Finite Group Databasebeta

← C2×C3×#64148 #13179544

Subgroups — C2×C3×#64148 #13179544

Order 1728.

3500 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×D4C211yesmaximal, characteristic, complemented
2C22×C3×#638C211yesmaximal, characteristic, complemented
3C22×C3×#638C211yesmaximal, characteristic, complemented
4C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×C3×#3757C211yesmaximal, characteristic, complemented
6C3×#64148C218yesmaximal, direct product factor
7C2×C3×#3752C211yesmaximal, characteristic, complemented
8C2×C3×#3750C211yesmaximal, characteristic, complemented
9C2×#64148C311yesmaximal, characteristic, direct product factor
10C2×C3×#939—34nomaximal
11C24×C33C2211yescharacteristic
12C24×C33C2211yescharacteristic
13C23×C3×C3⋊D3C2211yescharacteristic
14C2×C33×D4—24no—
15C2×C33×D4C2214yescomplemented
16C22×C33×C4C2211yescharacteristic
17C2×C3×#638—24no—
18C2×C3×#638—24no—
19C2×C3×#638C2214yescomplemented
20C2×C3×#638C2214yescomplemented
21C22×C3×C3⋊Dic3—24no—
22C22×C3×C3⋊Dic3C2211yescharacteristic
23C22×C3×C3⋊Dic3C2211yescharacteristic
24C22×C3×C3⋊Dic3C2211yescharacteristic
25C22×C3×C3⋊Dic3C2214yescomplemented
26C3×#3757C2214yescomplemented
27C3×#3752C2214yescomplemented
28C3×#3750C2214yescomplemented
29C22×C32×D4—24no—
30C22×C32×D4D314yescomplemented
31C22×C32×D4C2×C311yescharacteristic, complemented
32C22×#638C2×C311yescharacteristic, complemented
33C22×#638C2×C311yescharacteristic, complemented
34C23×C3⋊Dic3C2×C311yescharacteristic, complemented
35C22×C3×#67—34no—
36C22×C3×#67—34no—
37C23×C3×Dic3—34no—
38C2×#3757C2×C311yescharacteristic, complemented
39#64148C2×C318yesdirect product factor
40C2×#3752C2×C311yescharacteristic, complemented
41C2×#3750C2×C311yescharacteristic, complemented
42C2×C3×#217—34no—
43C3×#939—332no—
44C2×C3×(C2×D3)⋊C4—34no—
45C2×C3×Dic3⋊C4—34no—
46C23×C33—22no—
47C23×C33—22no—
48C23×C33—22no—
49C23×C33—22no—
50C23×C33—22no—
51C23×C33D414yescomplemented
52C23×C33C2312yes—
53C23×C33C2312yes—
54C23×C33C2311yescharacteristic
55C22×C3×C3⋊D3—22no—
56C22×C3×C3⋊D3—22no—
57C22×C3×C3⋊D3—22no—
58C22×C3×C3⋊D3C2312yes—
59C33×D4—28no—
60C2×C33×C4—22no—
61C2×C33×C4C2312yes—
62C3×#638—28no—
63C3×#638—28no—
64C2×C3×C3⋊Dic3—22no—
65C2×C3×C3⋊Dic3—22no—
66C2×C3×C3⋊Dic3—28no—
67C2×C3×C3⋊Dic3D414yescomplemented
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×#121—91no—
72C2×#939—34no—
73C24×C32—24no—
74C24×C32—24no—
75C24×C32C2×D314yes—
76C24×C32C2×D314yes—
77C24×C32C22×C311yescharacteristic
78C24×C32C22×C311yescharacteristic
79C23×C3⋊D3C22×C311yescharacteristic
80C23×C3×D3—34no—
81C2×C32×D4—232no—
82C2×C32×D4—216no—
83C2×C32×D4—216no—
84C2×C32×D4—24no—
85C2×C32×D4C2×D3116yescomplemented
86C2×C32×D4C22×C314yescomplemented
87C22×C32×C4—24no—
88C22×C32×C4C2×D314yes—
89C22×C32×C4C22×C311yescharacteristic
90C2×#638—24no—
91C2×#638—24no—
92C2×#638C22×C314yescomplemented
93C2×#638C22×C314yescomplemented
94C22×C3⋊Dic3—24no—
95C22×C3⋊Dic3C22×C311yescharacteristic
96C22×C3⋊Dic3C22×C311yescharacteristic
97C22×C3⋊Dic3C22×C311yescharacteristic
98C22×C3⋊Dic3C22×C314yescomplemented
99C2×C3×#67—316no—
100C2×C3×#67—316no—
101C2×C3×#67—616no—
102C2×C3×#67—616no—
103C22×C3×Dic3—34no—
104C22×C3×Dic3—34no—
105C22×C3×Dic3—34no—
106C22×C3×Dic3—316no—
107C22×C3×Dic3—616no—
108#3757C22×C314yescomplemented
109#3752C22×C314yescomplemented
110#3750C22×C314yescomplemented
111C3×#217—316no—
112C3×(C2×D3)⋊C4—316no—
113C3×Dic3⋊C4—316no—
114C22×C33—28no—
115C22×C33—22no—
116C22×C33—22no—
117C22×C33—22no—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—44no—
121C22×C33C2×D412yes—
122C22×C33C2×D412yes—
123C22×C33C2×D414yescomplemented
124C22×C33#4112yes—
125C22×C33C2411yescharacteristic
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—44no—
130C33×C4—22no—
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3—22no—
133C3×C3⋊Dic3C2×D414yescomplemented
134C22×C3×D4—24no—
135C22×C3×D4—91no—
136C22×C3×D4—91no—
137C22×C3×D4C3×D314yescomplemented
138C22×C3×D4C3⋊D311yescharacteristic, complemented
139C23×C3×C4—91no—
140C22×#67—34no—
141C22×#67—34no—
142C23×Dic3—34no—
143C3×#121—98no—
144C2×C3×C4⋊C4—91no—
145C2×C3×C22⋊C4—91no—
146C2×C3×C22⋊C4—91no—
147C2×#217—34no—
148#939—332no—
149C2×(C2×D3)⋊C4—34no—
150C2×Dic3⋊C4—34no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—216no—
154C23×C32—28no—
155C23×C32—22no—
156C23×C32—22no—
157C23×C32—22no—
158C23×C32—216no—
159C23×C32—24no—
160C23×C32—28no—
161C23×C32—28no—
162C23×C32—28no—
163C23×C32—28no—
164C23×C32—28no—
165C23×C32—28no—
166C23×C32—48no—
167C23×C32—48no—
168C23×C32—48no—
169C23×C32—48no—
170C23×C32#67116yescomplemented
171C23×C32C3×D414yescomplemented
172C23×C32C22×D318yes—
173C23×C32C22×D318yes—
174C23×C32C22×D314yes—
175C23×C32C23×C312yes—
176C23×C32C23×C312yes—
177C23×C32C23×C311yescharacteristic
178C22×C3⋊D3—22no—
179C22×C3⋊D3—22no—
180C22×C3⋊D3—22no—
181C22×C3⋊D3C23×C312yes—
182C22×C3×D3—38no—
183C22×C3×D3—68no—
184C22×C3×D3—68no—
185C22×C3×D3—68no—
186C32×D4—264no—
187C32×D4—232no—
188C32×D4—28no—
189C2×C32×C4—216no—
190C2×C32×C4—28no—
191C2×C32×C4—22no—
192C2×C32×C4—28no—
193C2×C32×C4C22×D318yes—
194C2×C32×C4C23×C312yes—
195#638—28no—
196#638—28no—
197C2×C3⋊Dic3—22no—
198C2×C3⋊Dic3—22no—
199C2×C3⋊Dic3—28no—
200C2×C3⋊Dic3C3×D414yescomplemented
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C312yes—
204C3×#67—632no—
205C3×#67—632no—
206C2×C3×Dic3—316no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—38no—
210C2×C3×Dic3—632no—
211C2×C3×Dic3—68no—
212C2×C3×Dic3—68no—
213C2×#121—91noSylow
214C2×C33—24no—
215C2×C33—42no—
216C2×C33#12114yescomplemented
217C2×C33C22×D411yescharacteristic
218C2×C33C22×D411yescharacteristic
219C2×C33C2×#4111yescharacteristic
220C3×C3⋊D3—42no—
221C24×C3—24no—
222C24×C3—24no—
223C24×C3—91no—
224C24×C3C2×C3×D314yes—
225C24×C3C2×C3×D314yes—
226C24×C3C2×C3⋊D311yescharacteristic
227C24×C3C2×C3⋊D311yescharacteristic
228C23×D3—34no—
229C2×C3×D4—232no—
230C2×C3×D4—216no—
231C2×C3×D4—24no—
232C2×C3×D4—216no—
233C2×C3×D4—94no—
234C2×C3×D4—94no—
235C2×C3×D4—184no—
236C2×C3×D4—184no—
237C2×C3×D4C2×C3×D3116yescomplemented
238C2×C3×D4C2×C3⋊D314yescomplemented
239C22×C3×C4—24no—
240C22×C3×C4—91no—
241C22×C3×C4—94no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C22×C3×C4—184no—
245C22×C3×C4C2×C3×D314yes—
246C22×C3×C4C2×C3⋊D311yescharacteristic
247C2×#67—316no—
248C2×#67—316no—
249C2×#67—616no—
250C2×#67—616no—
251C22×Dic3—34no—
252C22×Dic3—34no—
253C22×Dic3—34no—
254C22×Dic3—316no—
255C22×Dic3—616no—
256C3×C4⋊C4—94no—
257C3×C22⋊C4—94no—
258C3×C22⋊C4—94no—
259#217—316no—
260(C2×D3)⋊C4—316no—
261Dic3⋊C4—316no—
262C22×C32—28no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—216no—
266C22×C32—28no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—28no—
272C22×C32—28no—
273C22×C32—28no—
274C22×C32—216no—
275C22×C32—232no—
276C22×C32—28no—
277C22×C32—28no—
278C22×C32—28no—
279C22×C32—28no—
280C22×C32—432no—
281C22×C32—48no—
282C22×C32—48no—
283C22×C32—48no—
284C22×C32—48no—
285C22×C32—432no—
286C22×C32—416no—
287C22×C32—44no—
288C22×C32D3×D418yes—
289C22×C32#23718yes—
290C22×C32C2×#6718yes—
291C22×C32C2×#67116yescomplemented
292C22×C32C2×C3×D412yes—
293C22×C32C2×C3×D412yes—
294C22×C32C2×C3×D414yescomplemented
295C22×C32C3×#4112yes—
296C22×C32C23×D314yes—
297C22×C32C24×C311yescharacteristic
298C2×C3⋊D3—22no—
299C2×C3⋊D3—22no—
300C2×C3⋊D3—22no—
301C2×C3⋊D3—44no—
302C2×C3×D3—68no—
303C2×C3×D3—68no—
304C2×C3×D3—68no—
305C2×C3×D3—1216no—
306C32×C4—216no—
307C32×C4—28no—
308C32×C4—22no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3C2×C3×D414yescomplemented
312C3×Dic3—316no—
313C3×Dic3—68no—
314C3×Dic3—68no—
315C22×D4—91no—
316C22×D4—91no—
317C22×D4C3×C3⋊D311yescharacteristic, complemented
318C23×C4—91no—
319#121—98no—
320C2×C4⋊C4—91no—
321C2×C22⋊C4—91no—
322C2×C22⋊C4—91no—
323C33C2×#12111yescharacteristic, complemented, Sylow
324C23×C3—28no—
325C23×C3—28no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—216no—
329C23×C3—28no—
330C23×C3—28no—
331C23×C3—28no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—216no—
336C23×C3—24no—
337C23×C3—28no—
338C23×C3—28no—
339C23×C3—48no—
340C23×C3—48no—
341C23×C3—48no—
342C23×C3—48no—
343C23×C3—92no—
344C23×C3—182no—
345C23×C3—182no—
346C23×C3—182no—
347C23×C3C3×#67116yescomplemented
348C23×C3#63814yescomplemented
349C23×C3C22×C3×D318yes—
350C23×C3C22×C3×D318yes—
351C23×C3C22×C3×D314yes—
352C23×C3C22×C3⋊D312yes—
353C23×C3C22×C3⋊D312yes—
354C23×C3C22×C3⋊D311yescharacteristic
355C22×D3—38no—
356C22×D3—68no—
357C22×D3—68no—
358C22×D3—68no—
359C3×D4—264no—
360C3×D4—232no—
361C3×D4—28no—
362C3×D4—188no—
363C3×D4—188no—
364C2×C3×C4—216no—
365C2×C3×C4—28no—
366C2×C3×C4—22no—
367C2×C3×C4—28no—
368C2×C3×C4—92no—
369C2×C3×C4—94no—
370C2×C3×C4—92no—
371C2×C3×C4—92no—
372C2×C3×C4—188no—
373C2×C3×C4—182no—
374C2×C3×C4—182no—
375C2×C3×C4C22×C3×D318yes—
376C2×C3×C4C22×C3⋊D312yes—
377#67—632no—
378#67—632no—
379C2×Dic3—38no—
380C2×Dic3—316no—
381C2×Dic3—38no—
382C2×Dic3—38no—
383C2×Dic3—632no—
384C2×Dic3—68no—
385C2×Dic3—68no—
386C2×C32—24no—
387C2×C32—216no—
388C2×C32—24no—
389C2×C32—24no—
390C2×C32—24no—
391C2×C32—216no—
392C2×C32—416no—
393C2×C32—416no—
394C2×C32—48no—
395C2×C32—42no—
396C2×C32#939116yescomplemented
397C2×C32C3×#12114yescomplemented
398C2×C32C2×D3×D414yes—
399C2×C32C2×#23714yes—
400C2×C32C22×#6714yes—
401C2×C32C22×C3×D411yescharacteristic
402C2×C32C22×C3×D411yescharacteristic
403C2×C32C2×C3×#4111yescharacteristic
404C3⋊D3—42no—
405C3×D3—128no—
406C24—91no—
407C24C2×C3×C3⋊D311yescharacteristic
408C24C2×C3×C3⋊D311yescharacteristic
409C2×D4—24no—
410C2×D4—94no—
411C2×D4—94no—
412C2×D4—184no—
413C2×D4—184no—
414C2×D4C2×C3×C3⋊D314yescomplemented
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—94no—
419C22×C4—184no—
420C22×C4C2×C3×C3⋊D311yescharacteristic
421C4⋊C4—94no—
422C22⋊C4—94no—
423C22⋊C4—94no—
424C22×C3—232no—
425C22×C3—28no—
426C22×C3—28no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—28no—
430C22×C3—216no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—28no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—28no—
439C22×C3—28no—
440C22×C3—28no—
441C22×C3—216no—
442C22×C3—432no—
443C22×C3—48no—
444C22×C3—48no—
445C22×C3—48no—
446C22×C3—48no—
447C22×C3—432no—
448C22×C3—416no—
449C22×C3—44no—
450C22×C3—182no—
451C22×C3—182no—
452C22×C3—182no—
453C22×C3—364no—
454C22×C3C3×D3×D418yes—
455C22×C3C3×#23718yes—
456C22×C3C2×C3×#6718yes—
457C22×C3C2×C3×#67116yescomplemented
458C22×C3D4×C3⋊D312yes—
459C22×C3#383012yes—
460C22×C3C2×#63812yes—
461C22×C3C2×#63814yescomplemented
462C22×C3C23×C3×D314yes—
463C22×C3C23×C3⋊D311yescharacteristic
464C2×D3—68no—
465C2×D3—68no—
466C2×D3—68no—
467C2×D3—1216no—
468C3×C4—216no—
469C3×C4—28no—
470C3×C4—22no—
471C3×C4—94no—
472C3×C4—182no—
473C3×C4—182no—
474Dic3—316no—
475Dic3—68no—
476Dic3—68no—
477C32—24no—
478C32C2×#93914yescomplemented
479C32C2×C3×#12111yescharacteristic, complemented
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—92no—
486C23—182no—
487C23—182no—
488C23—182no—
489C23C3×#63814yescomplemented
490C23C22×C3×C3⋊D312yes—
491C23C22×C3×C3⋊D312yes—
492C23C22×C3×C3⋊D311yescharacteristic
493D4—28no—
494D4—188no—
495D4—188no—
496C2×C4—22no—
497C2×C4—92no—
498C2×C4—94no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—188no—
502C2×C4—182no—
503C2×C4—182no—
504C2×C4C22×C3×C3⋊D312yes—
505C2×C3—216no—
506C2×C3—24no—
507C2×C3—216no—
508C2×C3—24no—
509C2×C3—24no—
510C2×C3—24no—
511C2×C3—416no—
512C2×C3—416no—
513C2×C3—48no—
514C2×C3—42no—
515C2×C3—362no—
516C2×C3C3×#939116yescomplemented
517C2×C3#6414814yesdirect product factor
518C2×C3C2×C3×D3×D414yes—
519C2×C3C2×C3×#23714yes—
520C2×C3C22×C3×#6714yes—
521C2×C3C2×D4×C3⋊D311yescharacteristic
522C2×C3C2×#383011yescharacteristic
523C2×C3C22×#63811yescharacteristic
524D3—128no—
525C22—28no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—44no—
532C22—182no—
533C22—182no—
534C22—182no—
535C22—364no—
536C22C3×D4×C3⋊D312yes—
537C22C3×#383012yes—
538C22C2×C3×#63812yes—
539C22C2×C3×#63814yescomplemented
540C22C23×C3×C3⋊D311yescharacteristic
541C4—22no—
542C4—94no—
543C4—182no—
544C4—182no—
545C3—24no—
546C3C2×C3×#93914yescomplemented
547C3C2×#6414811yescharacteristic, direct product factor
548C2—24no—
549C2—42no—
550C2—362no—
551C2C3×#6414814yesdirect product factor
552C2C2×C3×D4×C3⋊D311yescharacteristic
553C2C2×C3×#383011yescharacteristic
554C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC22×C33×D4C2
socleC23×C33C23
hypercenterC22×C3×D4C3⋊D3
hypocenterC32C2×C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C33×D4C2semidirect
2. C22×C3×#638C2semidirect
3. C22×C3×#638C2semidirect
4. C23×C3×C3⋊Dic3C2semidirect
5. C2×C3×#3757C2semidirect
6. C3×#64148C2direct (N × H)
7. C2×C3×#3752C2semidirect
8. C2×C3×#3750C2semidirect
9. C2×#64148C3direct (N × H)
15. C2×C33×D4C22semidirect
19. C2×C3×#638C22semidirect
20. C2×C3×#638C22semidirect
25. C22×C3×C3⋊Dic3C22semidirect
26. C3×#3757C22semidirect
27. C3×#3752C22semidirect
28. C3×#3750C22semidirect
30. C22×C32×D4D3semidirect
31. C22×C32×D4C2×C3semidirect
32. C22×#638C2×C3semidirect
33. C22×#638C2×C3semidirect
34. C23×C3⋊Dic3C2×C3semidirect
38. C2×#3757C2×C3semidirect
39. #64148C2×C3direct (N × H)
40. C2×#3752C2×C3semidirect
41. C2×#3750C2×C3semidirect
51. C23×C33D4semidirect
67. C2×C3×C3⋊Dic3D4semidirect
85. C2×C32×D4C2×D3semidirect
86. C2×C32×D4C22×C3semidirect
92. C2×#638C22×C3semidirect
93. C2×#638C22×C3semidirect
98. C22×C3⋊Dic3C22×C3semidirect
108. #3757C22×C3semidirect
109. #3752C22×C3semidirect
110. #3750C22×C3semidirect
123. C22×C33C2×D4semidirect
133. C3×C3⋊Dic3C2×D4semidirect
137. C22×C3×D4C3×D3semidirect
138. C22×C3×D4C3⋊D3semidirect
170. C23×C32#67semidirect
171. C23×C32C3×D4semidirect
200. C2×C3⋊Dic3C3×D4semidirect
216. C2×C33#121semidirect
237. C2×C3×D4C2×C3×D3semidirect
238. C2×C3×D4C2×C3⋊D3semidirect
291. C22×C32C2×#67semidirect
294. C22×C32C2×C3×D4semidirect
311. C3⋊Dic3C2×C3×D4semidirect
317. C22×D4C3×C3⋊D3semidirect
323. C33C2×#121semidirect
347. C23×C3C3×#67semidirect
348. C23×C3#638semidirect
396. C2×C32#939semidirect
397. C2×C32C3×#121semidirect
414. C2×D4C2×C3×C3⋊D3semidirect
457. C22×C3C2×C3×#67semidirect
461. C22×C3C2×#638semidirect
478. C32C2×#939semidirect
479. C32C2×C3×#121semidirect
489. C23C3×#638semidirect
516. C2×C3C3×#939semidirect
517. C2×C3#64148direct (N × H)
539. C22C2×C3×#638semidirect
546. C3C2×C3×#939semidirect
547. C3C2×#64148direct (N × H)
551. C2C3×#64148direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×D42
C22×C3×#6382
C23×C3×C3⋊Dic32
C2×C3×#37572
C3×#641482
C2×C3×#37522
C2×C3×#37502
C2×#641483
C2×C3×#9393