—
147C3×#67—224no—
148C2×C3×Dic3—26no—
149C2×C3×Dic3—26no—
150C2×C3×Dic3—43no—
151C2×C3×Dic3C2413yes—
152C2×C3×Dic3C2416yes—
153C2×C3×Dic3C2411yescharacteristic
154C3×D12—46no—
155C3×D12—46no—
156C3×C4×D3—212no—
157C3×C4×D3—46no—
158#543C3×D311yescharacteristic, complemented
159#539—36no—
160#538—36no—
161#534—33no—
162#529—36no—
163#525—36no—
164C2×C4×D4—33no—
165C24×C3—21no—
166C24×C3—33no—
167C24×C3C22×D311yescharacteristic
168C24×C3C23×C311yescharacteristic
169C23×D3C23×C313yes—
170C3×#41—124no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8C22×D312yes—
174C2×C3×Q8C22×D312yes—
175C2×C3×Q8C23×C312yes—
176C2×C3×Q8C23×C312yes—
177C2×C3×D4—312no—
178C2×C3×D4—36no—
179C2×C3×D4—36no—
180C2×C3×D4—612no—
181C22×C3×C4—23no—
182C22×C3×C4—23no—
183C22×C3×C4—33no—
184C22×C3×C4—33no—
185C22×C3×C4—33no—
186C22×C3×C4—36no—
187C22×C3×C4—36no—
188C22×C3×C4—126no—
189C22×C3×C4—122no—
190C22×C3×C4C22×D313yes—
191C22×C3×C4C22×D313yes—
192C22×C3×C4C23×C313yes—
193C22×C3×C4C23×C313yes—
194C2×#67—212no—
195C2×#67C23×C3112yes—
196C22×Dic3—42no—
197C22×Dic3C23×C313yes—
198C22×Dic3C23×C313yes—
199C22×Dic3C23×C313yes—
200#239—44no—
201C2×D12C23×C316yes—
202C2×D12C23×C316yes—
203C2×C4×D3—46no—
204C2×C4×D3C23×C316yes—
205C2×C4×D3C23×C316yes—
206C3×C4⋊C4—26no—
207C3×C4⋊C4—26no—
208C3×C4⋊C4—312no—
209C3×C4⋊C4—36no—
210C3×C4⋊C4—36no—
211C3×C4⋊C4C22×D316yes—
212C3×C4⋊C4C22×D316yes—
213C3×C4⋊C4C23×C316yes—
214C3×C4⋊C4C23×C316yes—
215C3×C22⋊C4—212no—
216C3×C22⋊C4—312no—
217C3×C22⋊C4—312no—
218C3×C22⋊C4—312no—
219C3×C22⋊C4C22×D3112yes—
220C3×C22⋊C4C23×C3112yes—
221C3×C42—36no—
222C3×C42—36no—
223(C2×D3)⋊C4C23×C3112yes—
224(C2×D3)⋊C4C23×C3112yes—
225(C2×D3)⋊C4C23×C3112yes—
226C4⋊Dic3C23×C316yes—
227C4⋊Dic3C23×C316yes—
228Dic3⋊C4C23×C3112yes—
229C4×Dic3C23×C316yes—
230C4×Dic3C23×C316yes—
231C22×C32—23no—
232C22×C32—23no—
233C22×C32—44no—
234C22×C32C2×#4116yes—
235C22×C32C2511yescharacteristic
236C2×C3×D3—26no—
237C2×C3×D3—26no—
238C2×C3×D3—26no—
239C2×C3×D3—412no—
240C32×C4—26no—
241C32×C4—26no—
242C3×Dic3—26no—
243C3×Dic3—23no—
244C3×Dic3H(5,𝔽2)12yes—
245C2×#41—32no—
246C22×D4—33no—
247C23×C4—31no—
248C23×C4—33no—
249#126—312no—
250#125—36no—
251#124—36no—
252#123—312no—
253#123—312no—
254C22⋊Q8—36no—
255C22⋊Q8C2×C3×D316yescomplemented
256C22⋊Q8C2×C3×D316yescomplemented
257#121—36no—
258#121—36no—
259#121—36no—
260#120—312no—
261C4×Q8—32no—
262C4×D4—36no—
263C4×D4—312no—
264C4×D4—312no—
265#117—36no—
266C2×C4⋊C4—33no—
267C2×C22⋊C4—33no—
268C2×C22⋊C4C2×C3×D313yes—
269C2×C42—33no—
270C23×C3—23no—
271C23×C3—24no—
272C23×C3—36no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—66no—
278C23×C3—81no—
279C23×C3—81no—
280C23×C3—123no—
281C23×C3—123no—
282C23×C3#23914yescomplemented
283C23×C3C3×#4114yescomplemented
284C23×C3C23×D313yes—
285C23×C3C24×C313yes—
286C22×D3—26no—
287C22×D3—43no—
288C22×D3—43no—
289C22×D3C24×C316yes—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×Q8—82no—
295C3×Q8—82no—
296C3×D4—624no—
297C3×D4—126no—
298C3×D4—126no—
299C2×C3×C4—26no—
300C2×C3×C4—26no—
301C2×C3×C4—31no—
302C2×C3×C4—36no—
303C2×C3×C4—33no—
304C2×C3×C4—43no—
305C2×C3×C4—43no—
306C2×C3×C4—43no—
307C2×C3×C4—43no—
308C2×C3×C4—612no—
309C2×C3×C4—66no—
310C2×C3×C4—66no—
311C2×C3×C4—83no—
312C2×C3×C4—83no—
313C2×C3×C4—126no—
314C2×C3×C4—123no—
315C2×C3×C4C23×D316yes—
316C2×C3×C4C23×D316yes—
317C2×C3×C4C24×C316yes—
318C2×C3×C4C24×C316yes—
319#67—224no—
320C2×Dic3—26no—
321C2×Dic3—26no—
322C2×Dic3—43no—
323C2×Dic3C24×C313yes—
324C2×Dic3C24×C316yes—
325C2×Dic3C24×C311yescharacteristic
326D12—46no—
327D12—46no—
328C4×D3—212no—
329C4×D3—46no—
330C2×C32—26no—
331C2×C32C22×#4111yescharacteristic
332C2×C32C2×H(5,𝔽2)11yescharacteristic
333C2×C32#58511yescharacteristic
334C3×D3—46no—
335C24—33no—
336C24C22×C3×D311yescharacteristic
337#41—124no—
338C2×Q8C22×C3×D312yes—
339C2×Q8C22×C3×D312yes—
340C2×D4—312no—
341C2×D4—36no—
342C2×D4—36no—
343C2×D4—612no—
344C22×C4—33no—
345C22×C4—33no—
346C22×C4—33no—
347C22×C4—36no—
348C22×C4—36no—
349C22×C4—126no—
350C22×C4—122no—
351C22×C4C22×C3×D313yes—
352C22×C4C22×C3×D313yes—
353C4⋊C4—312no—
354C4⋊C4—36no—
355C4⋊C4—36no—
356C4⋊C4C22×C3×D316yes—
357C4⋊C4C22×C3×D316yes—
358C22⋊C4—312no—
359C22⋊C4—312no—
360C22⋊C4—312no—
361C22⋊C4C22×C3×D3112yes—
362C42—36no—
363C42—36no—
364C22×C3—23no—
365C22×C3—23no—
366C22×C3—23no—
367C22×C3—23no—
368C22×C3—26no—
369C22×C3—21no—
370C22×C3—43no—
371C22×C3—43no—
372C22×C3—44no—
373C22×C3—44no—
374C22×C3—66no—
375C22×C3—66no—
376C22×C3—66no—
377C22×C3—84no—
378C22×C3—1212no—
379C22×C3C2×#23916yes—
380C22×C3C2×C3×#4116yes—
381C22×C3C24×D311yescharacteristic
382C22×C3C25×C311yescharacteristic
383C2×D3—26no—
384C2×D3—26no—
385C2×D3—26no—
386C2×D3—412no—
387C3×C4—26no—
388C3×C4—26no—
389C3×C4—26no—
390C3×C4—26no—
391C3×C4—32no—
392C3×C4—46no—
393C3×C4—46no—
394C3×C4—66no—
395C3×C4—63no—
396Dic3—26no—
397Dic3—23no—
398Dic3C3×H(5,𝔽2)12yes—
399C32#345911yescharacteristic, complemented, Sylow
400C23—36no—
401C23—41no—
402C23—41no—
403C23—66no—
404C23—123no—
405C23—123no—
406C23C3×#23914yescomplemented
407C23C23×C3×D313yes—
408Q8—42no—
409Q8—42no—
410D4—624no—
411D4—126no—
412D4—126no—
413C2×C4—31no—
414C2×C4—36no—
415C2×C4—33no—
416C2×C4—43no—
417C2×C4—43no—
418C2×C4—612no—
419C2×C4—66no—
420C2×C4—66no—
421C2×C4—126no—
422C2×C4—123no—
423C2×C4C23×C3×D316yes—
424C2×C4C23×C3×D316yes—
425C2×C3—26no—
426C2×C3—26no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—46no—
431C2×C3—126no—
432C2×C3C22×#23911yescharacteristic
433C2×C3D3×H(5,𝔽2)11yescharacteristic
434C2×C3#596611yescharacteristic
435C2×C3C22×C3×#4111yescharacteristic
436C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
437C2×C3C3×#58511yescharacteristic
438D3—46no—
439C22—23no—
440C22—23no—
441C22—44no—
442C22—66no—
443C22—66no—
444C22—66no—
445C22—1212no—
446C22C2×C3×#23916yes—
447C22C24×C3×D311yescharacteristic
448C4—26no—
449C4—26no—
450C4—32no—
451C4—66no—
452C4—63no—
453C3—21no—
454C3#8657211yescharacteristic, direct product factor
455C3C3×#345911yescharacteristic, complemented
456C2—26no—
457C2—126no—
458C2C22×C3×#23911yescharacteristic
459C2C3×D3×H(5,𝔽2)11yescharacteristic
460C2C3×#596611yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#543C2
socleC22×C32C25
hypercenterC3×#543D3
hypocenterC3C3×#3459
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#543

Complements

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

NH ≅ G/Ntype
1. C32×#543C2semidirect
2. C3×#5645C2semidirect
3. C3×#5644C2semidirect
4. C3×#5627C2semidirect
5. C3×#5624C2semidirect
6. C3×#5493C2semidirect
7. C2×C3×#67⋊C4C2semidirect
9. #86572C3direct (N × H)
12. C2×C3×#239C22semidirect
13. C22×C3×C4×D3C22semidirect
14. C32×C22⋊Q8C22semidirect
15. C32×C22⋊Q8C22semidirect
17. C3×#947C22semidirect
18. C3×#946C22semidirect
19. C3×Q8×Dic3C22semidirect
22. C2×C3×(C2×D3)⋊C4C22semidirect
23. C2×C3×Dic3⋊C4C22semidirect
24. C2×C3×C4×Dic3C22semidirect
25. C3×#898C22semidirect
26. C3×#895C22semidirect
27. C3×#894C22semidirect
28. C3×#893C22semidirect
29. C3×#892C22semidirect
30. C3×#890C22semidirect
31. C3×#886C22semidirect
32. C3×#882C22semidirect
33. C3×#67⋊C4C22semidirect
34. C3×#67⋊C4C22semidirect
36. C3×#543D3semidirect
37. C3×#543C2×C3semidirect
44. #5645C2×C3semidirect
45. #5644C2×C3semidirect
46. #5627C2×C3semidirect
47. #5624C2×C3semidirect
48. #5493C2×C3semidirect
49. C2×#67⋊C4C2×C3semidirect
86. C2×#239C22×C3semidirect
87. C22×C4×D3C22×C3semidirect
96. C3×C22⋊Q8C2×D3semidirect
97. C3×C22⋊Q8C2×D3semidirect
98. C3×C22⋊Q8C22×C3semidirect
99. C3×C22⋊Q8C22×C3semidirect
115. #947C22×C3semidirect
116. #946C22×C3semidirect
117. Q8×Dic3C22×C3semidirect
120. C2×(C2×D3)⋊C4C22×C3semidirect
121. C2×Dic3⋊C4C22×C3semidirect
122. C2×C4×Dic3C22×C3semidirect
123. #898C22×C3semidirect
124. #895C22×C3semidirect
125. #894C22×C3semidirect
126. #893C22×C3semidirect
127. #892C22×C3semidirect
128. #890C22×C3semidirect
129. #886C22×C3semidirect
130. #882C22×C3semidirect
131. #67⋊C4C22×C3semidirect
132. #67⋊C4C22×C3semidirect
135. C23×C32#41semidirect
158. #543C3×D3semidirect
255. C22⋊Q8C2×C3×D3semidirect
256. C22⋊Q8C2×C3×D3semidirect
282. C23×C3#239semidirect
283. C23×C3C3×#41semidirect
399. C32#3459semidirect
406. C23C3×#239semidirect
454. C3#86572direct (N × H)
455. C3C3×#3459semidirect

Maximal subgroups

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

subgroupindex
C32×#5432
C3×#56452
C3×#56442
C3×#56272
C3×#56242
C3×#54932
C2×C3×#67⋊C42
C3×#34593
#865723