Finite Group Databasebeta

← C2×C3×#63855 #13174382

Subgroups — C2×C3×#63855 #13174382

Order 1728.

1826 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
5C2×C3×(C2×D3)⋊Dic3C212yesmaximal
6C3×#63855C218yesmaximal, direct product factor
7C2×C3×#926—31nomaximal
8C2×C3×#872—31nomaximal
9C2×#63855C311yesmaximal, characteristic, direct product factor
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2211yescharacteristic
15C2×C32×C4×D3C418yescomplemented
16C2×C32×C4×D3C2214yes—
17C3×C4×C3⋊Dic3C2214yescomplemented
18C32×C4×Dic3C2214yescomplemented
19C3×Dic3⋊Dic3C2218yescomplemented
20C3×(C2×D3)⋊Dic3C2218yes—
21C23×C32×C4—31no—
22C22×C3×C4×D3—21no—
23C22×C3×C4×D3D311yescharacteristic
24C22×C3×C4×D3C2×C311yescharacteristic
25C2×C32×C42—31no—
26C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×C3×#217—32no—
28C2×C3×(C2×D3)⋊C4—32no—
29C3×#926—38no—
30C2×C3×C4⋊Dic3—32no—
31C2×C3×Dic3⋊C4—32no—
32C2×C3×C4×Dic3—21no—
33C2×C3×C4×Dic3—31no—
34C2×C3×C4×Dic3—31no—
35C2×C3×C4×Dic3—61no—
36C2×C3×C4×Dic3D311yescharacteristic, complemented
37C2×C3×C4×Dic3C2×C311yescharacteristic, complemented
38C3×#872—38no—
39C2×Dic3⋊Dic3C2×C312yescomplemented
40C2×(C2×D3)⋊Dic3C2×C312yes—
41#63855C2×C318yesdirect product factor
42C23×C33C2311yescharacteristic
43C22×C32×D3—24no—
44C22×C32×D3C2×C414yes—
45C22×C32×D3C2312yes—
46C2×C33×C4C2×C414yes—
47C2×C33×C4C2312yes—
48C2×C3×C3⋊Dic3—24no—
49C2×C3×C3⋊Dic3C2314yes—
50C2×C32×Dic3—24no—
51C2×C32×Dic3C2×C414yes—
52C2×C32×Dic3C2314yes—
53C2×C32×Dic3C2312yes—
54C32×C4×D3C2×C4116yescomplemented
55C2×C3×#117—91no—
56C2×#926—31no—
57C2×#872—31no—
58C24×C32—31no—
59C23×C3×D3—21no—
60C23×C3×D3C2×D311yescharacteristic
61C23×C3×D3C22×C311yescharacteristic
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—31no—
66C22×C32×C4—38no—
67C22×C32×C4—34no—
68C22×C32×C4—32no—
69C22×C32×C4—41no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C312yes—
74C22×C3×Dic3—22no—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—62no—
79C22×C3×Dic3C2×D312yes—
80C22×C3×Dic3C2×D311yescharacteristic
81C22×C3×Dic3C22×C312yes—
82C22×C3×Dic3C22×C311yescharacteristic
83C2×C3×C4×D3—28no—
84C2×C3×C4×D3—24no—
85C2×C3×C4×D3Dic318yescomplemented
86C2×C3×C4×D3C3×C418yescomplemented
87C2×C3×C4×D3C2×D314yes—
88C2×C3×C4×D3C22×C314yes—
89C32×C42—34no—
90C4×C3⋊Dic3C22×C314yescomplemented
91C3×#217—38no—
92C3×(C2×D3)⋊C4—38no—
93C3×C4⋊Dic3—38no—
94C3×Dic3⋊C4—38no—
95C3×C4×Dic3—24no—
96C3×C4×Dic3—34no—
97C3×C4×Dic3—34no—
98C3×C4×Dic3—64no—
99C3×C4×Dic3C2×D314yescomplemented
100C3×C4×Dic3C22×C314yescomplemented
101Dic3⋊Dic3C22×C318yescomplemented
102(C2×D3)⋊Dic3C22×C318yes—
103C22×C33C22×C412yes—
104C22×C33#4114yes—
105C22×C33C2411yescharacteristic
106C2×C32×D3—28no—
107C2×C32×D3—24no—
108C2×C32×D3C22×C414yes—
109C33×C4C22×C414yes—
110C3×C3⋊Dic3—24no—
111C32×Dic3—24no—
112C32×Dic3C22×C414yes—
113C23×C3×C4—31no—
114C23×C3×C4—31no—
115C23×C3×C4—61no—
116C22×C4×D3C3×D311yescharacteristic
117C3×#117—98no—
118C2×C3×C4⋊C4—92no—
119C2×C3×C22⋊C4—92no—
120C2×C3×C42—31no—
121C2×C3×C42—31no—
122C2×C3×C42—61no—
123C2×C3×C42—91no—
124C2×#217—32no—
125C2×(C2×D3)⋊C4—32no—
126#926—38no—
127C2×C4⋊Dic3—32no—
128C2×Dic3⋊C4—32no—
129C2×C4×Dic3—31no—
130C2×C4×Dic3—31no—
131C2×C4×Dic3—61no—
132C2×C4×Dic3C3×D311yescharacteristic, complemented
133#872—38no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—34no—
138C23×C32—32no—
139C23×C32—41no—
140C23×C32—64no—
141C23×C32C22×D311yescharacteristic
142C23×C32C22×D311yescharacteristic
143C23×C32C23×C311yescharacteristic
144C22×C3×D3—28no—
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3—24no—
148C22×C3×D3—22no—
149C22×C3×D3C2×Dic314yes—
150C22×C3×D3C2×C3×C414yes—
151C22×C3×D3C22×D312yes—
152C22×C3×D3C23×C312yes—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—34no—
160C2×C32×C4—32no—
161C2×C32×C4—316no—
162C2×C32×C4—34no—
163C2×C32×C4—44no—
164C2×C32×C4—42no—
165C2×C32×C4—64no—
166C2×C32×C4C4×D314yes—
167C2×C32×C4C2×Dic314yes—
168C2×C32×C4C2×C3×C414yes—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C23×C312yes—
172C2×C3⋊Dic3—24no—
173C2×C3⋊Dic3C23×C314yes—
174C2×C3×Dic3—28no—
175C2×C3×Dic3—24no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—24no—
178C2×C3×Dic3—24no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—34no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3—68no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3C2×Dic314yes—
187C2×C3×Dic3C2×C3×C414yes—
188C2×C3×Dic3C22×D314yes—
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C23×C314yes—
191C2×C3×Dic3C23×C312yes—
192C3×C4×D3—216no—
193C3×C4×D3C2×Dic3116yescomplemented
194C3×C4×D3C2×C3×C4116yescomplemented
195C2×#117—91noSylow
196C2×C33#11714yescomplemented
197C2×C33C23×C411yescharacteristic
198C2×C33C2×#4112yes—
199C32×D3—24no—
200C24×C3—31no—
201C24×C3—31no—
202C24×C3—61no—
203C23×D3C2×C3×D311yescharacteristic
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—38no—
210C22×C3×C4—38no—
211C22×C3×C4—34no—
212C22×C3×C4—34no—
213C22×C3×C4—32no—
214C22×C3×C4—32no—
215C22×C3×C4—41no—
216C22×C3×C4—68no—
217C22×C3×C4—64no—
218C22×C3×C4—62no—
219C22×C3×C4—61no—
220C22×C3×C4—92no—
221C22×C3×C4D3211yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C22×C3×C4C2×C3×D311yescharacteristic
224C22×Dic3—32no—
225C22×Dic3—32no—
226C22×Dic3—62no—
227C22×Dic3C2×C3×D312yes—
228C22×Dic3C2×C3×D311yescharacteristic
229C2×C4×D3C3×Dic318yescomplemented
230C2×C4×D3C2×C3×D314yes—
231C3×C4⋊C4—98no—
232C3×C22⋊C4—98no—
233C3×C42—34no—
234C3×C42—34no—
235C3×C42—64no—
236C3×C42—94no—
237#217—38no—
238(C2×D3)⋊C4—38no—
239C4⋊Dic3—38no—
240Dic3⋊C4—38no—
241C4×Dic3—34no—
242C4×Dic3—34no—
243C4×Dic3—64no—
244C4×Dic3C2×C3×D314yescomplemented
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—24no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—24no—
253C22×C32—22no—
254C22×C32—34no—
255C22×C32—42no—
256C22×C32—44no—
257C22×C32—41no—
258C22×C32—68no—
259C22×C32—64no—
260C22×C32C2×C4×D312yes—
261C22×C32#23514yes—
262C22×C32#23514yes—
263C22×C32C22×Dic312yes—
264C22×C32C22×C3×C412yes—
265C22×C32C3×#4114yes—
266C22×C32C23×D311yescharacteristic
267C22×C32C23×D311yescharacteristic
268C22×C32C24×C311yescharacteristic
269C2×C3×D3—216no—
270C2×C3×D3—28no—
271C2×C3×D3—28no—
272C2×C3×D3—28no—
273C2×C3×D3—24no—
274C2×C3×D3—24no—
275C2×C3×D3—24no—
276C2×C3×D3C22×Dic314yes—
277C2×C3×D3C22×C3×C414yes—
278C32×C4—24no—
279C32×C4—24no—
280C32×C4—24no—
281C32×C4—34no—
282C32×C4—44no—
283C32×C4—64no—
284C32×C4C2×C4×D314yes—
285C32×C4C22×Dic314yes—
286C32×C4C22×C3×C414yes—
287C3⋊Dic3—24no—
288C3×Dic3—28no—
289C3×Dic3—24no—
290C3×Dic3—24no—
291C3×Dic3—24no—
292C3×Dic3—68no—
293C3×Dic3—64no—
294C3×Dic3—64no—
295C3×Dic3C22×Dic314yes—
296C3×Dic3C22×C3×C414yes—
297C23×C4—31no—
298#117—98no—
299C2×C4⋊C4—92no—
300C2×C22⋊C4—92no—
301C2×C42—31no—
302C2×C42—91no—
303C33C2×#11711yescharacteristic, complemented, Sylow
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—32no—
308C23×C3—32no—
309C23×C3—34no—
310C23×C3—34no—
311C23×C3—41no—
312C23×C3—62no—
313C23×C3—64no—
314C23×C3—68no—
315C23×C3—64no—
316C23×C3—64no—
317C23×C3C2×D3211yescharacteristic
318C23×C3C22×C3×D311yescharacteristic
319C23×C3C22×C3×D311yescharacteristic
320C22×D3—24no—
321C22×D3C2×C3×Dic314yes—
322C22×D3C22×C3×D312yes—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—34no—
332C2×C3×C4—34no—
333C2×C3×C4—316no—
334C2×C3×C4—316no—
335C2×C3×C4—34no—
336C2×C3×C4—34no—
337C2×C3×C4—44no—
338C2×C3×C4—42no—
339C2×C3×C4—616no—
340C2×C3×C4—64no—
341C2×C3×C4—62no—
342C2×C3×C4—64no—
343C2×C3×C4—68no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4—94no—
347C2×C3×C4—184no—
348C2×C3×C4D3×Dic314yes—
349C2×C3×C4C3×C4×D314yes—
350C2×C3×C4C2×C3×Dic314yes—
351C2×C3×C4C2×D3212yes—
352C2×C3×C4C22×C3×D312yes—
353C2×C3×C4C22×C3×D312yes—
354C2×Dic3—24no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C2×Dic3—64no—
358C2×Dic3—68no—
359C2×Dic3—64no—
360C2×Dic3—64no—
361C2×Dic3C2×C3×Dic314yes—
362C2×Dic3C22×C3×D314yes—
363C2×Dic3C22×C3×D312yes—
364C4×D3C2×C3×Dic3116yescomplemented
365C2×C32—22no—
366C2×C32—24no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—22no—
370C2×C32—22no—
371C2×C32—21no—
372C2×C32—24no—
373C2×C32—24no—
374C2×C32—44no—
375C2×C32—41no—
376C2×C32—42no—
377C2×C32—64no—
378C2×C32#87214yescomplemented
379C2×C32#92614yescomplemented
380C2×C32C3×#11714yescomplemented
381C2×C32C22×C4×D311yescharacteristic
382C2×C32C2×#23512yes—
383C2×C32C2×#23512yes—
384C2×C32C23×Dic311yescharacteristic
385C2×C32C23×C3×C411yescharacteristic
386C2×C32C2×C3×#4112yes—
387C3×D3—28no—
388C3×D3—24no—
389C3×D3—24no—
390C24—31no—
391C22×C4—31no—
392C22×C4—38no—
393C22×C4—34no—
394C22×C4—32no—
395C22×C4—92no—
396C22×C4C3×D3211yescharacteristic
397C4⋊C4—98no—
398C22⋊C4—98no—
399C42—34no—
400C42—94no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—34no—
411C22×C3—34no—
412C22×C3—41no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—64no—
416C22×C3—616no—
417C22×C3—68no—
418C22×C3—64no—
419C22×C3—64no—
420C22×C3—68no—
421C22×C3—68no—
422C22×C3#379814yes—
423C22×C3C2×D3×Dic312yes—
424C22×C3C2×C3×C4×D312yes—
425C22×C3C3×#23514yes—
426C22×C3C3×#23514yes—
427C22×C3C22×C3×Dic312yes—
428C22×C3C22×D3211yescharacteristic
429C22×C3C23×C3×D311yescharacteristic
430C22×C3C23×C3×D311yescharacteristic
431C2×D3—28no—
432C2×D3—24no—
433C2×D3C22×C3×Dic314yes—
434C3×C4—24no—
435C3×C4—24no—
436C3×C4—24no—
437C3×C4—34no—
438C3×C4—34no—
439C3×C4—44no—
440C3×C4—64no—
441C3×C4—68no—
442C3×C4—64no—
443C3×C4—64no—
444C3×C4—184no—
445C3×C4C2×D3×Dic314yes—
446C3×C4C2×C3×C4×D314yes—
447C3×C4C22×C3×Dic314yes—
448Dic3—24no—
449Dic3—68no—
450Dic3—64no—
451Dic3—64no—
452Dic3C22×C3×Dic314yes—
453C32—21no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32C2×#87211yescharacteristic, complemented
458C32C2×#92611yescharacteristic, complemented
459C32C2×C3×#11711yescharacteristic, complemented
460C23—32no—
461C23—34no—
462C23—64no—
463C23C2×C3×D3211yescharacteristic
464C2×C4—32no—
465C2×C4—34no—
466C2×C4—316no—
467C2×C4—34no—
468C2×C4—64no—
469C2×C4—94no—
470C2×C4—184no—
471C2×C4C3×D3×Dic314yes—
472C2×C4C2×C3×D3212yes—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—24no—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—41no—
483C2×C3—44no—
484C2×C3—42no—
485C2×C3—68no—
486C2×C3—64no—
487C2×C3—64no—
488C2×C3#6385514yesdirect product factor
489C2×C3C3×#87214yescomplemented
490C2×C3C3×#92614yescomplemented
491C2×C3C2×#379812yes—
492C2×C3C22×D3×Dic311yescharacteristic
493C2×C3C22×C3×C4×D311yescharacteristic
494C2×C3C2×C3×#23512yes—
495C2×C3C2×C3×#23512yes—
496C2×C3C23×C3×Dic311yescharacteristic
497D3—24no—
498C22—34no—
499C22—68no—
500C22—64no—
501C22C3×#379814yes—
502C22C2×C3×D3×Dic312yes—
503C22C22×C3×D3211yescharacteristic
504C4—34no—
505C4—64no—
506C4—184no—
507C4C2×C3×D3×Dic314yes—
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3C2×#6385511yescharacteristic, direct product factor
513C3C2×C3×#87211yescharacteristic, complemented
514C3C2×C3×#92611yescharacteristic, complemented
515C2—64no—
516C2C3×#6385514yesdirect product factor
517C2C2×C3×#379812yes—
518C2C22×C3×D3×Dic311yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C32×D3C22
cosocleC22×C32C24×C3
hypocenterC32C2×C3×#117
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C3⋊Dic3C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. C2×C3×Dic3⋊Dic3C2semidirect
6. C3×#63855C2direct (N × H)
9. C2×#63855C3direct (N × H)
15. C2×C32×C4×D3C4semidirect
17. C3×C4×C3⋊Dic3C22semidirect
18. C32×C4×Dic3C22semidirect
19. C3×Dic3⋊Dic3C22semidirect
26. C2×C4×C3⋊Dic3C2×C3semidirect
36. C2×C3×C4×Dic3D3semidirect
37. C2×C3×C4×Dic3C2×C3semidirect
39. C2×Dic3⋊Dic3C2×C3semidirect
41. #63855C2×C3direct (N × H)
54. C32×C4×D3C2×C4semidirect
85. C2×C3×C4×D3Dic3semidirect
86. C2×C3×C4×D3C3×C4semidirect
90. C4×C3⋊Dic3C22×C3semidirect
99. C3×C4×Dic3C2×D3semidirect
100. C3×C4×Dic3C22×C3semidirect
101. Dic3⋊Dic3C22×C3semidirect
132. C2×C4×Dic3C3×D3semidirect
193. C3×C4×D3C2×Dic3semidirect
194. C3×C4×D3C2×C3×C4semidirect
196. C2×C33#117semidirect
229. C2×C4×D3C3×Dic3semidirect
244. C4×Dic3C2×C3×D3semidirect
303. C33C2×#117semidirect
364. C4×D3C2×C3×Dic3semidirect
378. C2×C32#872semidirect
379. C2×C32#926semidirect
380. C2×C32C3×#117semidirect
457. C32C2×#872semidirect
458. C32C2×#926semidirect
459. C32C2×C3×#117semidirect
488. C2×C3#63855direct (N × H)
489. C2×C3C3×#872semidirect
490. C2×C3C3×#926semidirect
512. C3C2×#63855direct (N × H)
513. C3C2×C3×#872semidirect
514. C3C2×C3×#926semidirect
516. C2C3×#63855direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C4×D32
C2×C3×C4×C3⋊Dic32
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C2×C3×(C2×D3)⋊Dic32
C3×#638552
C2×C3×#9263
C2×C3×#8723
C2×#638553