Finite Group Databasebeta

← C3×#10593863 #13173448

Subgroups — C3×#10593863 #13173448

Order 1728.

1110 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#634C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic
3C32×#869C211yesmaximal, characteristic, complemented
4C3×#63863C212yesmaximal
5C3×#63858C214yesmaximal, complemented
6C3×#63857C212yesmaximal, complemented
7C3×#63843C212yesmaximal, complemented
8C3×#63841C212yesmaximal, complemented
9C3×#5732—31nomaximal
10C3×#5542—31nomaximal
11#10593863C311yesmaximal, characteristic, direct product factor
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3C2212yes—
15C2×C32×Dic6C2212yes—
16C33×C42C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750C2212yes—
19C3×C4×C3⋊Dic3C2212yes—
20C32×(C2×D3)⋊C4C2212yes—
21C32×C4⋊Dic3C2212yes—
22C32×C4⋊Dic3C2211yescharacteristic
23C32×C4⋊Dic3C2212yes—
24C3×Dic3⋊Dic3C2214yes—
25C3×(C2×D3)⋊Dic3C2214yes—
26C3×(C2×D3)⋊Dic3C2214yes—
27C3×Dic32C2214yes—
28C32×C4⋊Q8—31no—
29C32×C4×D4—31no—
30C4×#634C2×C311yescharacteristic, complemented
31C3×#947—32no—
32C3×Q8×Dic3—32no—
33C3×#936—32no—
34C3×#926—32no—
35C3×#924—32no—
36C3×#898—34no—
37C3×#897—32no—
38C3×#892—32no—
39C3×#887—32no—
40C3×#879—34no—
41C3×C4×D12—21no—
42C3×C4×D12D311yescharacteristic
43C3×C4×D12C2×C311yescharacteristic
44C3×#869—21no—
45C3×#869D311yescharacteristic, complemented
46C3×#869C2×C311yescharacteristic, complemented
47C3×C4×Dic6—31no—
48C3×C4×Dic6—31no—
49C3×C4×Dic6—61no—
50#63863C2×C312yes—
51#63858C2×C314yescomplemented
52#63857C2×C312yescomplemented
53#63843C2×C312yescomplemented
54#63841C2×C312yescomplemented
55C22×C32×D3C2312yes—
56C2×C33×C4C2311yescharacteristic
57C2×C33×C4C2311yescharacteristic
58C2×C33×C4C2311yescharacteristic
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2312yes—
61C3×#634—22no—
62C2×C32×Dic3C2312yes—
63C2×C32×Dic3C2314yes—
64C32×D12—22no—
65C32×C4×D3—24no—
66C32×Dic6—24no—
67C3×#556—91no—
68#5732—31no—
69#5542—31no—
70C2×C32×Q8—32no—
71C2×C32×D4—31no—
72C22×C32×C4—32no—
73C2×#634C22×C311yescharacteristic
74C2×C3×D12—21no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C22×C311yescharacteristic
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3C2×D312yes—
79C2×C3×C4×D3C22×C312yes—
80C2×C3×Dic6—22no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D312yes—
85C2×C3×Dic6C22×C312yes—
86C32×C4⋊C4—31no—
87C32×C4⋊C4—32no—
88C32×C4⋊C4—32no—
89C32×C22⋊C4—32no—
90C32×C42—21no—
91C32×C42—21no—
92C32×C42—21no—
93C32×C42—41no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96C32×C42C22×C311yescharacteristic
97C4⋊(C3⋊Dic3)C22×C311yescharacteristic
98#3750C22×C312yes—
99C4×C3⋊Dic3C22×C312yes—
100C3×#217—34no—
101C3×#217—34no—
102C3×(C2×D3)⋊C4—22no—
103C3×(C2×D3)⋊C4—34no—
104C3×(C2×D3)⋊C4—34no—
105C3×(C2×D3)⋊C4C2×D312yes—
106C3×(C2×D3)⋊C4C22×C312yes—
107C3×C4⋊Dic3—22no—
108C3×C4⋊Dic3—22no—
109C3×C4⋊Dic3—21no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—34no—
113C3×C4⋊Dic3—61no—
114C3×C4⋊Dic3C2×D312yes—
115C3×C4⋊Dic3C2×D312yes—
116C3×C4⋊Dic3C2×D311yescharacteristic
117C3×C4⋊Dic3C22×C312yes—
118C3×C4⋊Dic3C22×C311yescharacteristic
119C3×C4⋊Dic3C22×C312yes—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—34no—
123C3×Dic3⋊C4—62no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—34no—
127C3×C4×Dic3—34no—
128C3×C4×Dic3—62no—
129Dic3⋊Dic3C22×C314yes—
130(C2×D3)⋊Dic3C22×C314yes—
131(C2×D3)⋊Dic3C22×C314yes—
132Dic32C22×C314yes—
133C22×C33C2411yescharacteristic
134C2×C32×D3—22no—
135C2×C32×D3—22no—
136C2×C32×D3—22no—
137C33×C4—21no—
138C33×C4#4112yes—
139C33×C4#4112yes—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—22no—
142C32×Dic3—24no—
143C32×Dic3—22no—
144C3×C4⋊Q8—31no—
145C3×C4⋊Q8—31no—
146C3×C4⋊Q8—61no—
147C3×#126—94no—
148C3×#124—92no—
149C3×C22⋊Q8—92no—
150C3×C4×Q8—92no—
151C3×C4×Q8—91no—
152C3×C4×D4—31no—
153C3×C4×D4—31no—
154C3×C4×D4—61no—
155C3×#117—92no—
156#947—32no—
157Q8×Dic3—32no—
158#936—32no—
159#926—32no—
160#924—32no—
161#898—34no—
162#897—32no—
163#892—32no—
164#887—32no—
165#879—34no—
166C4×D12C3×D311yescharacteristic
167#869C3×D311yescharacteristic, complemented
168C4×Dic6—31no—
169C4×Dic6—31no—
170C4×Dic6—61no—
171C23×C32—32no—
172C22×C3×D3—22no—
173C22×C3×D3C22×D312yes—
174C22×C3×D3C23×C312yes—
175C32×Q8—64no—
176C32×D4—62no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—32no—
187C2×C32×C4—34no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—41no—
191C2×C32×C4—64no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C32×C4C23×C311yescharacteristic
200C2×C32×C4C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C312yes—
203#634—22no—
204C2×C3×Dic3—24no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—62no—
212C2×C3×Dic3C22×D312yes—
213C2×C3×Dic3C22×D314yes—
214C2×C3×Dic3C23×C312yes—
215C2×C3×Dic3C23×C314yes—
216C3×D12—24no—
217C3×D12—22no—
218C3×D12—22no—
219C3×C4×D3—24no—
220C3×C4×D3—24no—
221C3×C4×D3—44no—
222C3×Dic6—24no—
223C3×Dic6—24no—
224C3×Dic6—44no—
225C3×Dic6—64no—
226C3×Dic6—62no—
227C3×Dic6—62no—
228#556—91noSylow
229C2×C33C2×#4111yescharacteristic
230C2×C33C2×#4111yescharacteristic
231C2×C332−1+411yescharacteristic
232C32×D3—42no—
233C2×C3×Q8—32no—
234C2×C3×Q8—32no—
235C2×C3×Q8—62no—
236C2×C3×Q8—91no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C22×C3×C4—32no—
241C22×C3×C4—32no—
242C22×C3×C4—62no—
243C2×D12C2×C3×D311yescharacteristic
244C2×C4×D3C2×C3×D312yes—
245C2×Dic6—31no—
246C2×Dic6—31no—
247C2×Dic6—61no—
248C2×Dic6C2×C3×D312yes—
249C3×C4⋊C4—32no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—32no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—32no—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—62no—
257C3×C4⋊C4—62no—
258C3×C4⋊C4—94no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—91no—
261C3×C22⋊C4—32no—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—94no—
265C3×C22⋊C4—94no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—41no—
270C3×C42—94no—
271C3×C42—92no—
272C3×C42D3211yescharacteristic
273C3×C42C2×C3×D311yescharacteristic
274C3×C42C2×C3×D311yescharacteristic
275#217—34no—
276#217—34no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—34no—
279(C2×D3)⋊C4C2×C3×D312yes—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—34no—
283C4⋊Dic3—61no—
284C4⋊Dic3C2×C3×D312yes—
285C4⋊Dic3C2×C3×D312yes—
286C4⋊Dic3C2×C3×D311yescharacteristic
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—34no—
290Dic3⋊C4—62no—
291C4×Dic3—32no—
292C4×Dic3—32no—
293C4×Dic3—34no—
294C4×Dic3—34no—
295C4×Dic3—62no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—41no—
300C22×C32—62no—
301C22×C32—62no—
302C22×C32—62no—
303C22×C32C23×D311yescharacteristic
304C22×C32C23×D311yescharacteristic
305C22×C32C24×C311yescharacteristic
306C2×C3×D3—24no—
307C2×C3×D3—22no—
308C2×C3×D3—22no—
309C2×C3×D3—22no—
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C2×C3×D3—22no—
313C2×C3×D3—42no—
314C2×C3×D3—42no—
315C32×C4—21no—
316C32×C4—22no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—22no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—22no—
325C32×C4—42no—
326C32×C4—42no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—42no—
330C32×C4—64no—
331C32×C4—62no—
332C32×C4#23512yes—
333C32×C4#23712yes—
334C32×C4#23712yes—
335C32×C4#23912yes—
336C32×C4C3×#4112yes—
337C32×C4C3×#4112yes—
338C3⋊Dic3—22no—
339C3⋊Dic3—22no—
340C3×Dic3—24no—
341C3×Dic3—22no—
342C3×Dic3—24no—
343C3×Dic3—22no—
344C3×Dic3—44no—
345C3×Dic3—42no—
346C3×Dic3—64no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—122no—
352C4⋊Q8—31no—
353#126—94no—
354#124—92no—
355C22⋊Q8—92no—
356C4×Q8—92no—
357C4×Q8—91no—
358C4×D4—31no—
359#117—92no—
360C33#55611yescharacteristic, complemented, Sylow
361C23×C3—32no—
362C23×C3—32no—
363C23×C3—62no—
364C22×D3C22×C3×D312yes—
365C3×Q8—64no—
366C3×Q8—64no—
367C3×Q8—124no—
368C3×Q8—182no—
369C3×D4—64no—
370C3×D4—62no—
371C3×D4—62no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—32no—
382C2×C3×C4—34no—
383C2×C3×C4—32no—
384C2×C3×C4—34no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—62no—
389C2×C3×C4—64no—
390C2×C3×C4—64no—
391C2×C3×C4—64no—
392C2×C3×C4—92no—
393C2×C3×C4—92no—
394C2×C3×C4—124no—
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—62no—
409C2×Dic3—62no—
410C2×Dic3C22×C3×D312yes—
411C2×Dic3C22×C3×D314yes—
412D12—22no—
413C4×D3—24no—
414Dic6—24no—
415Dic6—64no—
416Dic6—62no—
417Dic6—62no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—122no—
431C2×C32C2×#23511yescharacteristic
432C2×C32C2×#23711yescharacteristic
433C2×C32C2×#23711yescharacteristic
434C2×C32C2×#23911yescharacteristic
435C2×C32#100711yescharacteristic
436C2×C32#101011yescharacteristic
437C2×C32C2×C3×#4111yescharacteristic
438C2×C32C2×C3×#4111yescharacteristic
439C2×C32C3×2−1+411yescharacteristic
440C3×D3—44no—
441C3×D3—42no—
442C3×D3—42no—
443C2×Q8—32no—
444C2×Q8—91no—
445C2×D4—31no—
446C22×C4—32no—
447C4⋊C4—32no—
448C4⋊C4—31no—
449C4⋊C4—32no—
450C4⋊C4—94no—
451C4⋊C4—91no—
452C4⋊C4—92no—
453C22⋊C4—32no—
454C22⋊C4—94no—
455C22⋊C4—94no—
456C42—94no—
457C42—92no—
458C42C3×D3211yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—64no—
464C22×C3—62no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3C22×D3211yescharacteristic
473C22×C3C23×C3×D311yescharacteristic
474C22×C3C23×C3×D311yescharacteristic
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—22no—
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—64no—
494C3×C4—62no—
495C3×C4—64no—
496C3×C4—62no—
497C3×C4—122no—
498C3×C4—124no—
499C3×C4—182no—
500C3×C4—182no—
501C3×C4#379512yes—
502C3×C4#379612yes—
503C3×C4C3×#23512yes—
504C3×C4C3×#23712yes—
505C3×C4C3×#23712yes—
506C3×C4C3×#23912yes—
507Dic3—24no—
508Dic3—22no—
509Dic3—64no—
510Dic3—62no—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—122no—
515C32—21no—
516C32—21no—
517C32—21no—
518C32—41no—
519C32#554211yescharacteristic, complemented
520C32#573211yescharacteristic, complemented
521C32C3×#55611yescharacteristic, complemented
522C23—32no—
523Q8—64no—
524Q8—182no—
525D4—62no—
526C2×C4—32no—
527C2×C4—34no—
528C2×C4—64no—
529C2×C4—92no—
530C2×C4—92no—
531C2×C4C2×C3×D3211yescharacteristic
532C2×C4C2×C3×D3211yescharacteristic
533C2×C4C2×C3×D3211yescharacteristic
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—124no—
547C2×C3—122no—
548C2×C3—122no—
549C2×C3C2×#379511yescharacteristic
550C2×C3C2×#379611yescharacteristic
551C2×C3#6429711yescharacteristic
552C2×C3C2×C3×#23511yescharacteristic
553C2×C3C2×C3×#23711yescharacteristic
554C2×C3C2×C3×#23711yescharacteristic
555C2×C3C2×C3×#23911yescharacteristic
556C2×C3C3×#100711yescharacteristic
557C2×C3C3×#101011yescharacteristic
558D3—42no—
559C22—62no—
560C22—62no—
561C22—62no—
562C22C22×C3×D3211yescharacteristic
563C4—21no—
564C4—64no—
565C4—62no—
566C4—182no—
567C4—182no—
568C4C3×#379512yes—
569C4C3×#379612yes—
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3#1059386311yescharacteristic, direct product factor
575C3C3×#554211yescharacteristic, complemented
576C3C3×#573211yescharacteristic, complemented
577C2—122no—
578C2C2×C3×#379511yescharacteristic
579C2C2×C3×#379611yescharacteristic
580C2C3×#6429711yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC33×C42C22
socleC22×C33C24
prime radicalC2×C32×D12C22
hypercenterC3×C42D32
hypocenterC32C3×#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#634C2semidirect
3. C32×#869C2semidirect
5. C3×#63858C2semidirect
6. C3×#63857C2semidirect
7. C3×#63843C2semidirect
8. C3×#63841C2semidirect
11. #10593863C3direct (N × H)
30. C4×#634C2×C3semidirect
45. C3×#869D3semidirect
46. C3×#869C2×C3semidirect
51. #63858C2×C3semidirect
52. #63857C2×C3semidirect
53. #63843C2×C3semidirect
54. #63841C2×C3semidirect
167. #869C3×D3semidirect
360. C33#556semidirect
519. C32#5542semidirect
520. C32#5732semidirect
521. C32C3×#556semidirect
574. C3#10593863direct (N × H)
575. C3C3×#5542semidirect
576. C3C3×#5732semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×C4×D122
C32×#8692
C3×#638632
C3×#638582
C3×#638572
C3×#638432
C3×#638412
C3×#57323
C3×#55423
#105938633