Finite Group Databasebeta

← C2×C3×#63927 #13174803

Subgroups — C2×C3×#63927 #13174803

Order 1728.

1562 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×C32×#213C212yesmaximal, complemented
4C3×#63927C218yesmaximal, direct product factor
5C2×C3×#3716C212yesmaximal, complemented
6C2×C3×#3712C211yesmaximal, characteristic, complemented
7C2×C3×#911—32nomaximal
8C2×#63927C311yesmaximal, characteristic, direct product factor
9C22×C3×D32—21no—
10C2×C33×D4C2211yescharacteristic, complemented
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830C2214yescomplemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic, complemented
17C2×C32×D12C2212yes—
18C2×C3×#625—21no—
19C3×#3802—22no—
20C3×#3802C2214yescomplemented
21C32×#213C2218yescomplemented
22C2×C32×C3⋊C8C2212yescomplemented
23C3×#3716C2218yescomplemented
24C3×#3712C2214yescomplemented
25C2×#3830C2×C311yescharacteristic, complemented
26C2×C3×#237—32no—
27C2×C3×#237—61no—
28C2×C3×D3×D4—32no—
29C2×#3802C2×C311yescharacteristic, complemented
30C2×C32×D8—32no—
31C2×C3×#214—32no—
32C2×C3×#213—22no—
33C2×C3×#213D312yescomplemented
34C2×C3×#213C2×C312yescomplemented
35C3×#911—316no—
36C2×C3×#204—32no—
37C2×C3×ΓL(1,𝔽25)—32no—
38#63927C2×C318yesdirect product factor
39C2×#3716C2×C312yescomplemented
40C2×#3712C2×C311yescharacteristic, complemented
41C23×C33—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—22no—
44C2×C3×D32—24no—
45C2×C3×D32—44no—
46C33×D4—21no—
47C33×D4C2312yescomplemented
48C2×C33×C4C2311yescharacteristic
49C3×#638—24no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—24no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3D412yes—
54C3×C4×C3⋊D3C2312yes—
55C3×#634—21no—
56C3×#634C2312yescomplemented
57C32×D12—22no—
58C32×D12C2314yes—
59C3×#625—24no—
60C32×C3⋊C8C2314yescomplemented
61C2×C3×AGL(1,ℤ/8)—91no—
62C2×#911—32no—
63C23×C3×D3—62no—
64C22×D32—21no—
65C2×C32×D4—21no—
66C2×C32×D4—22no—
67C2×C32×D4—32no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D312yescomplemented
70C2×C32×D4C22×C311yescharacteristic, complemented
71C2×#638—21no—
72C22×C3⋊Dic3—21no—
73#3830—22no—
74#3830C22×C314yescomplemented
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic, complemented
77C2×C3×#67—62no—
78C2×C3×#67—62no—
79C2×C3×#67—62no—
80C22×C3×Dic3—62no—
81C22×C3×Dic3—62no—
82C3×#237—38no—
83C3×#237—64no—
84C3×#237—64no—
85C3×#237—64no—
86C3×D3×D4—38no—
87C3×D3×D4—64no—
88C2×C3×D12—22no—
89C2×C3×D12C2×D312yes—
90C2×C3×D12C22×C312yes—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—61no—
93C2×C3×Dic6—32no—
94C2×C3×Dic6—61no—
95C2×#625—21no—
96#3802—22no—
97#3802C22×C314yescomplemented
98C32×D8—38no—
99C2×C32×C8—32no—
100C3×#214—38no—
101C3×#213—28no—
102C3×#213C2×D318yescomplemented
103C3×#213C22×C318yescomplemented
104C2×C3×C3⋊C8—22no—
105C2×C3×C3⋊C8C2×D312yescomplemented
106C2×C3×C3⋊C8C22×C312yescomplemented
107C3×#204—38no—
108C3×ΓL(1,𝔽25)—38no—
109#3716C22×C318yescomplemented
110#3712C22×C314yescomplemented
111C22×C33—22no—
112C22×C33—42no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3C2×D412yes—
116C2×C32×D3—24no—
117C2×C32×D3—44no—
118C3×D32—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3C2×D412yes—
123C2×C3×#41—91no—
124C22×C3×D4—91no—
125C2×#237—32no—
126C2×#237—61no—
127C2×D3×D4—32no—
128C3×AGL(1,ℤ/8)—98no—
129C2×C3×SD16—92no—
130C2×C3×D8—32no—
131C2×C3×D8—32no—
132C2×C3×D8—62no—
133C2×C3×M16—91no—
134C2×#214—32no—
135C2×#213C3×D312yescomplemented
136#911—316no—
137C2×#204—32no—
138C2×ΓL(1,𝔽25)—32no—
139C23×C32—21no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—42no—
143C23×C32—42no—
144C23×C32—62no—
145C22×C3⋊D3C3×D411yescharacteristic
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—32no—
149C22×C3×D3—42no—
150C22×C3×D3—61no—
151C22×C3×D3—68no—
152C22×C3×D3—128no—
153C2×D32—24no—
154C2×D32—44no—
155C32×D4—21no—
156C32×D4—22no—
157C32×D4—22no—
158C32×D4—22no—
159C32×D4—24no—
160C32×D4—34no—
161C32×D4—42no—
162C32×D4—42no—
163C32×D4—42no—
164C32×D4—62no—
165C32×D4C22×D314yescomplemented
166C32×D4C23×C312yescomplemented
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—41no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C23×C311yescharacteristic
172#638—24no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—24no—
175C2×C3⋊Dic3C3×D411yescharacteristic
176C4×C3⋊D3C3×D412yes—
177C4×C3⋊D3C23×C312yes—
178#634—21no—
179#634C23×C312yescomplemented
180C3×#67—68no—
181C3×#67—68no—
182C3×#67—68no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—68no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—68no—
188C2×C3×Dic3—62no—
189C3×D12—22no—
190C3×D12—22no—
191C3×D12—24no—
192C3×D12—42no—
193C3×D12C22×D314yes—
194C3×D12C23×C314yes—
195C3×C4×D3—34no—
196C3×C4×D3—34no—
197C3×C4×D3—62no—
198C3×C4×D3—62no—
199C3×Dic6—34no—
200C3×Dic6—62no—
201C3×Dic6—62no—
202C3×Dic6—62no—
203#625—24no—
204C32×C8—34no—
205C3×C3⋊C8—24no—
206C3×C3⋊C8C22×D314yescomplemented
207C3×C3⋊C8C23×C314yescomplemented
208C2×AGL(1,ℤ/8)—91noSylow
209C2×C33—42no—
210C2×C33AGL(1,ℤ/8)12yescomplemented
211C2×C33C22×D411yescharacteristic
212C3×C3⋊D3—22no—
213C32×D3—44no—
214C24×C3—181no—
215C23×D3—62no—
216C3×#41—94no—
217C3×#41—182no—
218C2×C3×Q8—91no—
219C2×C3×D4—21no—
220C2×C3×D4—22no—
221C2×C3×D4—32no—
222C2×C3×D4—32no—
223C2×C3×D4—41no—
224C2×C3×D4—62no—
225C2×C3×D4—94no—
226C2×C3×D4—182no—
227C2×C3×D4—181no—
228C2×C3×D4—181no—
229C2×C3×D4D3211yescharacteristic, complemented
230C2×C3×D4C2×C3×D312yescomplemented
231C22×C3×C4—91no—
232C22×C3×C4—181no—
233C2×#67—62no—
234C2×#67—62no—
235C2×#67—62no—
236C22×Dic3—62no—
237C22×Dic3—62no—
238#237—38no—
239#237—64no—
240#237—64no—
241#237—64no—
242D3×D4—38no—
243D3×D4—64no—
244C2×D12C2×C3×D312yes—
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×Dic6—32no—
248C2×Dic6—61no—
249C3×SD16—98no—
250C3×D8—38no—
251C3×D8—38no—
252C3×D8—68no—
253C3×M16—94no—
254C2×C3×C8—32no—
255C2×C3×C8—32no—
256C2×C3×C8—62no—
257#214—38no—
258#213C2×C3×D318yescomplemented
259C2×C3⋊C8C2×C3×D312yescomplemented
260#204—38no—
261ΓL(1,𝔽25)—38no—
262C22×C32—22no—
263C22×C32—24no—
264C22×C32—24no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—41no—
268C22×C32—44no—
269C22×C32—44no—
270C22×C32—42no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32—64no—
274C22×C32—84no—
275C22×C32—84no—
276C22×C32—124no—
277C22×C32D3×D412yes—
278C22×C32C2×C3×D411yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3C2×C3×D412yes—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—34no—
284C2×C3×D3—44no—
285C2×C3×D3—44no—
286C2×C3×D3—44no—
287C2×C3×D3—62no—
288C2×C3×D3—64no—
289C2×C3×D3—84no—
290C2×C3×D3—122no—
291C2×C3×D3—1216no—
292D32—48no—
293C32×C4—21no—
294C32×C4—22no—
295C32×C4—21no—
296C32×C4—22no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4D3×D412yes—
300C32×C4C2×C3×D411yescharacteristic
301C32×C4C23×D312yes—
302C32×C4C24×C311yescharacteristic
303C3⋊Dic3—22no—
304C3⋊Dic3C2×C3×D412yes—
305C3×Dic3—34no—
306C3×Dic3—62no—
307C3×Dic3—64no—
308C3×Dic3—64no—
309C2×#41—91no—
310C22×D4—91no—
311AGL(1,ℤ/8)—98no—
312C2×SD16—92no—
313C2×D8—32no—
314C2×M16—91no—
315C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—22no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—91no—
324C23×C3—122no—
325C23×C3—184no—
326C23×C3—364no—
327C22×D3—22no—
328C22×D3—32no—
329C22×D3—61no—
330C22×D3—68no—
331C22×D3—128no—
332C3×Q8—92no—
333C3×Q8—181no—
334C3×D4—22no—
335C3×D4—21no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—24no—
339C3×D4—34no—
340C3×D4—34no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—64no—
345C3×D4—62no—
346C3×D4—62no—
347C3×D4—122no—
348C3×D4—184no—
349C3×D4—184no—
350C3×D4C2×D3212yescomplemented
351C3×D4C22×C3×D314yescomplemented
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—184no—
359C2×C3×C4—181no—
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C22×C3×D312yes—
362#67—68no—
363#67—68no—
364#67—68no—
365C2×Dic3—32no—
366C2×Dic3—61no—
367C2×Dic3—68no—
368C2×Dic3—62no—
369C2×Dic3—68no—
370C2×Dic3—62no—
371D12—22no—
372D12C22×C3×D314yes—
373C4×D3—34no—
374C4×D3—34no—
375C4×D3—62no—
376C4×D3—62no—
377Dic6—34no—
378Dic6—62no—
379Dic6—62no—
380Dic6—62no—
381C3×C8—34no—
382C3×C8—34no—
383C3×C8—64no—
384C3⋊C8C22×C3×D314yescomplemented
385C2×C32—24no—
386C2×C32—21no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—42no—
390C2×C32—41no—
391C2×C32—42no—
392C2×C32—44no—
393C2×C32—44no—
394C2×C32—84no—
395C2×C32—84no—
396C2×C32—124no—
397C2×C32#91114yescomplemented
398C2×C32C3×AGL(1,ℤ/8)12yescomplemented
399C2×C32C2×D3×D412yes—
400C2×C32C22×C3×D411yescharacteristic
401C3⋊D3—22no—
402C3×D3—44no—
403C3×D3—44no—
404C3×D3—64no—
405C3×D3—84no—
406C3×D3—122no—
407C24—181no—
408#41—94no—
409#41—182no—
410C2×Q8—91no—
411C2×D4—32no—
412C2×D4—94no—
413C2×D4—182no—
414C2×D4—181no—
415C2×D4—181no—
416C2×D4C3×D3211yescharacteristic, complemented
417C22×C4—91no—
418C22×C4—181no—
419SD16—98no—
420D8—38no—
421M16—94no—
422C2×C8—32no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—24no—
426C22×C3—21no—
427C22×C3—22no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—64no—
435C22×C3—64no—
436C22×C3—84no—
437C22×C3—84no—
438C22×C3—92no—
439C22×C3—124no—
440C22×C3—124no—
441C22×C3—124no—
442C22×C3—182no—
443C22×C3—244no—
444C22×C3—368no—
445C22×C3#381111yescharacteristic
446C22×C3C3×D3×D412yes—
447C2×D3—24no—
448C2×D3—34no—
449C2×D3—44no—
450C2×D3—62no—
451C2×D3—64no—
452C2×D3—122no—
453C2×D3—1216no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—92no—
461C3×C4—182no—
462C3×C4#381111yescharacteristic
463C3×C4C3×D3×D412yes—
464C3×C4C22×D3211yescharacteristic
465C3×C4C23×C3×D312yes—
466Dic3—34no—
467Dic3—62no—
468Dic3—64no—
469Dic3—64no—
470C32—22no—
471C32—21no—
472C32—41no—
473C32C2×#91112yescomplemented
474C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
475C23—21no—
476C23—62no—
477C23—91no—
478C23—184no—
479C23—364no—
480Q8—92no—
481Q8—181no—
482D4—21no—
483D4—34no—
484D4—62no—
485D4—184no—
486D4—184no—
487D4C2×C3×D3212yescomplemented
488C2×C4—91no—
489C2×C4—92no—
490C2×C4—92no—
491C2×C4—184no—
492C2×C4—181no—
493C2×C4C2×C3×D3211yescharacteristic
494C8—34no—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—24no—
499C2×C3—44no—
500C2×C3—44no—
501C2×C3—42no—
502C2×C3—41no—
503C2×C3—42no—
504C2×C3—84no—
505C2×C3—84no—
506C2×C3—124no—
507C2×C3—124no—
508C2×C3—182no—
509C2×C3—244no—
510C2×C3#6392712yesdirect product factor
511C2×C3C3×#91114yescomplemented
512C2×C3C2×#381111yescharacteristic
513C2×C3C2×C3×D3×D412yes—
514D3—44no—
515D3—64no—
516D3—122no—
517C22—22no—
518C22—42no—
519C22—64no—
520C22—92no—
521C22—124no—
522C22—182no—
523C22—368no—
524C22C3×#381111yescharacteristic
525C4—92no—
526C4—182no—
527C4C3×#381111yescharacteristic
528C4C22×C3×D3211yescharacteristic
529C3—21no—
530C3—22no—
531C3—41no—
532C3C2×#6392711yescharacteristic, direct product factor
533C3C2×C3×#91112yescomplemented
534C2—42no—
535C2—124no—
536C2—182no—
537C2C3×#6392712yesdirect product factor
538C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C2×C3×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×C32×#213C2semidirect
4. C3×#63927C2direct (N × H)
5. C2×C3×#3716C2semidirect
6. C2×C3×#3712C2semidirect
8. C2×#63927C3direct (N × H)
10. C2×C33×D4C22semidirect
14. C3×#3830C22semidirect
16. C2×C3×#634C22semidirect
20. C3×#3802C22semidirect
21. C32×#213C22semidirect
22. C2×C32×C3⋊C8C22semidirect
23. C3×#3716C22semidirect
24. C3×#3712C22semidirect
25. C2×#3830C2×C3semidirect
29. C2×#3802C2×C3semidirect
33. C2×C3×#213D3semidirect
34. C2×C3×#213C2×C3semidirect
38. #63927C2×C3direct (N × H)
39. C2×#3716C2×C3semidirect
40. C2×#3712C2×C3semidirect
47. C33×D4C23semidirect
56. C3×#634C23semidirect
60. C32×C3⋊C8C23semidirect
69. C2×C32×D4C2×D3semidirect
70. C2×C32×D4C22×C3semidirect
74. #3830C22×C3semidirect
76. C2×#634C22×C3semidirect
97. #3802C22×C3semidirect
102. C3×#213C2×D3semidirect
103. C3×#213C22×C3semidirect
105. C2×C3×C3⋊C8C2×D3semidirect
106. C2×C3×C3⋊C8C22×C3semidirect
109. #3716C22×C3semidirect
110. #3712C22×C3semidirect
135. C2×#213C3×D3semidirect
165. C32×D4C22×D3semidirect
166. C32×D4C23×C3semidirect
179. #634C23×C3semidirect
206. C3×C3⋊C8C22×D3semidirect
207. C3×C3⋊C8C23×C3semidirect
210. C2×C33AGL(1,ℤ/8)semidirect
229. C2×C3×D4D32semidirect
230. C2×C3×D4C2×C3×D3semidirect
258. #213C2×C3×D3semidirect
259. C2×C3⋊C8C2×C3×D3semidirect
315. C33C2×AGL(1,ℤ/8)semidirect
350. C3×D4C2×D32semidirect
351. C3×D4C22×C3×D3semidirect
384. C3⋊C8C22×C3×D3semidirect
397. C2×C32#911semidirect
398. C2×C32C3×AGL(1,ℤ/8)semidirect
416. C2×D4C3×D32semidirect
473. C32C2×#911semidirect
474. C32C2×C3×AGL(1,ℤ/8)semidirect
487. D4C2×C3×D32semidirect
510. C2×C3#63927direct (N × H)
511. C2×C3C3×#911semidirect
532. C3C2×#63927direct (N × H)
533. C3C2×C3×#911semidirect
537. C2C3×#63927direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C3×#38022
C2×C32×#2132
C3×#639272
C2×C3×#37162
C2×C3×#37122
C2×C3×#9113
C2×#639273