Finite Group Databasebeta

← C2×C3×#63573 #13158153

Subgroups — C2×C3×#63573 #13158153

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×#63573C214yesmaximal, direct product factor
5C2×C3×#817—31nomaximal
6C2×C3×#808—31nomaximal
7C2×#63573C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C22×C32×Dic3—21no—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C32×C4⋊Dic3—21no—
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C414yescomplemented
16C2×C32×C3⋊C8C2212yes—
17C22×C32×C8—31no—
18C2×C32×C4⋊C4—31no—
19C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
20C2×C3×C4⋊Dic3—21no—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×C4⋊Dic3—31no—
23C2×C3×C4⋊Dic3—61no—
24C2×C3×C4⋊Dic3D311yescharacteristic
25C2×C3×C4⋊Dic3C2×C311yescharacteristic
26C22×C3×C3⋊C8—21no—
27C22×C3×C3⋊C8D311yescharacteristic
28C22×C3×C3⋊C8C2×C311yescharacteristic
29C3×#817—34no—
30C3×#808—34no—
31#63573C2×C314yesdirect product factor
32C23×C33D411yescharacteristic
33C2×C33×C4C2×C412yes—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4Q812yes—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—42no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—42no—
41C32×C3⋊C8C2×C414yescomplemented
42C2×C3×#106—91no—
43C2×#817—31no—
44C2×#808—31no—
45C22×C32×C4—21no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4—41no—
49C22×C32×C4—61no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C22×C3⋊Dic3—21no—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—21no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3—62no—
58C22×C3×Dic3—61no—
59C22×C3×Dic3—61no—
60C2×C32×C8—34no—
61C2×C32×C8—32no—
62C32×C4⋊C4—32no—
63C32×C4⋊C4—61no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)C22×C312yes—
66C3×C4⋊Dic3—22no—
67C3×C4⋊Dic3—22no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3—62no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—61no—
76C3×C4⋊Dic3C2×D312yes—
77C3×C4⋊Dic3C22×C312yes—
78C2×C3×C3⋊C8—24no—
79C2×C3×C3⋊C8—22no—
80C2×C3×C3⋊C8Dic314yescomplemented
81C2×C3×C3⋊C8C3×C414yescomplemented
82C2×C3×C3⋊C8C2×D312yes—
83C2×C3×C3⋊C8C22×C312yes—
84C22×C33C4⋊C412yes—
85C22×C33SD1612yes—
86C22×C33SD1612yes—
87C22×C33C2×D411yescharacteristic
88C33×C4C4⋊C412yes—
89C33×C4C22×C411yescharacteristic
90C33×C4C2×Q811yescharacteristic
91C3×C3⋊Dic3—42no—
92C32×Dic3—42no—
93C22×C3×C8—31no—
94C22×C3×C8—31no—
95C22×C3×C8—61no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊C4—61no—
99C2×C3×C4⋊C4—91no—
100C2×C4⋊Dic3—31no—
101C2×C4⋊Dic3—31no—
102C2×C4⋊Dic3—61no—
103C2×C4⋊Dic3C3×D311yescharacteristic
104C22×C3⋊C8C3×D311yescharacteristic
105C3×#106—94no—
106#817—34no—
107#808—34no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—41no—
112C23×C32D1211yescharacteristic
113C23×C32#6711yescharacteristic
114C23×C32C3×D411yescharacteristic
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—42no—
128C2×C32×C4—41no—
129C2×C32×C4—42no—
130C2×C32×C4—41no—
131C2×C32×C4—62no—
132C2×C32×C4—122no—
133C2×C32×C4Dic612yes—
134C2×C32×C4Dic612yes—
135C2×C32×C4C4×D312yes—
136C2×C32×C4D1211yescharacteristic
137C2×C32×C4C2×Dic312yes—
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4C2×C3×C412yes—
140C2×C32×C4C3×D411yescharacteristic
141C2×C32×C4C3×Q812yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—42no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—44no—
151C2×C3×Dic3—42no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—124no—
157C2×C3×Dic3—122no—
158C2×C3×Dic3—122no—
159C32×C8—34no—
160C3×C3⋊C8—24no—
161C3×C3⋊C8C2×Dic314yescomplemented
162C3×C3⋊C8C2×C3×C414yescomplemented
163C2×#106—91noSylow
164C2×C33#10614yescomplemented
165C2×C33C2×C4⋊C411yescharacteristic
166C2×C33C2×SD1611yescharacteristic
167C2×C33C2×SD1611yescharacteristic
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—41no—
172C22×C3×C4—62no—
173C22×C3×C4—61no—
174C22×C3×C4—61no—
175C22×C3×C4—181no—
176C22×C3×C4D3211yescharacteristic
177C22×C3×C4C2×C3×D311yescharacteristic
178C22×C3×C4C2×C3×D311yescharacteristic
179C22×Dic3—21no—
180C22×Dic3—62no—
181C22×Dic3—61no—
182C22×Dic3—61no—
183C2×C3×C8—34no—
184C2×C3×C8—34no—
185C2×C3×C8—32no—
186C2×C3×C8—32no—
187C2×C3×C8—64no—
188C2×C3×C8—62no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—62no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—92no—
196C3×C4⋊C4—181no—
197C4⋊Dic3—21no—
198C4⋊Dic3—32no—
199C4⋊Dic3—32no—
200C4⋊Dic3—62no—
201C4⋊Dic3—62no—
202C4⋊Dic3—61no—
203C4⋊Dic3—61no—
204C4⋊Dic3C2×C3×D312yes—
205C2×C3⋊C8C3×Dic314yescomplemented
206C2×C3⋊C8C2×C3×D312yes—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32#20412yes—
224C22×C32#20412yes—
225C22×C32Dic3⋊C412yes—
226C22×C32C4⋊Dic312yes—
227C22×C32#21412yes—
228C22×C32#21512yes—
229C22×C32C3×C4⋊C412yes—
230C22×C32C3×SD1612yes—
231C22×C32C3×SD1612yes—
232C22×C32C2×D1211yescharacteristic
233C22×C32C2×#6711yescharacteristic
234C22×C32C2×C3×D411yescharacteristic
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—42no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—122no—
248C32×C4Dic3⋊C412yes—
249C32×C4C4⋊Dic312yes—
250C32×C4C3×C4⋊C412yes—
251C32×C4C2×Dic611yescharacteristic
252C32×C4C2×Dic611yescharacteristic
253C32×C4C2×C4×D311yescharacteristic
254C32×C4C22×Dic311yescharacteristic
255C32×C4C22×C3×C411yescharacteristic
256C32×C4C2×C3×Q811yescharacteristic
257C3⋊Dic3—42no—
258C3×Dic3—44no—
259C3×Dic3—42no—
260C3×Dic3—42no—
261C3×Dic3—124no—
262C3×Dic3—122no—
263C3×Dic3—122no—
264C22×C8—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—91no—
267#106—94no—
268C33C2×#10611yescharacteristic, complemented, Sylow
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—41no—
273C23×C3#62611yescharacteristic
274C23×C3C3×D1211yescharacteristic
275C23×C3C3×#6711yescharacteristic
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—64no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—124no—
296C2×C3×C4—122no—
297C2×C3×C4—122no—
298C2×C3×C4—182no—
299C2×C3×C4—362no—
300C2×C3×C4D3×Dic312yes—
301C2×C3×C4#62611yescharacteristic
302C2×C3×C4#62712yes—
303C2×C3×C4C3×Dic612yes—
304C2×C3×C4C3×Dic612yes—
305C2×C3×C4C3×C4×D312yes—
306C2×C3×C4C3×D1211yescharacteristic
307C2×C3×C4C2×C3×Dic312yes—
308C2×C3×C4C3×#6711yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310C2×C3×C4C22×C3×D311yescharacteristic
311C2×C3×C4C22×C3×D311yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—42no—
314C2×Dic3—64no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—124no—
318C2×Dic3—122no—
319C2×Dic3—122no—
320C3×C8—34no—
321C3×C8—34no—
322C3×C8—64no—
323C3⋊C8C2×C3×Dic314yescomplemented
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—24no—
332C2×C32—21no—
333C2×C32—24no—
334C2×C32—24no—
335C2×C32—21no—
336C2×C32—44no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32#80814yescomplemented
341C2×C32#81714yescomplemented
342C2×C32C3×#10614yescomplemented
343C2×C32C2×#20411yescharacteristic
344C2×C32C2×#20411yescharacteristic
345C2×C32C2×Dic3⋊C411yescharacteristic
346C2×C32C2×C4⋊Dic311yescharacteristic
347C2×C32C2×#21411yescharacteristic
348C2×C32C2×#21511yescharacteristic
349C2×C32C2×C3×C4⋊C411yescharacteristic
350C2×C32C2×C3×SD1611yescharacteristic
351C2×C32C2×C3×SD1611yescharacteristic
352C22×C4—61no—
353C22×C4—181no—
354C22×C4C3×D3211yescharacteristic
355C2×C8—34no—
356C2×C8—32no—
357C4⋊C4—32no—
358C4⋊C4—61no—
359C4⋊C4—92no—
360C4⋊C4—181no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3#371612yes—
378C22×C3#371712yes—
379C22×C3Dic3⋊Dic312yes—
380C22×C3C3×#20412yes—
381C22×C3C3×#20412yes—
382C22×C3C3×Dic3⋊C412yes—
383C22×C3C3×C4⋊Dic312yes—
384C22×C3C3×#21412yes—
385C22×C3C3×#21512yes—
386C22×C3C2×#62611yescharacteristic
387C22×C3C2×C3×D1211yescharacteristic
388C22×C3C2×C3×#6711yescharacteristic
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—42no—
401C3×C4—124no—
402C3×C4—122no—
403C3×C4—122no—
404C3×C4—362no—
405C3×C4Dic3⋊Dic312yes—
406C3×C4C3×Dic3⋊C412yes—
407C3×C4C3×C4⋊Dic312yes—
408C3×C4C2×D3×Dic311yescharacteristic
409C3×C4C2×#62711yescharacteristic
410C3×C4C2×C3×Dic611yescharacteristic
411C3×C4C2×C3×Dic611yescharacteristic
412C3×C4C2×C3×C4×D311yescharacteristic
413C3×C4C22×C3×Dic311yescharacteristic
414Dic3—42no—
415Dic3—124no—
416Dic3—122no—
417Dic3—122no—
418C32—21no—
419C32—21no—
420C32—21no—
421C32—41no—
422C32C2×#80811yescharacteristic, complemented
423C32C2×#81711yescharacteristic, complemented
424C32C2×C3×#10611yescharacteristic, complemented
425C23C3×#62611yescharacteristic
426C2×C4—62no—
427C2×C4—122no—
428C2×C4—182no—
429C2×C4—362no—
430C2×C4C3×D3×Dic312yes—
431C2×C4C3×#62611yescharacteristic
432C2×C4C3×#62712yes—
433C2×C4C2×C3×D3211yescharacteristic
434C8—34no—
435C2×C3—24no—
436C2×C3—24no—
437C2×C3—24no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—44no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3#6357314yesdirect product factor
452C2×C3C3×#80814yescomplemented
453C2×C3C3×#81714yescomplemented
454C2×C3C2×#371611yescharacteristic
455C2×C3C2×#371711yescharacteristic
456C2×C3C2×Dic3⋊Dic311yescharacteristic
457C2×C3C2×C3×#20411yescharacteristic
458C2×C3C2×C3×#20411yescharacteristic
459C2×C3C2×C3×Dic3⋊C411yescharacteristic
460C2×C3C2×C3×C4⋊Dic311yescharacteristic
461C2×C3C2×C3×#21411yescharacteristic
462C2×C3C2×C3×#21511yescharacteristic
463C22C3×#371612yes—
464C22C3×#371712yes—
465C22C3×Dic3⋊Dic312yes—
466C22C2×C3×#62611yescharacteristic
467C4—122no—
468C4—362no—
469C4C3×Dic3⋊Dic312yes—
470C4C2×C3×D3×Dic311yescharacteristic
471C4C2×C3×#62711yescharacteristic
472C3—21no—
473C3—21no—
474C3—21no—
475C3—41no—
476C3C2×#6357311yescharacteristic, direct product factor
477C3C2×C3×#80811yescharacteristic, complemented
478C3C2×C3×#81711yescharacteristic, complemented
479C2C3×#6357314yesdirect product factor
480C2C2×C3×#371611yescharacteristic
481C2C2×C3×#371711yescharacteristic
482C2C2×C3×Dic3⋊Dic311yescharacteristic
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×#106
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×#63573C2direct (N × H)
7. C2×#63573C3direct (N × H)
15. C2×C32×C3⋊C8C4semidirect
31. #63573C2×C3direct (N × H)
41. C32×C3⋊C8C2×C4semidirect
80. C2×C3×C3⋊C8Dic3semidirect
81. C2×C3×C3⋊C8C3×C4semidirect
161. C3×C3⋊C8C2×Dic3semidirect
162. C3×C3⋊C8C2×C3×C4semidirect
164. C2×C33#106semidirect
205. C2×C3⋊C8C3×Dic3semidirect
268. C33C2×#106semidirect
323. C3⋊C8C2×C3×Dic3semidirect
340. C2×C32#808semidirect
341. C2×C32#817semidirect
342. C2×C32C3×#106semidirect
422. C32C2×#808semidirect
423. C32C2×#817semidirect
424. C32C2×C3×#106semidirect
451. C2×C3#63573direct (N × H)
452. C2×C3C3×#808semidirect
453. C2×C3C3×#817semidirect
476. C3C2×#63573direct (N × H)
477. C3C2×C3×#808semidirect
478. C3C2×C3×#817semidirect
479. C2C3×#63573direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4⋊Dic32
C22×C32×C3⋊C82
C3×#635732
C2×C3×#8173
C2×C3×#8083
C2×#635733