Finite Group Databasebeta

← C3×#86573 #11904962

Subgroups — C3×#86573 #11904962

Order 1152.

2225 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#543C211yesmaximal, characteristic, complemented
2C3×#5645C214yesmaximal, complemented
3C3×#5642C218yesmaximal, complemented
4C3×#5637C218yesmaximal, complemented
5C3×#5624C214yesmaximal, complemented
6C3×#5493C211yesmaximal, characteristic
7C2×C3×#886C212yesmaximal, complemented
8C3×#5486C212yesmaximal, complemented
9C2×C3×#877C211yesmaximal, characteristic, complemented
10C3×#3491—31nomaximal
11#86573C311yesmaximal, characteristic, direct product factor
12C22×C3×#67C2212yes—
13C23×C3×Dic3C2211yescharacteristic
14C32×C22⋊Q8C2214yescomplemented
15C32×C22⋊Q8C2218yescomplemented
16C2×C32×C22⋊C4C2212yes—
17C2×C32×C22⋊C4C2211yescharacteristic
18C2×C3×#217C2211yescharacteristic
19C3×#947C2214yescomplemented
20C3×#946C2218yescomplemented
21C3×Q8×Dic3C2214yescomplemented
22C3×#930C2214yes—
23C3×C4×#67C2218yes—
24C2×C3×(C2×D3)⋊C4C2214yescomplemented
25C2×C3×C4⋊Dic3C2214yescomplemented
26C2×C3×Dic3⋊C4C2212yescomplemented
27C2×C3×C4×Dic3C2212yescomplemented
28C3×#898C2218yescomplemented
29C3×#898C2218yescomplemented
30C3×#894C2218yescomplemented
31C3×#893C2214yescomplemented
32C3×#892C2218yescomplemented
33C3×#890C2214yescomplemented
34C3×#890C2218yescomplemented
35C3×#886C2218yes—
36C3×#886C2214yes—
37C3×#886C2214yescomplemented
38C3×#886C2214yescomplemented
39C3×#882C2214yes—
40C3×#67⋊C4C2218yescomplemented
41C3×D3×C22⋊C4C2214yescomplemented
42C3×#877C2218yescomplemented
43C3×#877C2214yescomplemented
44C3×#552—38no—
45C3×#551—38no—
46C3×#543—21no—
47C3×#543D311yescharacteristic, complemented
48C3×#543C2×C311yescharacteristic, complemented
49C3×#539—34no—
50C3×#534—31no—
51C3×#529—34no—
52C2×C3×#123—32no—
53C3×#518—32no—
54C2×C3×#117—31no—
55#5645C2×C314yescomplemented
56#5642C2×C318yescomplemented
57#5637C2×C318yescomplemented
58#5624C2×C314yescomplemented
59#5493C2×C311yescharacteristic
60C2×#886C2×C312yescomplemented
61#5486C2×C312yescomplemented
62C2×#877C2×C311yescharacteristic, complemented
63C24×C32C2311yescharacteristic
64C23×C3×D3C2312yes—
65C2×C32×Q8C2314yes—
66C22×C32×C4C2312yes—
67C22×C32×C4C2314yes—
68C2×C3×#67—28no—
69C2×C3×#67C2314yes—
70C2×C3×#67C2314yes—
71C22×C3×Dic3—24no—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2314yes—
74C22×C3×Dic3C2311yescharacteristic
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2312yes—
77C2×C3×D12C2314yes—
78C2×C3×C4×D3C2318yes—
79C32×C4⋊C4C2318yes—
80C32×C4⋊C4C2314yes—
81C32×C22⋊C4C2314yes—
82C32×C22⋊C4C2314yes—
83C32×C22⋊C4C2314yes—
84C3×#217C2314yes—
85C3×(C2×D3)⋊C4C2318yes—
86C3×(C2×D3)⋊C4C2318yes—
87C3×(C2×D3)⋊C4C2318yes—
88C3×(C2×D3)⋊C4C2318yes—
89C3×C4⋊Dic3C2314yes—
90C3×C4⋊Dic3C2318yes—
91C3×C4⋊Dic3C2318yes—
92C3×Dic3⋊C4C2318yes—
93C3×Dic3⋊C4C2318yes—
94C3×C4×Dic3C2314yes—
95C3×C4×Dic3C2314yes—
96C3×C4×Dic3C2318yes—
97#3491—31noSylow
98C22×C3×D4—32no—
99C23×C3×C4—31no—
100C22×#67C22×C312yes—
101C23×Dic3C22×C311yescharacteristic
102C3×#126—38no—
103C3×#126—38no—
104C3×#125—34no—
105C3×#125—38no—
106C3×#124—34no—
107C3×#123—38no—
108C3×#123—38no—
109C3×#123—34no—
110C3×#123—34no—
111C3×#123—34no—
112C3×C22⋊Q8—24no—
113C3×C22⋊Q8—28no—
114C3×C22⋊Q8—38no—
115C3×C22⋊Q8C2×D314yescomplemented
116C3×C22⋊Q8C2×D318yescomplemented
117C3×C22⋊Q8C22×C318yescomplemented
118C3×C22⋊Q8C22×C314yescomplemented
119C3×#121—34no—
120C3×#120—34no—
121C3×C4×Q8—34no—
122C3×C4×D4—38no—
123C3×C4×D4—34no—
124C3×C4×D4—38no—
125C3×#117—38no—
126C3×#117—38no—
127C3×#117—34no—
128C2×C3×C4⋊C4—32no—
129C2×C3×C4⋊C4—34no—
130C2×C3×C22⋊C4—21no—
131C2×C3×C22⋊C4—22no—
132C2×C3×C22⋊C4—31no—
133C2×C3×C22⋊C4—34no—
134C2×C3×C22⋊C4—34no—
135C2×C3×C22⋊C4C2×D311yescharacteristic
136C2×C3×C22⋊C4C2×D312yes—
137C2×C3×C22⋊C4C22×C311yescharacteristic
138C2×C3×C22⋊C4C22×C312yes—
139C2×C3×C42—32no—
140C2×#217C22×C311yescharacteristic
141#947C22×C314yescomplemented
142#946C22×C318yescomplemented
143Q8×Dic3C22×C314yescomplemented
144#930C22×C314yes—
145C4×#67C22×C318yes—
146C2×(C2×D3)⋊C4C22×C314yescomplemented
147C2×C4⋊Dic3C22×C314yescomplemented
148C2×Dic3⋊C4C22×C312yescomplemented
149C2×C4×Dic3C22×C312yescomplemented
150#898C22×C318yescomplemented
151#898C22×C318yescomplemented
152#894C22×C318yescomplemented
153#893C22×C314yescomplemented
154#892C22×C318yescomplemented
155#890C22×C318yescomplemented
156#890C22×C314yescomplemented
157#886C22×C318yes—
158#886C22×C314yes—
159#886C22×C314yescomplemented
160#886C22×C314yescomplemented
161#882C22×C314yes—
162#67⋊C4C22×C318yescomplemented
163D3×C22⋊C4C22×C314yescomplemented
164#877C22×C318yescomplemented
165#877C22×C314yescomplemented
166C23×C32—42no—
167C23×C32#4114yes—
168C23×C32C2412yes—
169C23×C32C2411yescharacteristic
170C22×C3×D3—24no—
171C22×C3×D3—44no—
172C22×C3×D3C2414yes—
173C32×Q8—44no—
174C2×C32×C4—44no—
175C2×C32×C4—42no—
176C2×C32×C4C2414yes—
177C2×C32×C4C2418yes—
178C3×#67—28no—
179C3×#67—28no—
180C2×C3×Dic3—28no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—42no—
185C2×C3×Dic3C2414yes—
186C2×C3×Dic3C2412yes—
187C2×C3×Dic3C2414yes—
188C2×C3×Dic3C2412yes—
189C3×D12—44no—
190C3×C4×D3—48no—
191#552—38no—
192#551—38no—
193#543C3×D311yescharacteristic, complemented
194#539—34no—
195#534—31no—
196#529—34no—
197C2×#123—32no—
198#518—32no—
199C2×#117—31no—
200C24×C3—21no—
201C24×C3—32no—
202C24×C3C22×D311yescharacteristic
203C24×C3C23×C311yescharacteristic
204C23×D3C23×C312yes—
205C2×C3×Q8—24no—
206C2×C3×Q8C22×D314yes—
207C2×C3×Q8C23×C314yes—
208C2×C3×D4—34no—
209C2×C3×D4—34no—
210C2×C3×D4—34no—
211C2×C3×D4—68no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—31no—
215C22×C3×C4—34no—
216C22×C3×C4—32no—
217C22×C3×C4—31no—
218C22×C3×C4—32no—
219C22×C3×C4—38no—
220C22×C3×C4—64no—
221C22×C3×C4C22×D312yes—
222C22×C3×C4C22×D314yes—
223C22×C3×C4C23×C312yes—
224C22×C3×C4C23×C314yes—
225C2×#67—28no—
226C2×#67C23×C314yes—
227C2×#67C23×C314yes—
228C22×Dic3—24no—
229C22×Dic3C23×C312yes—
230C22×Dic3C23×C314yes—
231C22×Dic3C23×C311yescharacteristic
232C22×Dic3C23×C311yescharacteristic
233C22×Dic3C23×C312yes—
234C2×D12C23×C314yes—
235C2×C4×D3C23×C318yes—
236C3×C4⋊C4—28no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—38no—
239C3×C4⋊C4—38no—
240C3×C4⋊C4—38no—
241C3×C4⋊C4—38no—
242C3×C4⋊C4—34no—
243C3×C4⋊C4C22×D318yes—
244C3×C4⋊C4C22×D314yes—
245C3×C4⋊C4C23×C318yes—
246C3×C4⋊C4C23×C314yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—34no—
251C3×C22⋊C4—38no—
252C3×C22⋊C4—38no—
253C3×C22⋊C4—38no—
254C3×C22⋊C4—38no—
255C3×C22⋊C4C22×D314yes—
256C3×C22⋊C4C22×D314yes—
257C3×C22⋊C4C22×D314yes—
258C3×C22⋊C4C23×C314yes—
259C3×C22⋊C4C23×C314yes—
260C3×C22⋊C4C23×C314yes—
261C3×C42—34no—
262C3×C42—34no—
263C3×C42—38no—
264#217C23×C314yes—
265(C2×D3)⋊C4C23×C318yes—
266(C2×D3)⋊C4C23×C318yes—
267(C2×D3)⋊C4C23×C318yes—
268(C2×D3)⋊C4C23×C318yes—
269C4⋊Dic3C23×C314yes—
270C4⋊Dic3C23×C318yes—
271C4⋊Dic3C23×C318yes—
272Dic3⋊C4C23×C318yes—
273Dic3⋊C4C23×C318yes—
274C4×Dic3C23×C314yes—
275C4×Dic3C23×C314yes—
276C4×Dic3C23×C318yes—
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—44no—
280C22×C32C2×#4114yes—
281C22×C32C2×#4112yes—
282C22×C32C2511yescharacteristic
283C2×C3×D3—24no—
284C2×C3×D3—28no—
285C2×C3×D3—84no—
286C32×C4—28no—
287C32×C4—24no—
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—24no—
291C3×Dic3—24no—
292C22×D4—32no—
293C23×C4—31no—
294#126—38no—
295#126—38no—
296#125—34no—
297#125—38no—
298#124—34no—
299#123—38no—
300#123—34no—
301#123—34no—
302#123—38no—
303#123—34no—
304C22⋊Q8—38no—
305C22⋊Q8C2×C3×D314yescomplemented
306C22⋊Q8C2×C3×D318yescomplemented
307#121—34no—
308#120—34no—
309C4×Q8—34no—
310C4×D4—38no—
311C4×D4—34no—
312C4×D4—38no—
313#117—38no—
314#117—38no—
315#117—34no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—34no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4C2×C3×D311yescharacteristic
322C2×C22⋊C4C2×C3×D312yes—
323C2×C42—32no—
324C23×C3—22no—
325C23×C3—21no—
326C23×C3—24no—
327C23×C3—34no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—64no—
331C23×C3—82no—
332C23×C3—124no—
333C23×C3#23914yes—
334C23×C3C3×#4114yes—
335C23×C3C23×D312yes—
336C23×C3C23×D311yescharacteristic
337C23×C3C24×C312yes—
338C23×C3C24×C311yescharacteristic
339C22×D3—24no—
340C22×D3—44no—
341C22×D3C24×C314yes—
342C3×Q8—44no—
343C3×Q8—44no—
344C3×Q8—84no—
345C3×D4—68no—
346C3×D4—68no—
347C3×D4—124no—
348C2×C3×C4—24no—
349C2×C3×C4—28no—
350C2×C3×C4—32no—
351C2×C3×C4—34no—
352C2×C3×C4—34no—
353C2×C3×C4—32no—
354C2×C3×C4—44no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—42no—
358C2×C3×C4—68no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—64no—
362C2×C3×C4—84no—
363C2×C3×C4—82no—
364C2×C3×C4—128no—
365C2×C3×C4—122no—
366C2×C3×C4C23×D314yes—
367C2×C3×C4C23×D318yes—
368C2×C3×C4C24×C314yes—
369C2×C3×C4C24×C318yes—
370#67—28no—
371#67—28no—
372C2×Dic3—28no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—42no—
377C2×Dic3C24×C314yes—
378C2×Dic3C24×C312yes—
379C2×Dic3C24×C314yes—
380C2×Dic3C24×C312yes—
381D12—44no—
382C4×D3—48no—
383C2×C32—24no—
384C2×C32—22no—
385C2×C32C22×#4111yescharacteristic
386C2×C32#58512yes—
387C3×D3—44no—
388C24—32no—
389C24C22×C3×D311yescharacteristic
390C2×Q8C22×C3×D314yes—
391C2×D4—34no—
392C2×D4—34no—
393C2×D4—34no—
394C2×D4—68no—
395C22×C4—31no—
396C22×C4—34no—
397C22×C4—32no—
398C22×C4—31no—
399C22×C4—32no—
400C22×C4—38no—
401C22×C4—64no—
402C22×C4C22×C3×D312yes—
403C22×C4C22×C3×D314yes—
404C4⋊C4—38no—
405C4⋊C4—38no—
406C4⋊C4—38no—
407C4⋊C4—38no—
408C4⋊C4—34no—
409C4⋊C4C22×C3×D318yes—
410C4⋊C4C22×C3×D314yes—
411C22⋊C4—34no—
412C22⋊C4—38no—
413C22⋊C4—38no—
414C22⋊C4—38no—
415C22⋊C4—38no—
416C22⋊C4C22×C3×D314yes—
417C22⋊C4C22×C3×D314yes—
418C22⋊C4C22×C3×D314yes—
419C42—34no—
420C42—34no—
421C42—38no—
422C22×C3—22no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—42no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—64no—
434C22×C3—68no—
435C22×C3—84no—
436C22×C3—244no—
437C22×C3C2×#23914yes—
438C22×C3C2×#23912yes—
439C22×C3C2×C3×#4114yes—
440C22×C3C2×C3×#4112yes—
441C22×C3C24×D311yescharacteristic
442C22×C3C25×C311yescharacteristic
443C2×D3—24no—
444C2×D3—28no—
445C2×D3—84no—
446C3×C4—28no—
447C3×C4—24no—
448C3×C4—28no—
449C3×C4—24no—
450C3×C4—48no—
451C3×C4—44no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—64no—
455C3×C4—64no—
456Dic3—22no—
457Dic3—22no—
458Dic3—24no—
459Dic3—24no—
460C32#349111yescharacteristic, complemented, Sylow
461C23—34no—
462C23—42no—
463C23—64no—
464C23—124no—
465C23C3×#23914yes—
466C23C23×C3×D312yes—
467C23C23×C3×D311yescharacteristic
468Q8—44no—
469D4—68no—
470D4—68no—
471D4—124no—
472C2×C4—32no—
473C2×C4—34no—
474C2×C4—34no—
475C2×C4—32no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—68no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—128no—
483C2×C4—122no—
484C2×C4C23×C3×D314yes—
485C2×C4C23×C3×D318yes—
486C2×C3—24no—
487C2×C3—22no—
488C2×C3—24no—
489C2×C3—22no—
490C2×C3—21no—
491C2×C3—22no—
492C2×C3—44no—
493C2×C3—42no—
494C2×C3—124no—
495C2×C3C22×#23911yescharacteristic
496C2×C3#596612yes—
497C2×C3C22×C3×#4111yescharacteristic
498C2×C3C3×#58512yes—
499D3—44no—
500C22—22no—
501C22—24no—
502C22—44no—
503C22—64no—
504C22—68no—
505C22—244no—
506C22C2×C3×#23914yes—
507C22C2×C3×#23912yes—
508C22C24×C3×D311yescharacteristic
509C4—28no—
510C4—24no—
511C4—62no—
512C4—62no—
513C4—64no—
514C4—64no—
515C3—21no—
516C3#8657311yescharacteristic, direct product factor
517C3C3×#349111yescharacteristic, complemented
518C2—24no—
519C2—22no—
520C2—124no—
521C2C22×C3×#23911yescharacteristic
522C2C3×#596612yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#543C2
socleC22×C32C25
prime radicalC3×#5493C2
hypercenterC3×#543D3
hypocenterC3C3×#3491
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×#5642C2semidirect
4. C3×#5637C2semidirect
5. C3×#5624C2semidirect
7. C2×C3×#886C2semidirect
8. C3×#5486C2semidirect
9. C2×C3×#877C2semidirect
11. #86573C3direct (N × H)
14. C32×C22⋊Q8C22semidirect
15. C32×C22⋊Q8C22semidirect
19. C3×#947C22semidirect
20. C3×#946C22semidirect
21. C3×Q8×Dic3C22semidirect
24. C2×C3×(C2×D3)⋊C4C22semidirect
25. C2×C3×C4⋊Dic3C22semidirect
26. C2×C3×Dic3⋊C4C22semidirect
27. C2×C3×C4×Dic3C22semidirect
28. C3×#898C22semidirect
29. C3×#898C22semidirect
30. C3×#894C22semidirect
31. C3×#893C22semidirect
32. C3×#892C22semidirect
33. C3×#890C22semidirect
34. C3×#890C22semidirect
37. C3×#886C22semidirect
38. C3×#886C22semidirect
40. C3×#67⋊C4C22semidirect
41. C3×D3×C22⋊C4C22semidirect
42. C3×#877C22semidirect
43. C3×#877C22semidirect
47. C3×#543D3semidirect
48. C3×#543C2×C3semidirect
55. #5645C2×C3semidirect
56. #5642C2×C3semidirect
57. #5637C2×C3semidirect
58. #5624C2×C3semidirect
60. C2×#886C2×C3semidirect
61. #5486C2×C3semidirect
62. C2×#877C2×C3semidirect
115. C3×C22⋊Q8C2×D3semidirect
116. C3×C22⋊Q8C2×D3semidirect
117. C3×C22⋊Q8C22×C3semidirect
118. C3×C22⋊Q8C22×C3semidirect
141. #947C22×C3semidirect
142. #946C22×C3semidirect
143. Q8×Dic3C22×C3semidirect
146. C2×(C2×D3)⋊C4C22×C3semidirect
147. C2×C4⋊Dic3C22×C3semidirect
148. C2×Dic3⋊C4C22×C3semidirect
149. C2×C4×Dic3C22×C3semidirect
150. #898C22×C3semidirect
151. #898C22×C3semidirect
152. #894C22×C3semidirect
153. #893C22×C3semidirect
154. #892C22×C3semidirect
155. #890C22×C3semidirect
156. #890C22×C3semidirect
159. #886C22×C3semidirect
160. #886C22×C3semidirect
162. #67⋊C4C22×C3semidirect
163. D3×C22⋊C4C22×C3semidirect
164. #877C22×C3semidirect
165. #877C22×C3semidirect
193. #543C3×D3semidirect
305. C22⋊Q8C2×C3×D3semidirect
306. C22⋊Q8C2×C3×D3semidirect
460. C32#3491semidirect
516. C3#86573direct (N × H)
517. C3C3×#3491semidirect

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×#56422
C3×#56372
C3×#56242
C3×#54932
C2×C3×#8862
C3×#54862
C2×C3×#8772
C3×#34913
#865733