Finite Group Databasebeta

← C2×C3×#63955 #13175072

Subgroups — C2×C3×#63955 #13175072

Order 1728.

1954 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×Dic3C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
5C3×#63955C218yesmaximal, direct product factor
6C2×C3×Dic32C212yesmaximal, complemented
7C2×C3×#926—31nomaximal
8C2×C3×#877—31nomaximal
9C2×#63955C311yesmaximal, characteristic, direct product factor
10C24×C33C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C418yescomplemented
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2214yes—
17C3×#3757C2214yes—
18C32×#217C2214yes—
19C3×Dic3⋊Dic3C2218yescomplemented
20C3×Dic32C2218yescomplemented
21C23×C32×C4—31no—
22C23×C3×Dic3—21no—
23C23×C3×Dic3D311yescharacteristic
24C23×C3×Dic3C2×C311yescharacteristic
25C2×C32×C22⋊C4—31no—
26C2×#3757C2×C311yescharacteristic
27C2×C3×#217—21no—
28C2×C3×#217—31no—
29C2×C3×#217—31no—
30C2×C3×#217—61no—
31C2×C3×#217D311yescharacteristic
32C2×C3×#217C2×C311yescharacteristic
33C3×#926—38no—
34C2×C3×C4⋊Dic3—32no—
35C2×C3×Dic3⋊C4—32no—
36C2×C3×C4×Dic3—32no—
37C2×C3×C4×Dic3—32no—
38C3×#877—38no—
39C2×Dic3⋊Dic3C2×C312yescomplemented
40#63955C2×C318yesdirect product factor
41C2×Dic32C2×C312yescomplemented
42C23×C33—24no—
43C23×C33C2×C414yes—
44C23×C33C2312yes—
45C23×C33C2311yescharacteristic
46C2×C3×C3⋊Dic3—24no—
47C2×C3×C3⋊Dic3C2314yes—
48C2×C32×Dic3—24no—
49C2×C32×Dic3C2×C414yes—
50C2×C32×Dic3C2×C414yes—
51C2×C32×Dic3C2×C4116yescomplemented
52C2×C32×Dic3C2314yes—
53C2×C32×Dic3C2312yes—
54C2×C32×Dic3C2312yes—
55C2×C3×#117—91no—
56C2×#926—31no—
57C2×#877—31no—
58C24×C32—21no—
59C24×C32—21no—
60C24×C32—21no—
61C24×C32—41no—
62C24×C32C2×D311yescharacteristic
63C24×C32C2×D311yescharacteristic
64C24×C32C22×C311yescharacteristic
65C22×C32×C4—38no—
66C22×C32×C4—31no—
67C22×C32×C4—31no—
68C22×C32×C4—34no—
69C22×C32×C4—32no—
70C22×C3⋊Dic3C22×C312yes—
71C22×C3×Dic3—28no—
72C22×C3×Dic3—21no—
73C22×C3×Dic3—21no—
74C22×C3×Dic3—24no—
75C22×C3×Dic3—22no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—62no—
79C22×C3×Dic3Dic318yescomplemented
80C22×C3×Dic3C3×C418yescomplemented
81C22×C3×Dic3C2×D312yes—
82C22×C3×Dic3C2×D311yescharacteristic
83C22×C3×Dic3C2×D311yescharacteristic
84C22×C3×Dic3C2×D314yes—
85C22×C3×Dic3C22×C312yes—
86C22×C3×Dic3C22×C311yescharacteristic
87C22×C3×Dic3C22×C311yescharacteristic
88C22×C3×Dic3C22×C314yes—
89C32×C22⋊C4—34no—
90#3757C22×C314yes—
91C3×#217—24no—
92C3×#217—34no—
93C3×#217—34no—
94C3×#217—64no—
95C3×#217C2×D314yes—
96C3×#217C22×C314yes—
97C3×C4⋊Dic3—38no—
98C3×Dic3⋊C4—38no—
99C3×C4×Dic3—38no—
100C3×C4×Dic3—38no—
101Dic3⋊Dic3C22×C318yescomplemented
102Dic32C22×C318yescomplemented
103C22×C33—28no—
104C22×C33—24no—
105C22×C33C22×C414yes—
106C22×C33C22×C412yes—
107C22×C33#4114yes—
108C22×C33C2411yescharacteristic
109C3×C3⋊Dic3—24no—
110C32×Dic3—24no—
111C32×Dic3C22×C414yes—
112C32×Dic3C22×C414yes—
113C23×C3×C4—31no—
114C23×C3×C4—31no—
115C23×C3×C4—61no—
116C23×Dic3C3×D311yescharacteristic
117C3×#117—98no—
118C2×C3×C4⋊C4—92no—
119C2×C3×C22⋊C4—31no—
120C2×C3×C22⋊C4—31no—
121C2×C3×C22⋊C4—61no—
122C2×C3×C22⋊C4—91no—
123C2×C3×C42—92no—
124C2×#217—31no—
125C2×#217—31no—
126C2×#217—61no—
127C2×#217C3×D311yescharacteristic
128#926—38no—
129C2×C4⋊Dic3—32no—
130C2×Dic3⋊C4—32no—
131C2×C4×Dic3—32no—
132C2×C4×Dic3—32no—
133#877—38no—
134C23×C32—24no—
135C23×C32—28no—
136C23×C32—24no—
137C23×C32—24no—
138C23×C32—24no—
139C23×C32—24no—
140C23×C32—24no—
141C23×C32—22no—
142C23×C32—21no—
143C23×C32—22no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—22no—
147C23×C32—44no—
148C23×C32—41no—
149C23×C32—42no—
150C23×C32—44no—
151C23×C32—44no—
152C23×C32—48no—
153C23×C32C4×D314yes—
154C23×C32C2×Dic314yes—
155C23×C32C2×C3×C414yes—
156C23×C32C22×D312yes—
157C23×C32C22×D312yes—
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C23×C32C23×C312yes—
161C23×C32C23×C311yescharacteristic
162C2×C32×C4—34no—
163C2×C32×C4—34no—
164C2×C32×C4—316no—
165C2×C32×C4—32no—
166C2×C32×C4—32no—
167C2×C32×C4—34no—
168C2×C32×C4—64no—
169C2×C3⋊Dic3—24no—
170C2×C3⋊Dic3C23×C314yes—
171C2×C3×Dic3—24no—
172C2×C3×Dic3—24no—
173C2×C3×Dic3—24no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—216no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—24no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—44no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—124no—
186C2×C3×Dic3C2×Dic314yes—
187C2×C3×Dic3C2×Dic314yes—
188C2×C3×Dic3C2×Dic3116yescomplemented
189C2×C3×Dic3C2×C3×C414yes—
190C2×C3×Dic3C2×C3×C414yes—
191C2×C3×Dic3C2×C3×C4116yescomplemented
192C2×C3×Dic3C22×D314yes—
193C2×C3×Dic3C22×D312yes—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C23×C314yes—
196C2×C3×Dic3C23×C312yes—
197C2×C3×Dic3C23×C312yes—
198C2×#117—91noSylow
199C2×C33—24no—
200C2×C33#11714yescomplemented
201C2×C33C23×C411yescharacteristic
202C2×C33C2×#4112yes—
203C24×C3—21no—
204C24×C3—21no—
205C24×C3—21no—
206C24×C3—41no—
207C24×C3D3211yescharacteristic
208C24×C3C2×C3×D311yescharacteristic
209C24×C3C2×C3×D311yescharacteristic
210C22×C3×C4—38no—
211C22×C3×C4—38no—
212C22×C3×C4—34no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—34no—
218C22×C3×C4—32no—
219C22×C3×C4—32no—
220C22×C3×C4—68no—
221C22×C3×C4—61no—
222C22×C3×C4—61no—
223C22×C3×C4—64no—
224C22×C3×C4—62no—
225C22×C3×C4—92no—
226C22×Dic3—32no—
227C22×Dic3—32no—
228C22×Dic3—62no—
229C22×Dic3C3×Dic318yescomplemented
230C22×Dic3C2×C3×D312yes—
231C22×Dic3C2×C3×D311yescharacteristic
232C22×Dic3C2×C3×D311yescharacteristic
233C22×Dic3C2×C3×D314yes—
234C3×C4⋊C4—98no—
235C3×C22⋊C4—34no—
236C3×C22⋊C4—34no—
237C3×C22⋊C4—64no—
238C3×C22⋊C4—94no—
239C3×C42—98no—
240#217—34no—
241#217—34no—
242#217—64no—
243#217C2×C3×D314yes—
244C4⋊Dic3—38no—
245Dic3⋊C4—38no—
246C4×Dic3—38no—
247C4×Dic3—38no—
248C22×C32—28no—
249C22×C32—24no—
250C22×C32—216no—
251C22×C32—28no—
252C22×C32—28no—
253C22×C32—28no—
254C22×C32—24no—
255C22×C32—24no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—22no—
260C22×C32—24no—
261C22×C32—24no—
262C22×C32—22no—
263C22×C32—24no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—22no—
268C22×C32—42no—
269C22×C32—44no—
270C22×C32—44no—
271C22×C32—41no—
272C22×C32—48no—
273C22×C32—44no—
274C22×C32—44no—
275C22×C32—48no—
276C22×C32—416no—
277C22×C32—48no—
278C22×C32C2×C4×D314yes—
279C22×C32C2×C4×D312yes—
280C22×C32#23514yes—
281C22×C32#23714yes—
282C22×C32C22×Dic314yes—
283C22×C32C22×Dic312yes—
284C22×C32C22×C3×C414yes—
285C22×C32C22×C3×C412yes—
286C22×C32C3×#4114yes—
287C22×C32C23×D311yescharacteristic
288C22×C32C23×D311yescharacteristic
289C22×C32C24×C311yescharacteristic
290C32×C4—34no—
291C32×C4—34no—
292C32×C4—64no—
293C3⋊Dic3—24no—
294C3×Dic3—24no—
295C3×Dic3—24no—
296C3×Dic3—24no—
297C3×Dic3—24no—
298C3×Dic3—44no—
299C3×Dic3—64no—
300C3×Dic3—64no—
301C3×Dic3—124no—
302C3×Dic3C22×Dic314yes—
303C3×Dic3C22×Dic314yes—
304C3×Dic3C22×C3×C414yes—
305C3×Dic3C22×C3×C414yes—
306C23×C4—31no—
307#117—98no—
308C2×C4⋊C4—92no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—91no—
311C2×C42—92no—
312C33C2×#11711yescharacteristic, complemented, Sylow
313C23×C3—28no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—22no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—44no—
327C23×C3—44no—
328C23×C3—48no—
329C23×C3—44no—
330C23×C3—42no—
331C23×C3—41no—
332C23×C3D3×Dic314yes—
333C23×C3C3×C4×D314yes—
334C23×C3C2×C3×Dic314yes—
335C23×C3C2×D3212yes—
336C23×C3C2×D3211yescharacteristic
337C23×C3C22×C3×D312yes—
338C23×C3C22×C3×D312yes—
339C23×C3C22×C3×D311yescharacteristic
340C23×C3C22×C3×D311yescharacteristic
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—32no—
345C2×C3×C4—34no—
346C2×C3×C4—34no—
347C2×C3×C4—316no—
348C2×C3×C4—34no—
349C2×C3×C4—34no—
350C2×C3×C4—316no—
351C2×C3×C4—34no—
352C2×C3×C4—34no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—64no—
356C2×C3×C4—64no—
357C2×C3×C4—616no—
358C2×C3×C4—64no—
359C2×C3×C4—64no—
360C2×C3×C4—64no—
361C2×C3×C4—94no—
362C2×C3×C4—124no—
363C2×C3×C4—184no—
364C2×Dic3—24no—
365C2×Dic3—34no—
366C2×Dic3—34no—
367C2×Dic3—64no—
368C2×Dic3—64no—
369C2×Dic3—64no—
370C2×Dic3—124no—
371C2×Dic3C2×C3×Dic314yes—
372C2×Dic3C2×C3×Dic314yes—
373C2×Dic3C2×C3×Dic3116yescomplemented
374C2×Dic3C22×C3×D314yes—
375C2×Dic3C22×C3×D312yes—
376C2×Dic3C22×C3×D312yes—
377C2×C32—24no—
378C2×C32—28no—
379C2×C32—24no—
380C2×C32—24no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32—21no—
384C2×C32—22no—
385C2×C32—21no—
386C2×C32—22no—
387C2×C32—22no—
388C2×C32—21no—
389C2×C32—24no—
390C2×C32—44no—
391C2×C32—41no—
392C2×C32—42no—
393C2×C32—44no—
394C2×C32—44no—
395C2×C32—48no—
396C2×C32#87714yescomplemented
397C2×C32#92614yescomplemented
398C2×C32C3×#11714yescomplemented
399C2×C32C22×C4×D311yescharacteristic
400C2×C32C2×#23512yes—
401C2×C32C2×#23712yes—
402C2×C32C23×Dic311yescharacteristic
403C2×C32C23×C3×C411yescharacteristic
404C2×C32C2×C3×#4112yes—
405C24C3×D3211yescharacteristic
406C22×C4—38no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—92no—
412C4⋊C4—98no—
413C22⋊C4—34no—
414C22⋊C4—94no—
415C42—98no—
416C22×C3—28no—
417C22×C3—216no—
418C22×C3—28no—
419C22×C3—28no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—28no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—24no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—48no—
439C22×C3—48no—
440C22×C3—48no—
441C22×C3—416no—
442C22×C3—41no—
443C22×C3—44no—
444C22×C3—42no—
445C22×C3—44no—
446C22×C3C2×D3×Dic314yes—
447C22×C3C2×D3×Dic312yes—
448C22×C3#380414yes—
449C22×C3C2×C3×C4×D314yes—
450C22×C3C2×C3×C4×D312yes—
451C22×C3C3×#23514yes—
452C22×C3C3×#23714yes—
453C22×C3C22×C3×Dic314yes—
454C22×C3C22×C3×Dic312yes—
455C22×C3C22×D3211yescharacteristic
456C22×C3C23×C3×D311yescharacteristic
457C22×C3C23×C3×D311yescharacteristic
458C3×C4—34no—
459C3×C4—34no—
460C3×C4—34no—
461C3×C4—34no—
462C3×C4—64no—
463C3×C4—64no—
464C3×C4—64no—
465C3×C4—64no—
466C3×C4—124no—
467C3×C4—184no—
468Dic3—24no—
469Dic3—64no—
470Dic3—64no—
471Dic3—124no—
472Dic3C22×C3×Dic314yes—
473Dic3C22×C3×Dic314yes—
474C32—21no—
475C32—21no—
476C32—21no—
477C32—41no—
478C32C2×#87711yescharacteristic, complemented
479C32C2×#92611yescharacteristic, complemented
480C32C2×C3×#11711yescharacteristic, complemented
481C23—24no—
482C23C3×D3×Dic314yes—
483C23C2×C3×D3212yes—
484C23C2×C3×D3211yescharacteristic
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—34no—
488C2×C4—34no—
489C2×C4—316no—
490C2×C4—34no—
491C2×C4—64no—
492C2×C4—94no—
493C2×C4—184no—
494C2×C3—28no—
495C2×C3—24no—
496C2×C3—24no—
497C2×C3—24no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—24no—
505C2×C3—24no—
506C2×C3—24no—
507C2×C3—44no—
508C2×C3—44no—
509C2×C3—48no—
510C2×C3—41no—
511C2×C3—42no—
512C2×C3—44no—
513C2×C3#6395514yesdirect product factor
514C2×C3C3×#87714yescomplemented
515C2×C3C3×#92614yescomplemented
516C2×C3C22×D3×Dic311yescharacteristic
517C2×C3C2×#380412yes—
518C2×C3C22×C3×C4×D311yescharacteristic
519C2×C3C2×C3×#23512yes—
520C2×C3C2×C3×#23712yes—
521C2×C3C23×C3×Dic311yescharacteristic
522C22—24no—
523C22—28no—
524C22C2×C3×D3×Dic314yes—
525C22C2×C3×D3×Dic312yes—
526C22C3×#380414yes—
527C22C22×C3×D3211yescharacteristic
528C4—34no—
529C4—34no—
530C4—64no—
531C4—184no—
532C3—21no—
533C3—21no—
534C3—21no—
535C3—41no—
536C3C2×#6395511yescharacteristic, direct product factor
537C3C2×C3×#87711yescharacteristic, complemented
538C3C2×C3×#92611yescharacteristic, complemented
539C2—24no—
540C2C3×#6395514yesdirect product factor
541C2C22×C3×D3×Dic311yescharacteristic
542C2C2×C3×#380412yes—
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
Fitting, prime radicalC24×C33C22
socleC23×C33C23
cosocleC22×C32C24×C3
hypercenterC24×C3D32
hypocenterC32C2×C3×#117
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
4. C2×C3×Dic3⋊Dic3C2semidirect
5. C3×#63955C2direct (N × H)
6. C2×C3×Dic32C2semidirect
9. C2×#63955C3direct (N × H)
12. C22×C32×Dic3C4semidirect
19. C3×Dic3⋊Dic3C22semidirect
20. C3×Dic32C22semidirect
39. C2×Dic3⋊Dic3C2×C3semidirect
40. #63955C2×C3direct (N × H)
41. C2×Dic32C2×C3semidirect
51. C2×C32×Dic3C2×C4semidirect
79. C22×C3×Dic3Dic3semidirect
80. C22×C3×Dic3C3×C4semidirect
101. Dic3⋊Dic3C22×C3semidirect
102. Dic32C22×C3semidirect
188. C2×C3×Dic3C2×Dic3semidirect
191. C2×C3×Dic3C2×C3×C4semidirect
200. C2×C33#117semidirect
229. C22×Dic3C3×Dic3semidirect
312. C33C2×#117semidirect
373. C2×Dic3C2×C3×Dic3semidirect
396. C2×C32#877semidirect
397. C2×C32#926semidirect
398. C2×C32C3×#117semidirect
478. C32C2×#877semidirect
479. C32C2×#926semidirect
480. C32C2×C3×#117semidirect
513. C2×C3#63955direct (N × H)
514. C2×C3C3×#877semidirect
515. C2×C3C3×#926semidirect
536. C3C2×#63955direct (N × H)
537. C3C2×C3×#877semidirect
538. C3C2×C3×#926semidirect
540. C2C3×#63955direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×Dic32
C2×C3×#37572
C2×C32×#2172
C2×C3×Dic3⋊Dic32
C3×#639552
C2×C3×Dic322
C2×C3×#9263
C2×C3×#8773
C2×#639553