Finite Group Databasebeta

← C3×#10591317 #13158294

Subgroups — C3×#10591317 #13158294

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 606 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×Dic3⋊Dic3C212yesmaximal, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal
5C32×C3⋊#95C211yesmaximal, characteristic, complemented
6C3×#4949—31nomaximal
7C3×#4677—31nomaximal
8#10591317C311yesmaximal, 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×Dic3⋊Dic3—24no—
17C3×(C2×D3)⋊Dic3—24no—
18C2×C32×C22⋊C4—31no—
19C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
20C2×C3×#217—32no—
21C2×C3×(C2×D3)⋊C4—21no—
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×(C2×D3)⋊C4D311yescharacteristic
24C2×C3×(C2×D3)⋊C4C2×C311yescharacteristic
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4⋊Dic3—32no—
28C2×C3×C4⋊Dic3—61no—
29C2×C3×Dic3⋊C4—32no—
30C2×Dic3⋊Dic3C2×C312yescomplemented
31C2×(C2×D3)⋊Dic3C2×C312yes—
32C32×#95—31no—
33C3×C3⋊#95—21no—
34C3×C3⋊#95D311yescharacteristic, complemented
35C3×C3⋊#95C2×C311yescharacteristic, complemented
36C23×C33C2311yescharacteristic
37C22×C32×D3—21no—
38C22×C32×D3—22no—
39C22×C32×D3—21no—
40C22×C32×D3—22no—
41C22×C32×D3Q812yes—
42C2×C33×C4—22no—
43C2×C33×C4D412yes—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—22no—
51C2×C32×Dic3D414yes—
52C3×#399—91no—
53#4949—31no—
54#4677—31no—
55C24×C32—31no—
56C23×C3×D3—21no—
57C23×C3×D3C2×D311yescharacteristic
58C23×C3×D3C22×C311yescharacteristic
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—32no—
63C22×C32×C4—31no—
64C22×C32×C4—41no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C22×C3⋊Dic3C22×C312yes—
69C22×C3×Dic3—21no—
70C22×C3×Dic3—22no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—62no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D312yes—
76C22×C3×Dic3C22×C311yescharacteristic
77C22×C3×Dic3C22×C312yes—
78C32×C22⋊C4—62no—
79C4⋊(C3⋊Dic3)—22no—
80C3×#217—64no—
81C3×(C2×D3)⋊C4—24no—
82C3×(C2×D3)⋊C4—22no—
83C3×(C2×D3)⋊C4—22no—
84C3×(C2×D3)⋊C4—64no—
85C3×C4⋊Dic3—64no—
86C3×C4⋊Dic3—64no—
87C3×C4⋊Dic3—62no—
88C3×C4⋊Dic3—62no—
89C3×Dic3⋊C4—64no—
90Dic3⋊Dic3—24no—
91(C2×D3)⋊Dic3—24no—
92C22×C33C2×D412yes—
93C22×C33C2×D411yescharacteristic
94C22×C33C2×Q811yescharacteristic
95C22×C33#4111yescharacteristic
96C22×C33#4112yes—
97C2×C32×D3—22no—
98C2×C32×D3—24no—
99C2×C32×D3—42no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C33×C4—22no—
103C3×C3⋊Dic3—42no—
104C32×Dic3—24no—
105C32×Dic3—41no—
106C2×C3×C4⋊C4—92no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C22⋊C4—31no—
110C2×C3×C22⋊C4—61no—
111C2×C3×C22⋊C4—92no—
112C2×#217—32no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4C3×D311yescharacteristic
115C2×C4⋊Dic3—31no—
116C2×C4⋊Dic3—31no—
117C2×C4⋊Dic3—32no—
118C2×C4⋊Dic3—61no—
119C2×Dic3⋊C4—32no—
120C3×#95—31no—
121C3×#95—31no—
122C3×#95—61no—
123C3⋊#95C3×D311yescharacteristic, complemented
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—32no—
128C23×C32—41no—
129C23×C32—61no—
130C23×C32—61no—
131C23×C32—62no—
132C23×C32—62no—
133C23×C32C22×D311yescharacteristic
134C23×C32C22×D311yescharacteristic
135C23×C32C23×C311yescharacteristic
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3—21no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—21no—
142C22×C3×D3—22no—
143C22×C3×D3—21no—
144C22×C3×D3—21no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—42no—
148C22×C3×D3—42no—
149C22×C3×D3Dic612yes—
150C22×C3×D3C3×Q812yes—
151C2×C32×C4—22no—
152C2×C32×C4—24no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—34no—
159C2×C32×C4—42no—
160C2×C32×C4—42no—
161C2×C32×C4—42no—
162C2×C32×C4—44no—
163C2×C32×C4—61no—
164C2×C32×C4—62no—
165C2×C32×C4—62no—
166C2×C32×C4—62no—
167C2×C32×C4D1212yes—
168C2×C32×C4#6712yes—
169C2×C32×C4C3×D412yes—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3—42no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—122no—
194C2×C3×Dic3—122no—
195C2×C3×Dic3D1214yes—
196C2×C3×Dic3C3×D414yes—
197#399—91noSylow
198C2×C33C22⋊Q812yes—
199C2×C33C22⋊Q811yescharacteristic
200C2×C33#12312yes—
201C2×C33#12311yescharacteristic
202C2×C33#12711yescharacteristic
203C32×D3—42no—
204C24×C3—31no—
205C24×C3—31no—
206C24×C3—61no—
207C23×D3C2×C3×D311yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—32no—
212C22×C3×C4—32no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—41no—
216C22×C3×C4—62no—
217C22×C3×C4—61no—
218C22×C3×C4—92no—
219C22×C3×C4D3211yescharacteristic
220C22×C3×C4C2×C3×D311yescharacteristic
221C22×C3×C4C2×C3×D311yescharacteristic
222C22×Dic3—32no—
223C22×Dic3—32no—
224C22×Dic3—62no—
225C22×Dic3C2×C3×D311yescharacteristic
226C22×Dic3C2×C3×D312yes—
227C3×C4⋊C4—184no—
228C3×C4⋊C4—182no—
229C3×C22⋊C4—64no—
230C3×C22⋊C4—62no—
231C3×C22⋊C4—62no—
232C3×C22⋊C4—184no—
233#217—64no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—64no—
236C4⋊Dic3—64no—
237C4⋊Dic3—64no—
238C4⋊Dic3—62no—
239C4⋊Dic3—62no—
240Dic3⋊C4—64no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—41no—
259C22×C32—42no—
260C22×C32—41no—
261C22×C32—62no—
262C22×C32—64no—
263C22×C32—121no—
264C22×C32—122no—
265C22×C32—121no—
266C22×C32C2×Dic611yescharacteristic
267C22×C32C2×D1212yes—
268C22×C32C2×D1211yescharacteristic
269C22×C32#23511yescharacteristic
270C22×C32D3×D412yes—
271C22×C32#23711yescharacteristic
272C22×C32#23712yes—
273C22×C32#23712yes—
274C22×C32D3×Q811yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C22×C32C2×C3×D412yes—
277C22×C32C2×C3×D411yescharacteristic
278C22×C32C2×C3×Q811yescharacteristic
279C22×C32C3×#4111yescharacteristic
280C22×C32C3×#4112yes—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—22no—
284C2×C3×D3—24no—
285C2×C3×D3—22no—
286C2×C3×D3—44no—
287C2×C3×D3—44no—
288C2×C3×D3—42no—
289C2×C3×D3—42no—
290C2×C3×D3—42no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3—42no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C32×C4—22no—
297C32×C4—24no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4—42no—
301C32×C4—42no—
302C32×C4—44no—
303C32×C4—64no—
304C32×C4—121no—
305C3⋊Dic3—42no—
306C3×Dic3—24no—
307C3×Dic3—24no—
308C3×Dic3—44no—
309C3×Dic3—42no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—124no—
313C3×Dic3—122no—
314C3×Dic3—122no—
315C2×C4⋊C4—92no—
316C2×C4⋊C4—91no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—92no—
319#95—31no—
320C33#39911yescharacteristic, complemented, Sylow
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—32no—
325C23×C3—32no—
326C23×C3—41no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—61no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—61no—
334C23×C3—62no—
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—62no—
338C23×C3—122no—
339C23×C3—122no—
340C23×C3C2×D3211yescharacteristic
341C23×C3C22×C3×D311yescharacteristic
342C23×C3C22×C3×D311yescharacteristic
343C22×D3—21no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—21no—
347C22×D3C3×Dic612yes—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—34no—
356C2×C3×C4—34no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—44no—
360C2×C3×C4—42no—
361C2×C3×C4—64no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—61no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4—122no—
372C2×C3×C4—122no—
373C2×C3×C4—122no—
374C2×C3×C4—182no—
375C2×C3×C4—182no—
376C2×C3×C4—182no—
377C2×C3×C4#62612yes—
378C2×C3×C4C3×D1212yes—
379C2×C3×C4C3×#6712yes—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—64no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—62no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—122no—
392C2×Dic3—122no—
393C2×Dic3C3×D1214yes—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—22no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—22no—
405C2×C32—22no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—42no—
411C2×C32—41no—
412C2×C32—42no—
413C2×C32—41no—
414C2×C32—122no—
415C2×C32#87411yescharacteristic
416C2×C32#87812yes—
417C2×C32#88312yes—
418C2×C32#88612yes—
419C2×C32#88611yescharacteristic
420C2×C32#89711yescharacteristic
421C2×C32#89712yes—
422C2×C32#92411yescharacteristic
423C2×C32#93511yescharacteristic
424C2×C32#94011yescharacteristic
425C2×C32C3×C22⋊Q812yes—
426C2×C32C3×C22⋊Q811yescharacteristic
427C2×C32C3×#12312yes—
428C2×C32C3×#12311yescharacteristic
429C2×C32C3×#12711yescharacteristic
430C3×D3—44no—
431C3×D3—42no—
432C3×D3—42no—
433C24—31no—
434C22×C4—32no—
435C22×C4—31no—
436C22×C4—92no—
437C22×C4C3×D3211yescharacteristic
438C4⋊C4—184no—
439C4⋊C4—182no—
440C22⋊C4—62no—
441C22⋊C4—184no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—64no—
463C22×C3—62no—
464C22×C3—64no—
465C22×C3—62no—
466C22×C3—64no—
467C22×C3—124no—
468C22×C3—122no—
469C22×C3—124no—
470C22×C3—122no—
471C22×C3—121no—
472C22×C3—122no—
473C22×C3—122no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3D3×Dic611yescharacteristic
478C22×C3#379511yescharacteristic
479C22×C3D3×D1212yes—
480C22×C3#380512yes—
481C22×C3C2×#62611yescharacteristic
482C22×C3C2×C3×Dic611yescharacteristic
483C22×C3C2×C3×D1212yes—
484C22×C3C2×C3×D1211yescharacteristic
485C22×C3C3×#23511yescharacteristic
486C22×C3C3×D3×D412yes—
487C22×C3C3×#23711yescharacteristic
488C22×C3C3×#23712yes—
489C22×C3C3×#23712yes—
490C22×C3C3×D3×Q811yescharacteristic
491C22×C3C2×C3×#6711yescharacteristic
492C2×D3—24no—
493C2×D3—22no—
494C2×D3—42no—
495C2×D3—41no—
496C2×D3—41no—
497C3×C4—24no—
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—44no—
504C3×C4—64no—
505C3×C4—64no—
506C3×C4—124no—
507C3×C4—122no—
508C3×C4—121no—
509C3×C4—121no—
510C3×C4—362no—
511Dic3—24no—
512Dic3—41no—
513Dic3—124no—
514Dic3—122no—
515Dic3—122no—
516C32—21no—
517C32—21no—
518C32—21no—
519C32—41no—
520C32#467711yescharacteristic, complemented
521C32#494911yescharacteristic, complemented
522C32C3×#39911yescharacteristic, complemented
523C23—32no—
524C23—61no—
525C23—62no—
526C23—62no—
527C23—61no—
528C23C2×C3×D3211yescharacteristic
529C2×C4—22no—
530C2×C4—34no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—182no—
536C2×C4—182no—
537C2×C4—182no—
538C2×C4C3×#62612yes—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—22no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—42no—
557C2×C3—42no—
558C2×C3—41no—
559C2×C3—124no—
560C2×C3—122no—
561C2×C3—122no—
562C2×C3#6385711yescharacteristic
563C2×C3#6389112yes—
564C2×C3#6389312yes—
565C2×C3#6391111yescharacteristic
566C2×C3#6396211yescharacteristic
567C2×C3C3×#87411yescharacteristic
568C2×C3C3×#87812yes—
569C2×C3C3×#88312yes—
570C2×C3C3×#88611yescharacteristic
571C2×C3C3×#88612yes—
572C2×C3C3×#89711yescharacteristic
573C2×C3C3×#89712yes—
574C2×C3C3×#92411yescharacteristic
575C2×C3C3×#93511yescharacteristic
576C2×C3C3×#94011yescharacteristic
577D3—42no—
578C22—62no—
579C22—64no—
580C22—121no—
581C22—122no—
582C22—121no—
583C22C3×D3×Dic611yescharacteristic
584C22C3×#379511yescharacteristic
585C22C3×D3×D1212yes—
586C22C3×#380512yes—
587C22C2×C3×#62611yescharacteristic
588C4—22no—
589C4—64no—
590C4—121no—
591C4—362no—
592C3—21no—
593C3—21no—
594C3—21no—
595C3—41no—
596C3#1059131711yescharacteristic, direct product factor
597C3C3×#467711yescharacteristic, complemented
598C3C3×#494911yescharacteristic, complemented
599C2—122no—
600C2C3×#6385711yescharacteristic
601C2C3×#6389112yes—
602C2C3×#6389312yes—
603C2C3×#6391111yescharacteristic
604C2C3×#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×#399
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
3. C2×C3×Dic3⋊Dic3C2semidirect
5. C32×C3⋊#95C2semidirect
8. #10591317C3direct (N × H)
19. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
30. C2×Dic3⋊Dic3C2×C3semidirect
34. C3×C3⋊#95D3semidirect
35. C3×C3⋊#95C2×C3semidirect
123. C3⋊#95C3×D3semidirect
320. C33#399semidirect
520. C32#4677semidirect
521. C32#4949semidirect
522. C32C3×#399semidirect
596. C3#10591317direct (N × H)
597. C3C3×#4677semidirect
598. C3C3×#4949semidirect

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×Dic3⋊Dic32
C2×C3×(C2×D3)⋊Dic32
C32×C3⋊#952
C3×#49493
C3×#46773
#105913173