Finite Group Databasebeta

← C2×C3×#63567 #13158135

Subgroups — C2×C3×#63567 #13158135

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×#63567C214yesmaximal, direct product factor
5C2×C3×#816—31nomaximal
6C2×C3×#810—31nomaximal
7C2×#63567C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C3×#634—21no—
11C2×C3×#634—22no—
12C2×C3×#634C414yescomplemented
13C2×C3×#634C2212yes—
14C22×C32×Dic3—21no—
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2212yes—
17C2×C32×C3⋊C8—22no—
18C2×C32×C3⋊C8C2212yes—
19C22×#634C2×C311yescharacteristic
20C22×C3×Dic6—31no—
21C22×C3×Dic6—31no—
22C22×C3×Dic6—61no—
23C22×C32×C8—31no—
24C2×C32×C4⋊C4—31no—
25C2×C3×C4⋊Dic3—21no—
26C2×C3×C4⋊Dic3D311yescharacteristic
27C2×C3×C4⋊Dic3C2×C311yescharacteristic
28C22×C3×C3⋊C8—21no—
29C22×C3×C3⋊C8D311yescharacteristic
30C22×C3×C3⋊C8C2×C311yescharacteristic
31C3×#816—34no—
32C3×#810—34no—
33#63567C2×C314yesdirect product factor
34C23×C33D411yescharacteristic
35C2×C33×C4C2×C412yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D412yes—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3—22no—
41C3×#634—24no—
42C3×#634—22no—
43C3×#634C2×C414yescomplemented
44C2×C32×Dic3—22no—
45C2×C32×Dic3—42no—
46C32×C3⋊C8—22no—
47C2×C3×#103—91no—
48C2×#816—31no—
49C2×#810—31no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—41no—
54C22×C32×C4—61no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3—21no—
59C2×#634—21no—
60C2×#634—22no—
61C2×#634C3×C414yescomplemented
62C2×#634C22×C312yes—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3—21no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3—61no—
68C22×C3×Dic3—61no—
69C2×C3×Dic6—34no—
70C2×C3×Dic6—34no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—64no—
74C2×C3×Dic6—62no—
75C2×C3×Dic6—62no—
76C2×C3×Dic6—64no—
77C2×C3×Dic6—61no—
78C2×C3×Dic6—62no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—62no—
81C2×C32×C8—32no—
82C2×C32×C8—62no—
83C32×C4⋊C4—32no—
84C32×C4⋊C4—61no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2×D312yes—
90C3×C4⋊Dic3C22×C312yes—
91C2×C3×C3⋊C8—24no—
92C2×C3×C3⋊C8—22no—
93C2×C3×C3⋊C8—22no—
94C2×C3×C3⋊C8—22no—
95C2×C3×C3⋊C8C2×D312yes—
96C2×C3×C3⋊C8C22×C312yes—
97C22×C33C22⋊C412yes—
98C22×C33SD1612yes—
99C22×C33Q1612yes—
100C22×C33C2×D411yescharacteristic
101C33×C4C22⋊C412yes—
102C33×C4C22×C411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3×C3⋊Dic3—24no—
105C32×Dic3—42no—
106C22×C3×Q8—91no—
107C22×Dic6—31no—
108C22×Dic6—31no—
109C22×Dic6—61no—
110C22×C3×C8—31no—
111C22×C3×C8—31no—
112C22×C3×C8—61no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C4⋊C4—31no—
115C2×C3×C4⋊C4—61no—
116C2×C4⋊Dic3C3×D311yescharacteristic
117C22×C3⋊C8C3×D311yescharacteristic
118C3×#103—94no—
119#816—34no—
120#810—34no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C23×C32D1211yescharacteristic
126C23×C32#6711yescharacteristic
127C23×C32C3×D411yescharacteristic
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—42no—
141C2×C32×C4—41no—
142C2×C32×C4—42no—
143C2×C32×C4—41no—
144C2×C32×C4—62no—
145C2×C32×C4—122no—
146C2×C32×C4C4×D312yes—
147C2×C32×C4C4×D312yes—
148C2×C32×C4D1211yescharacteristic
149C2×C32×C4D1212yes—
150C2×C32×C4#6711yescharacteristic
151C2×C32×C4#6712yes—
152C2×C32×C4C2×C3×C412yes—
153C2×C32×C4C3×D411yescharacteristic
154C2×C32×C4C3×D412yes—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3—24no—
159C2×C3⋊Dic3—22no—
160#634—24no—
161#634—22no—
162#634C2×C3×C414yescomplemented
163C2×C3×Dic3—24no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—44no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—42no—
169C2×C3×Dic3—68no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C3×Dic6—34no—
176C3×Dic6—34no—
177C3×Dic6—64no—
178C3×Dic6—68no—
179C3×Dic6—64no—
180C3×Dic6—64no—
181C3×Dic6—64no—
182C3×Dic6—62no—
183C3×Dic6—62no—
184C32×C8—62no—
185C3×C3⋊C8—24no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—22no—
188C2×#103—91noSylow
189C2×C33#10314yescomplemented
190C2×C33C2×C22⋊C411yescharacteristic
191C2×C33C2×SD1611yescharacteristic
192C2×C33C2×Q1611yescharacteristic
193C2×C3×Q8—94no—
194C2×C3×Q8—92no—
195C2×C3×Q8—182no—
196C2×C3×Q8—181no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—41no—
201C22×C3×C4—62no—
202C22×C3×C4—61no—
203C22×C3×C4—61no—
204C22×C3×C4—181no—
205C22×C3×C4D3211yescharacteristic
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—62no—
210C22×Dic3—61no—
211C22×Dic3—61no—
212C2×Dic6—34no—
213C2×Dic6—34no—
214C2×Dic6—32no—
215C2×Dic6—32no—
216C2×Dic6—64no—
217C2×Dic6—62no—
218C2×Dic6—62no—
219C2×Dic6—64no—
220C2×Dic6—61no—
221C2×Dic6—62no—
222C2×Dic6—61no—
223C2×Dic6—62no—
224C2×C3×C8—32no—
225C2×C3×C8—32no—
226C2×C3×C8—62no—
227C2×C3×C8—64no—
228C2×C3×C8—62no—
229C2×C3×C8—62no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—62no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—61no—
236C4⋊Dic3—21no—
237C4⋊Dic3C2×C3×D312yes—
238C2×C3⋊C8—22no—
239C2×C3⋊C8C2×C3×D312yes—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32#20412yes—
257C22×C32Dic1212yes—
258C22×C32(C2×D3)⋊C412yes—
259C22×C32(C2×D3)⋊C412yes—
260C22×C32#21412yes—
261C22×C32#21612yes—
262C22×C32C3×C22⋊C412yes—
263C22×C32C3×SD1612yes—
264C22×C32C3×Q1612yes—
265C22×C32C2×D1211yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C22×C32C2×C3×D411yescharacteristic
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—42no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—122no—
281C32×C4(C2×D3)⋊C412yes—
282C32×C4(C2×D3)⋊C412yes—
283C32×C4C3×C22⋊C412yes—
284C32×C4C2×C4×D311yescharacteristic
285C32×C4C2×C4×D311yescharacteristic
286C32×C4C2×D1211yescharacteristic
287C32×C4C2×#6711yescharacteristic
288C32×C4C22×C3×C411yescharacteristic
289C32×C4C2×C3×D411yescharacteristic
290C3⋊Dic3—24no—
291C3×Dic3—44no—
292C3×Dic3—42no—
293C3×Dic3—42no—
294C3×Dic3—68no—
295C3×Dic3—64no—
296C3×Dic3—64no—
297C22×Q8—91no—
298C22×C8—31no—
299C2×C4⋊C4—31no—
300#103—94no—
301C33C2×#10311yescharacteristic, complemented, Sylow
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—41no—
306C23×C3#62611yescharacteristic
307C23×C3C3×D1211yescharacteristic
308C23×C3C3×#6711yescharacteristic
309C3×Q8—94no—
310C3×Q8—184no—
311C3×Q8—182no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—64no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—124no—
332C2×C3×C4—122no—
333C2×C3×C4—122no—
334C2×C3×C4—184no—
335C2×C3×C4—182no—
336C2×C3×C4#62412yes—
337C2×C3×C4#62611yescharacteristic
338C2×C3×C4#62612yes—
339C2×C3×C4C3×C4×D312yes—
340C2×C3×C4C3×C4×D312yes—
341C2×C3×C4C3×D1211yescharacteristic
342C2×C3×C4C3×D1212yes—
343C2×C3×C4C3×#6711yescharacteristic
344C2×C3×C4C3×#6712yes—
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C22×C3×D311yescharacteristic
347C2×C3×C4C22×C3×D311yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—42no—
350C2×Dic3—64no—
351C2×Dic3—68no—
352C2×Dic3—64no—
353C2×Dic3—64no—
354C2×Dic3—62no—
355C2×Dic3—62no—
356Dic6—34no—
357Dic6—34no—
358Dic6—64no—
359Dic6—64no—
360Dic6—68no—
361Dic6—64no—
362Dic6—64no—
363Dic6—62no—
364Dic6—62no—
365C3×C8—64no—
366C3×C8—62no—
367C3×C8—62no—
368C3⋊C8—22no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—24no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—24no—
379C2×C32—24no—
380C2×C32—21no—
381C2×C32—44no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32#81014yescomplemented
386C2×C32#81614yescomplemented
387C2×C32C3×#10314yescomplemented
388C2×C32C2×#20411yescharacteristic
389C2×C32C2×Dic1211yescharacteristic
390C2×C32C2×(C2×D3)⋊C411yescharacteristic
391C2×C32C2×(C2×D3)⋊C411yescharacteristic
392C2×C32C2×#21411yescharacteristic
393C2×C32C2×#21611yescharacteristic
394C2×C32C2×C3×C22⋊C411yescharacteristic
395C2×C32C2×C3×SD1611yescharacteristic
396C2×C32C2×C3×Q1611yescharacteristic
397C2×Q8—94no—
398C2×Q8—92no—
399C2×Q8—181no—
400C2×Q8—182no—
401C22×C4—61no—
402C22×C4—181no—
403C22×C4C3×D3211yescharacteristic
404C2×C8—32no—
405C2×C8—62no—
406C4⋊C4—32no—
407C4⋊C4—61no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3#371612yes—
425C22×C3#371912yes—
426C22×C3#372212yes—
427C22×C3C3×#20412yes—
428C22×C3C3×Dic1212yes—
429C22×C3C3×(C2×D3)⋊C412yes—
430C22×C3C3×(C2×D3)⋊C412yes—
431C22×C3C3×#21412yes—
432C22×C3C3×#21612yes—
433C22×C3C2×#62611yescharacteristic
434C22×C3C2×C3×D1211yescharacteristic
435C22×C3C2×C3×#6711yescharacteristic
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—42no—
448C3×C4—124no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—184no—
452C3×C4#372212yes—
453C3×C4C3×(C2×D3)⋊C412yes—
454C3×C4C3×(C2×D3)⋊C412yes—
455C3×C4C2×#62411yescharacteristic
456C3×C4C2×#62611yescharacteristic
457C3×C4C2×C3×C4×D311yescharacteristic
458C3×C4C2×C3×C4×D311yescharacteristic
459C3×C4C2×C3×D1211yescharacteristic
460C3×C4C2×C3×#6711yescharacteristic
461Dic3—42no—
462Dic3—68no—
463Dic3—64no—
464Dic3—64no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32C2×#81011yescharacteristic, complemented
470C32C2×#81611yescharacteristic, complemented
471C32C2×C3×#10311yescharacteristic, complemented
472C23C3×#62611yescharacteristic
473Q8—94no—
474Q8—182no—
475Q8—184no—
476C2×C4—62no—
477C2×C4—122no—
478C2×C4—182no—
479C2×C4—184no—
480C2×C4C3×#62412yes—
481C2×C4C3×#62612yes—
482C2×C4C3×#62611yescharacteristic
483C2×C4C2×C3×D3211yescharacteristic
484C8—62no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—24no—
495C2×C3—24no—
496C2×C3—24no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—44no—
501C2×C3#6356714yesdirect product factor
502C2×C3C3×#81014yescomplemented
503C2×C3C3×#81614yescomplemented
504C2×C3C2×#371611yescharacteristic
505C2×C3C2×#371911yescharacteristic
506C2×C3C2×#372211yescharacteristic
507C2×C3C2×C3×#20411yescharacteristic
508C2×C3C2×C3×Dic1211yescharacteristic
509C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
510C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
511C2×C3C2×C3×#21411yescharacteristic
512C2×C3C2×C3×#21611yescharacteristic
513C22C3×#371612yes—
514C22C3×#371912yes—
515C22C3×#372212yes—
516C22C2×C3×#62611yescharacteristic
517C4—122no—
518C4—184no—
519C4C3×#372212yes—
520C4C2×C3×#62411yescharacteristic
521C4C2×C3×#62611yescharacteristic
522C3—21no—
523C3—21no—
524C3—21no—
525C3—41no—
526C3C2×#6356711yescharacteristic, direct product factor
527C3C2×C3×#81011yescharacteristic, complemented
528C3C2×C3×#81611yescharacteristic, complemented
529C2C3×#6356714yesdirect product factor
530C2C2×C3×#371611yescharacteristic
531C2C2×C3×#371911yescharacteristic
532C2C2×C3×#372211yescharacteristic
subgroupG/H
centerC23×C3#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33D4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#103
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63567C2direct (N × H)
7. C2×#63567C3direct (N × H)
12. C2×C3×#634C4semidirect
33. #63567C2×C3direct (N × H)
43. C3×#634C2×C4semidirect
61. C2×#634C3×C4semidirect
162. #634C2×C3×C4semidirect
189. C2×C33#103semidirect
301. C33C2×#103semidirect
385. C2×C32#810semidirect
386. C2×C32#816semidirect
387. C2×C32C3×#103semidirect
469. C32C2×#810semidirect
470. C32C2×#816semidirect
471. C32C2×C3×#103semidirect
501. C2×C3#63567direct (N × H)
502. C2×C3C3×#810semidirect
503. C2×C3C3×#816semidirect
526. C3C2×#63567direct (N × H)
527. C3C2×C3×#810semidirect
528. C3C2×C3×#816semidirect
529. C2C3×#63567direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C32×C4⋊Dic32
C22×C32×C3⋊C82
C3×#635672
C2×C3×#8163
C2×C3×#8103
C2×#635673