Finite Group Databasebeta

← C3×#10591315 #13158235

Subgroups — C3×#10591315 #13158235

Order 1728.

1034 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×C3×(C2×D3)⋊Dic3C212yesmaximal
4C32×C3⋊#95C211yesmaximal, characteristic, complemented
5C3×#63579C212yesmaximal, complemented
6C3×#4953—31nomaximal
7C3×#4674—31nomaximal
8#10591315C311yesmaximal, characteristic, direct product factor
9C23×C32×D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C3×C4⋊(C3⋊Dic3)—22no—
15C32×(C2×D3)⋊C4—22no—
16C3×(C2×D3)⋊Dic3—24no—
17C2×C32×C22⋊C4—31no—
18C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
19C2×C3×#217—32no—
20C2×C3×(C2×D3)⋊C4—21no—
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4D311yescharacteristic
23C2×C3×(C2×D3)⋊C4C2×C311yescharacteristic
24C2×C3×C4⋊Dic3—31no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—61no—
27C2×(C2×D3)⋊Dic3C2×C312yes—
28C32×#95—31no—
29C3×C3⋊#95—21no—
30C3×C3⋊#95—32no—
31C3×C3⋊#95—32no—
32C3×C3⋊#95D311yescharacteristic, complemented
33C3×C3⋊#95C2×C311yescharacteristic, complemented
34#63579C2×C312yescomplemented
35C23×C33C2311yescharacteristic
36C22×C32×D3—21no—
37C22×C32×D3—21no—
38C22×C32×D3—22no—
39C22×C32×D3—22no—
40C22×C32×D3D412yes—
41C2×C33×C4—22no—
42C2×C33×C4D412yes—
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—22no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—22no—
50C2×C32×Dic3—22no—
51C3×#397—91no—
52#4953—31no—
53#4674—31no—
54C24×C32—31no—
55C23×C3×D3—21no—
56C23×C3×D3C2×D311yescharacteristic
57C23×C3×D3C22×C311yescharacteristic
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—32no—
62C22×C32×C4—31no—
63C22×C32×C4—41no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C312yes—
68C22×C3×Dic3—21no—
69C22×C3×Dic3—22no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—62no—
73C22×C3×Dic3C2×D311yescharacteristic
74C22×C3×Dic3C2×D312yes—
75C22×C3×Dic3C22×C311yescharacteristic
76C22×C3×Dic3C22×C312yes—
77C32×C22⋊C4—62no—
78C4⋊(C3⋊Dic3)—22no—
79C3×#217—64no—
80C3×(C2×D3)⋊C4—24no—
81C3×(C2×D3)⋊C4—22no—
82C3×(C2×D3)⋊C4—22no—
83C3×(C2×D3)⋊C4—64no—
84C3×C4⋊Dic3—64no—
85C3×C4⋊Dic3—62no—
86C3×C4⋊Dic3—62no—
87(C2×D3)⋊Dic3—24no—
88C22×C33C2×D411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C22×C33#4111yescharacteristic
91C22×C33#4112yes—
92C22×C33#4112yes—
93C2×C32×D3—22no—
94C2×C32×D3—24no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—42no—
98C33×C4—22no—
99C3×C3⋊Dic3—42no—
100C32×Dic3—42no—
101C32×Dic3—41no—
102C2×C3×C4⋊C4—91no—
103C2×C3×C22⋊C4—31no—
104C2×C3×C22⋊C4—31no—
105C2×C3×C22⋊C4—61no—
106C2×C3×C22⋊C4—92no—
107C2×#217—32no—
108C2×(C2×D3)⋊C4—32no—
109C2×(C2×D3)⋊C4C3×D311yescharacteristic
110C2×C4⋊Dic3—31no—
111C2×C4⋊Dic3—31no—
112C2×C4⋊Dic3—61no—
113C3×#95—31no—
114C3×#95—31no—
115C3×#95—61no—
116C3×#95—92no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95C3×D311yescharacteristic, complemented
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—32no—
124C23×C32—41no—
125C23×C32—61no—
126C23×C32—61no—
127C23×C32—62no—
128C23×C32—62no—
129C23×C32C22×D311yescharacteristic
130C23×C32C22×D311yescharacteristic
131C23×C32C23×C311yescharacteristic
132C22×C3×D3—22no—
133C22×C3×D3—22no—
134C22×C3×D3—21no—
135C22×C3×D3—22no—
136C22×C3×D3—22no—
137C22×C3×D3—21no—
138C22×C3×D3—21no—
139C22×C3×D3—21no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—42no—
144C22×C3×D3—42no—
145C22×C3×D3D1212yes—
146C22×C3×D3C3×D412yes—
147C2×C32×C4—22no—
148C2×C32×C4—24no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—42no—
155C2×C32×C4—42no—
156C2×C32×C4—42no—
157C2×C32×C4—44no—
158C2×C32×C4—61no—
159C2×C32×C4—62no—
160C2×C32×C4—62no—
161C2×C32×C4—62no—
162C2×C32×C4—62no—
163C2×C32×C4D1212yes—
164C2×C32×C4#6712yes—
165C2×C32×C4C3×D412yes—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—24no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—42no—
182C2×C3×Dic3—42no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—122no—
192C2×C3×Dic3—122no—
193#397—91noSylow
194C2×C33#12111yescharacteristic
195C2×C33#12312yes—
196C2×C33#12311yescharacteristic
197C2×C33#12411yescharacteristic
198C2×C33#12612yes—
199C32×D3—42no—
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—32no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—32no—
211C22×C3×C4—41no—
212C22×C3×C4—62no—
213C22×C3×C4—61no—
214C22×C3×C4—92no—
215C22×C3×C4D3211yescharacteristic
216C22×C3×C4C2×C3×D311yescharacteristic
217C22×C3×C4C2×C3×D311yescharacteristic
218C22×Dic3—32no—
219C22×Dic3—32no—
220C22×Dic3—62no—
221C22×Dic3C2×C3×D311yescharacteristic
222C22×Dic3C2×C3×D312yes—
223C3×C4⋊C4—182no—
224C3×C22⋊C4—64no—
225C3×C22⋊C4—62no—
226C3×C22⋊C4—62no—
227C3×C22⋊C4—184no—
228#217—64no—
229(C2×D3)⋊C4—22no—
230(C2×D3)⋊C4—64no—
231C4⋊Dic3—64no—
232C4⋊Dic3—62no—
233C4⋊Dic3—62no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—22no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—41no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—41no—
254C22×C32—62no—
255C22×C32—64no—
256C22×C32—121no—
257C22×C32—121no—
258C22×C32—122no—
259C22×C32C2×D1211yescharacteristic
260C22×C32C2×D1211yescharacteristic
261C22×C32#23512yes—
262C22×C32#23511yescharacteristic
263C22×C32D3×D411yescharacteristic
264C22×C32#23712yes—
265C22×C32#23712yes—
266C22×C32#23712yes—
267C22×C32#23911yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C22×C32C2×C3×D411yescharacteristic
270C22×C32C2×C3×D411yescharacteristic
271C22×C32C3×#4112yes—
272C22×C32C3×#4111yescharacteristic
273C22×C32C3×#4112yes—
274C2×C3×D3—24no—
275C2×C3×D3—24no—
276C2×C3×D3—22no—
277C2×C3×D3—22no—
278C2×C3×D3—24no—
279C2×C3×D3—44no—
280C2×C3×D3—44no—
281C2×C3×D3—42no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C2×C3×D3—42no—
289C32×C4—22no—
290C32×C4—24no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—42no—
294C32×C4—42no—
295C32×C4—44no—
296C32×C4—121no—
297C32×C4—122no—
298C3⋊Dic3—42no—
299C3×Dic3—44no—
300C3×Dic3—42no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—124no—
306C3×Dic3—122no—
307C3×Dic3—122no—
308C2×C4⋊C4—91no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—92no—
311#95—31no—
312#95—92no—
313C33#39711yescharacteristic, complemented, Sylow
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—32no—
318C23×C3—32no—
319C23×C3—41no—
320C23×C3—62no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—62no—
324C23×C3—61no—
325C23×C3—62no—
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—61no—
331C23×C3—122no—
332C23×C3—122no—
333C23×C3C2×D3211yescharacteristic
334C23×C3C22×C3×D311yescharacteristic
335C23×C3C22×C3×D311yescharacteristic
336C22×D3—21no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—21no—
340C22×D3C3×D1212yes—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—44no—
351C2×C3×C4—42no—
352C2×C3×C4—62no—
353C2×C3×C4—64no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—61no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—122no—
365C2×C3×C4—122no—
366C2×C3×C4—122no—
367C2×C3×C4—182no—
368C2×C3×C4—182no—
369C2×C3×C4—182no—
370C2×C3×C4#62612yes—
371C2×C3×C4C3×D1212yes—
372C2×C3×C4C3×#6712yes—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—64no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—62no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—122no—
386C2×Dic3—122no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—21no—
390C2×C32—22no—
391C2×C32—21no—
392C2×C32—22no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—22no—
398C2×C32—22no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—42no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32—122no—
408C2×C32#87511yescharacteristic
409C2×C32#87912yes—
410C2×C32#88312yes—
411C2×C32#88612yes—
412C2×C32#88611yescharacteristic
413C2×C32#89511yescharacteristic
414C2×C32#89812yes—
415C2×C32#93011yescharacteristic
416C2×C32#93511yescharacteristic
417C2×C32#93611yescharacteristic
418C2×C32C3×#12111yescharacteristic
419C2×C32C3×#12312yes—
420C2×C32C3×#12311yescharacteristic
421C2×C32C3×#12411yescharacteristic
422C2×C32C3×#12612yes—
423C3×D3—44no—
424C3×D3—42no—
425C3×D3—42no—
426C24—31no—
427C22×C4—32no—
428C22×C4—31no—
429C22×C4—92no—
430C22×C4C3×D3211yescharacteristic
431C4⋊C4—182no—
432C22⋊C4—62no—
433C22⋊C4—184no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—42no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—41no—
454C22×C3—64no—
455C22×C3—64no—
456C22×C3—62no—
457C22×C3—64no—
458C22×C3—62no—
459C22×C3—124no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3—124no—
463C22×C3—121no—
464C22×C3—122no—
465C22×C3—121no—
466C22×C3—122no—
467C22×C3—121no—
468C22×C3—121no—
469C22×C3#379512yes—
470C22×C3#379911yescharacteristic
471C22×C3D3×D1211yescharacteristic
472C22×C3#380512yes—
473C22×C3C2×#62611yescharacteristic
474C22×C3C2×C3×D1211yescharacteristic
475C22×C3C2×C3×D1211yescharacteristic
476C22×C3C3×#23512yes—
477C22×C3C3×#23511yescharacteristic
478C22×C3C3×D3×D411yescharacteristic
479C22×C3C3×#23712yes—
480C22×C3C3×#23712yes—
481C22×C3C3×#23712yes—
482C22×C3C3×#23911yescharacteristic
483C22×C3C2×C3×#6711yescharacteristic
484C2×D3—24no—
485C2×D3—22no—
486C2×D3—42no—
487C2×D3—41no—
488C2×D3—41no—
489C3×C4—24no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—122no—
497C3×C4—124no—
498C3×C4—122no—
499C3×C4—121no—
500C3×C4—122no—
501C3×C4—121no—
502C3×C4—362no—
503Dic3—42no—
504Dic3—41no—
505Dic3—124no—
506Dic3—122no—
507Dic3—122no—
508C32—21no—
509C32—21no—
510C32—21no—
511C32—41no—
512C32#467411yescharacteristic, complemented
513C32#495311yescharacteristic, complemented
514C32C3×#39711yescharacteristic, complemented
515C23—32no—
516C23—62no—
517C23—61no—
518C23—62no—
519C23—61no—
520C23C2×C3×D3211yescharacteristic
521C2×C4—22no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—61no—
527C2×C4—182no—
528C2×C4—182no—
529C2×C4—182no—
530C2×C4C3×#62612yes—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—41no—
551C2×C3—124no—
552C2×C3—122no—
553C2×C3—122no—
554C2×C3#6386011yescharacteristic
555C2×C3#6386112yes—
556C2×C3#6389112yes—
557C2×C3#6390911yescharacteristic
558C2×C3#6396211yescharacteristic
559C2×C3C3×#87511yescharacteristic
560C2×C3C3×#87912yes—
561C2×C3C3×#88312yes—
562C2×C3C3×#88611yescharacteristic
563C2×C3C3×#88612yes—
564C2×C3C3×#89511yescharacteristic
565C2×C3C3×#89812yes—
566C2×C3C3×#93011yescharacteristic
567C2×C3C3×#93511yescharacteristic
568C2×C3C3×#93611yescharacteristic
569D3—42no—
570C22—64no—
571C22—62no—
572C22—121no—
573C22—121no—
574C22—122no—
575C22C3×#379512yes—
576C22C3×#379911yescharacteristic
577C22C3×D3×D1211yescharacteristic
578C22C3×#380512yes—
579C22C2×C3×#62611yescharacteristic
580C4—22no—
581C4—121no—
582C4—122no—
583C4—362no—
584C3—21no—
585C3—21no—
586C3—21no—
587C3—41no—
588C3#1059131511yescharacteristic, direct product factor
589C3C3×#467411yescharacteristic, complemented
590C3C3×#495311yescharacteristic, complemented
591C2—122no—
592C2C3×#6386011yescharacteristic
593C2C3×#6386112yes—
594C2C3×#6389112yes—
595C2C3×#6390911yescharacteristic
596C2C3×#6396211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C32×D3C22
hypercenterC22×C3×C4D32
hypocenterC32C3×#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
4. C32×C3⋊#95C2semidirect
5. C3×#63579C2semidirect
8. #10591315C3direct (N × H)
18. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
32. C3×C3⋊#95D3semidirect
33. C3×C3⋊#95C2×C3semidirect
34. #63579C2×C3semidirect
119. C3⋊#95C3×D3semidirect
313. C33#397semidirect
512. C32#4674semidirect
513. C32#4953semidirect
514. C32C3×#397semidirect
588. C3#10591315direct (N × H)
589. C3C3×#4674semidirect
590. C3C3×#4953semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×(C2×D3)⋊C42
C2×C3×(C2×D3)⋊Dic32
C32×C3⋊#952
C3×#635792
C3×#49533
C3×#46743
#105913153