Finite Group Databasebeta

← C2×C3×#63579 #13158708

Subgroups — C2×C3×#63579 #13158708

Order 1728.

2162 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C23×C32×Dic3C212yesmaximal
3C3×#63579C214yesmaximal, direct product factor
4C2×C3×C3⋊#95—32nomaximal
5C2×#63579C311yesmaximal, characteristic, direct product factor
6C24×C33C2211yescharacteristic
7C22×C3×C3⋊Dic3—24no—
8C22×C3×C3⋊Dic3C414yescomplemented
9C22×C3×C3⋊Dic3C2212yes—
10C22×C32×Dic3—24no—
11C22×C32×Dic3—24no—
12C22×C32×Dic3C418yescomplemented
13C22×C32×Dic3C2214yes—
14C23×C32×C4—32no—
15C23×C3⋊Dic3C2×C311yescharacteristic
16C23×C3×Dic3—22no—
17C23×C3×Dic3—32no—
18C23×C3×Dic3—61no—
19C23×C3×Dic3D312yes—
20C23×C3×Dic3C2×C312yes—
21C3×C3⋊#95—38no—
22#63579C2×C314yesdirect product factor
23C23×C33C2×C412yes—
24C23×C33C2×C414yes—
25C23×C33D414yes—
26C23×C33D412yes—
27C23×C33Q812yes—
28C23×C33C2311yescharacteristic
29C2×C3×C3⋊Dic3—28no—
30C2×C3×C3⋊Dic3—24no—
31C2×C3×C3⋊Dic3C2×C414yescomplemented
32C2×C32×Dic3—216no—
33C2×C32×Dic3—24no—
34C2×C32×Dic3—24no—
35C2×C32×Dic3C2×C418yescomplemented
36C2×C3×#95—91no—
37C2×C3⋊#95—32no—
38C24×C32—21no—
39C24×C32—22no—
40C24×C32—41no—
41C24×C32C2×D312yes—
42C24×C32C22×C311yescharacteristic
43C22×C32×C4—38no—
44C22×C32×C4—34no—
45C22×C32×C4—64no—
46C22×C32×C4—64no—
47C22×C3⋊Dic3—24no—
48C22×C3⋊Dic3C3×C414yescomplemented
49C22×C3⋊Dic3C22×C312yes—
50C22×C3×Dic3—28no—
51C22×C3×Dic3—28no—
52C22×C3×Dic3—28no—
53C22×C3×Dic3—24no—
54C22×C3×Dic3—24no—
55C22×C3×Dic3—24no—
56C22×C3×Dic3—24no—
57C22×C3×Dic3—24no—
58C22×C3×Dic3—38no—
59C22×C3×Dic3—34no—
60C22×C3×Dic3—64no—
61C22×C3×Dic3—62no—
62C22×C3×Dic3—68no—
63C22×C3×Dic3—64no—
64C22×C3×Dic3—64no—
65C22×C3×Dic3Dic318yescomplemented
66C22×C3×Dic3C3×C418yescomplemented
67C22×C3×Dic3C2×D314yes—
68C22×C3×Dic3C22×C314yes—
69C22×C33C4214yes—
70C22×C33C22⋊C414yes—
71C22×C33C22⋊C418yes—
72C22×C33C4⋊C414yes—
73C22×C33C4⋊C418yes—
74C22×C33C22×C411yescharacteristic
75C22×C33C22×C412yes—
76C22×C33C2×D412yes—
77C22×C33C2×D411yescharacteristic
78C22×C33C2×Q811yescharacteristic
79C3×C3⋊Dic3—24no—
80C32×Dic3—28no—
81C23×C3×C4—32no—
82C23×C3×C4—32no—
83C23×C3×C4—62no—
84C23×C3×C4—91no—
85C23×Dic3—32no—
86C23×Dic3—61no—
87C23×Dic3C3×D312yes—
88C3×#95—94no—
89C3⋊#95—38no—
90C23×C32—24no—
91C23×C32—22no—
92C23×C32—22no—
93C23×C32—24no—
94C23×C32—22no—
95C23×C32—24no—
96C23×C32—24no—
97C23×C32—24no—
98C23×C32—24no—
99C23×C32—24no—
100C23×C32—24no—
101C23×C32—24no—
102C23×C32—21no—
103C23×C32—22no—
104C23×C32—44no—
105C23×C32—42no—
106C23×C32—42no—
107C23×C32—44no—
108C23×C32—42no—
109C23×C32—41no—
110C23×C32Dic614yes—
111C23×C32C4×D314yes—
112C23×C32C4×D314yes—
113C23×C32D1214yes—
114C23×C32C2×Dic314yes—
115C23×C32#6714yes—
116C23×C32#6714yes—
117C23×C32C2×C3×C412yes—
118C23×C32C2×C3×C414yes—
119C23×C32C3×D414yes—
120C23×C32C3×D412yes—
121C23×C32C3×Q812yes—
122C23×C32C22×D312yes—
123C23×C32C23×C311yescharacteristic
124C2×C32×C4—38no—
125C2×C32×C4—616no—
126C2×C32×C4—64no—
127C2×C32×C4—64no—
128C2×C3⋊Dic3—28no—
129C2×C3⋊Dic3—24no—
130C2×C3⋊Dic3C2×C3×C414yescomplemented
131C2×C3×Dic3—232no—
132C2×C3×Dic3—28no—
133C2×C3×Dic3—28no—
134C2×C3×Dic3—216no—
135C2×C3×Dic3—216no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—28no—
141C2×C3×Dic3—38no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—616no—
144C2×C3×Dic3—68no—
145C2×C3×Dic3—616no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3C2×Dic318yescomplemented
149C2×C3×Dic3C2×C3×C418yescomplemented
150C2×#95—91noSylow
151C2×C33#9518yescomplemented
152C2×C33C2×C4211yescharacteristic
153C2×C33C2×C22⋊C411yescharacteristic
154C2×C33C2×C22⋊C412yes—
155C2×C33C2×C4⋊C411yescharacteristic
156C2×C33C2×C4⋊C412yes—
157C24×C3—22no—
158C24×C3—21no—
159C24×C3—41no—
160C24×C3D3211yescharacteristic
161C24×C3C2×C3×D312yes—
162C22×C3×C4—38no—
163C22×C3×C4—38no—
164C22×C3×C4—34no—
165C22×C3×C4—34no—
166C22×C3×C4—68no—
167C22×C3×C4—64no—
168C22×C3×C4—68no—
169C22×C3×C4—68no—
170C22×C3×C4—64no—
171C22×C3×C4—64no—
172C22×C3×C4—64no—
173C22×C3×C4—64no—
174C22×C3×C4—94no—
175C22×C3×C4—92no—
176C22×C3×C4—184no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—38no—
180C22×Dic3—34no—
181C22×Dic3—64no—
182C22×Dic3—62no—
183C22×Dic3—68no—
184C22×Dic3—64no—
185C22×Dic3—64no—
186C22×Dic3C3×Dic318yescomplemented
187C22×Dic3C2×C3×D314yes—
188C22×C32—24no—
189C22×C32—28no—
190C22×C32—24no—
191C22×C32—28no—
192C22×C32—24no—
193C22×C32—28no—
194C22×C32—28no—
195C22×C32—28no—
196C22×C32—28no—
197C22×C32—28no—
198C22×C32—28no—
199C22×C32—28no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—44no—
213C22×C32—48no—
214C22×C32—44no—
215C22×C32—48no—
216C22×C32—44no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—42no—
221C22×C32—41no—
222C22×C32C4×Dic318yes—
223C22×C32Dic3⋊C418yes—
224C22×C32Dic3⋊C418yes—
225C22×C32C4⋊Dic318yes—
226C22×C32(C2×D3)⋊C418yes—
227C22×C32(C2×D3)⋊C418yes—
228C22×C32#21718yes—
229C22×C32C3×C4214yes—
230C22×C32C3×C22⋊C414yes—
231C22×C32C3×C22⋊C418yes—
232C22×C32C3×C4⋊C414yes—
233C22×C32C3×C4⋊C418yes—
234C22×C32C2×Dic612yes—
235C22×C32C2×C4×D312yes—
236C22×C32C2×C4×D312yes—
237C22×C32C2×D1212yes—
238C22×C32C22×Dic312yes—
239C22×C32C2×#6712yes—
240C22×C32C2×#6712yes—
241C22×C32C22×C3×C411yescharacteristic
242C22×C32C22×C3×C412yes—
243C22×C32C2×C3×D412yes—
244C22×C32C2×C3×D411yescharacteristic
245C22×C32C2×C3×Q811yescharacteristic
246C32×C4—68no—
247C3⋊Dic3—24no—
248C3×Dic3—216no—
249C3×Dic3—28no—
250C3×Dic3—28no—
251C3×Dic3—68no—
252C3×Dic3—68no—
253C23×C4—32no—
254C23×C4—91no—
255#95—94no—
256C33C2×#9511yescharacteristic, complemented, Sylow
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—21no—
271C23×C3—44no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—44no—
275C23×C3—42no—
276C23×C3—41no—
277C23×C3D3×Dic314yes—
278C23×C3#62412yes—
279C23×C3#62512yes—
280C23×C3#62614yes—
281C23×C3#62712yes—
282C23×C3C3×Dic614yes—
283C23×C3C3×C4×D314yes—
284C23×C3C3×C4×D314yes—
285C23×C3C3×D1214yes—
286C23×C3C2×C3×Dic314yes—
287C23×C3C3×#6714yes—
288C23×C3C3×#6714yes—
289C23×C3C2×D3211yescharacteristic
290C23×C3C22×C3×D312yes—
291C2×C3×C4—38no—
292C2×C3×C4—38no—
293C2×C3×C4—68no—
294C2×C3×C4—68no—
295C2×C3×C4—68no—
296C2×C3×C4—632no—
297C2×C3×C4—616no—
298C2×C3×C4—616no—
299C2×C3×C4—64no—
300C2×C3×C4—64no—
301C2×C3×C4—64no—
302C2×C3×C4—64no—
303C2×C3×C4—94no—
304C2×C3×C4—188no—
305C2×C3×C4—184no—
306C2×Dic3—216no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—38no—
310C2×Dic3—64no—
311C2×Dic3—68no—
312C2×Dic3—616no—
313C2×Dic3—616no—
314C2×Dic3—64no—
315C2×Dic3—64no—
316C2×Dic3C2×C3×Dic318yescomplemented
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—22no—
322C2×C32—22no—
323C2×C32—22no—
324C2×C32—28no—
325C2×C32—216no—
326C2×C32—21no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—22no—
330C2×C32—21no—
331C2×C32—48no—
332C2×C32—41no—
333C2×C32—42no—
334C2×C32—41no—
335C2×C32—42no—
336C2×C32—41no—
337C2×C32C3⋊#95116yescomplemented
338C2×C32C3×#9518yescomplemented
339C2×C32C2×C4×Dic312yes—
340C2×C32C2×Dic3⋊C412yes—
341C2×C32C2×Dic3⋊C412yes—
342C2×C32C2×C4⋊Dic312yes—
343C2×C32C2×(C2×D3)⋊C412yes—
344C2×C32C2×(C2×D3)⋊C412yes—
345C2×C32C2×#21712yes—
346C2×C32C2×C3×C4211yescharacteristic
347C2×C32C2×C3×C22⋊C411yescharacteristic
348C2×C32C2×C3×C22⋊C412yes—
349C2×C32C2×C3×C4⋊C411yescharacteristic
350C2×C32C2×C3×C4⋊C412yes—
351C24C3×D3211yescharacteristic
352C22×C4—38no—
353C22×C4—34no—
354C22×C4—64no—
355C22×C4—64no—
356C22×C4—94no—
357C22×C4—92no—
358C22×C4—184no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—21no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—28no—
372C22×C3—24no—
373C22×C3—28no—
374C22×C3—28no—
375C22×C3—28no—
376C22×C3—24no—
377C22×C3—28no—
378C22×C3—28no—
379C22×C3—28no—
380C22×C3—28no—
381C22×C3—28no—
382C22×C3—24no—
383C22×C3—42no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—42no—
387C22×C3—41no—
388C22×C3—44no—
389C22×C3—48no—
390C22×C3—44no—
391C22×C3—48no—
392C22×C3—44no—
393C22×C3Dic3214yes—
394C22×C3(C2×D3)⋊Dic318yes—
395C22×C3#372214yes—
396C22×C3Dic3⋊Dic318yes—
397C22×C3#372414yes—
398C22×C3C3×C4×Dic318yes—
399C22×C3C3×Dic3⋊C418yes—
400C22×C3C3×Dic3⋊C418yes—
401C22×C3C3×C4⋊Dic318yes—
402C22×C3C3×(C2×D3)⋊C418yes—
403C22×C3C3×(C2×D3)⋊C418yes—
404C22×C3C3×#21718yes—
405C22×C3C2×D3×Dic312yes—
406C22×C3C2×#62411yescharacteristic
407C22×C3C2×#62511yescharacteristic
408C22×C3C2×#62612yes—
409C22×C3C2×#62711yescharacteristic
410C22×C3C2×C3×Dic612yes—
411C22×C3C2×C3×C4×D312yes—
412C22×C3C2×C3×C4×D312yes—
413C22×C3C2×C3×D1212yes—
414C22×C3C22×C3×Dic312yes—
415C22×C3C2×C3×#6712yes—
416C22×C3C2×C3×#6712yes—
417C3×C4—616no—
418C3×C4—68no—
419C3×C4—68no—
420C3×C4—184no—
421Dic3—28no—
422Dic3—68no—
423Dic3—68no—
424C32—22no—
425C32—21no—
426C32—41no—
427C32C2×C3⋊#9512yescomplemented
428C32C2×C3×#9511yescharacteristic, complemented
429C23C3×D3×Dic314yes—
430C23C3×#62412yes—
431C23C3×#62512yes—
432C23C3×#62614yes—
433C23C3×#62712yes—
434C23C2×C3×D3211yescharacteristic
435C2×C4—38no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—616no—
439C2×C4—94no—
440C2×C4—184no—
441C2×C4—188no—
442C2×C3—216no—
443C2×C3—28no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—22no—
451C2×C3—22no—
452C2×C3—22no—
453C2×C3—22no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—48no—
457C2×C3—41no—
458C2×C3—42no—
459C2×C3—41no—
460C2×C3—42no—
461C2×C3—41no—
462C2×C3#6357918yesdirect product factor
463C2×C3C3×C3⋊#95116yescomplemented
464C2×C3C2×Dic3211yescharacteristic
465C2×C3C2×(C2×D3)⋊Dic312yes—
466C2×C3C2×#372211yescharacteristic
467C2×C3C2×Dic3⋊Dic312yes—
468C2×C3C2×#372411yescharacteristic
469C2×C3C2×C3×C4×Dic312yes—
470C2×C3C2×C3×Dic3⋊C412yes—
471C2×C3C2×C3×Dic3⋊C412yes—
472C2×C3C2×C3×C4⋊Dic312yes—
473C2×C3C2×C3×(C2×D3)⋊C412yes—
474C2×C3C2×C3×(C2×D3)⋊C412yes—
475C2×C3C2×C3×#21712yes—
476C22C3×Dic3214yes—
477C22C3×(C2×D3)⋊Dic318yes—
478C22C3×#372214yes—
479C22C3×Dic3⋊Dic318yes—
480C22C3×#372414yes—
481C22C2×C3×D3×Dic312yes—
482C22C2×C3×#62411yescharacteristic
483C22C2×C3×#62511yescharacteristic
484C22C2×C3×#62612yes—
485C22C2×C3×#62711yescharacteristic
486C4—68no—
487C4—184no—
488C3—22no—
489C3—21no—
490C3—41no—
491C3C2×#6357911yescharacteristic, direct product factor
492C3C2×C3×C3⋊#9512yescomplemented
493C2C3×#6357918yesdirect product factor
494C2C2×C3×Dic3211yescharacteristic
495C2C2×C3×(C2×D3)⋊Dic312yes—
496C2C2×C3×#372211yescharacteristic
497C2C2×C3×Dic3⋊Dic312yes—
498C2C2×C3×#372411yescharacteristic
subgroupG/H
center, hypercenterC24×C3D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
Fitting, socle, prime radicalC24×C33C22
cosocleC23×C32C23×C3
hypocenterC32C2×C3×#95
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
3. C3×#63579C2direct (N × H)
5. C2×#63579C3direct (N × H)
8. C22×C3×C3⋊Dic3C4semidirect
12. C22×C32×Dic3C4semidirect
22. #63579C2×C3direct (N × H)
31. C2×C3×C3⋊Dic3C2×C4semidirect
35. C2×C32×Dic3C2×C4semidirect
48. C22×C3⋊Dic3C3×C4semidirect
65. C22×C3×Dic3Dic3semidirect
66. C22×C3×Dic3C3×C4semidirect
130. C2×C3⋊Dic3C2×C3×C4semidirect
148. C2×C3×Dic3C2×Dic3semidirect
149. C2×C3×Dic3C2×C3×C4semidirect
151. C2×C33#95semidirect
186. C22×Dic3C3×Dic3semidirect
256. C33C2×#95semidirect
316. C2×Dic3C2×C3×Dic3semidirect
337. C2×C32C3⋊#95semidirect
338. C2×C32C3×#95semidirect
427. C32C2×C3⋊#95semidirect
428. C32C2×C3×#95semidirect
462. C2×C3#63579direct (N × H)
463. C2×C3C3×C3⋊#95semidirect
491. C3C2×#63579direct (N × H)
492. C3C2×C3×C3⋊#95semidirect
493. C2C3×#63579direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C23×C32×Dic32
C3×#635792
C2×C3×C3⋊#953
C2×#635793