Finite Group Databasebeta

← C3×#10591326 #13158245

Subgroups — C3×#10591326 #13158245

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×#3752C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3722C212yesmaximal
4C32×C3⋊#95C211yesmaximal, characteristic, complemented
5C3×#63579C212yesmaximal, complemented
6C3×#4992—31nomaximal
7C3×#4674—31nomaximal
8#10591326C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C3×#3752—22no—
15C32×C4⋊Dic3—22no—
16C3×#3722—24no—
17C2×C32×C4⋊C4—31no—
18C2×#3752C2×C311yescharacteristic
19C2×C3×(C2×D3)⋊C4—31no—
20C2×C3×(C2×D3)⋊C4—31no—
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×(C2×D3)⋊C4—61no—
24C2×C3×C4⋊Dic3—21no—
25C2×C3×C4⋊Dic3D311yescharacteristic, complemented
26C2×C3×C4⋊Dic3C2×C311yescharacteristic, complemented
27C2×#3722C2×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×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3—21no—
38C22×C3×C3⋊D3—22no—
39C22×C3×C3⋊D3—22no—
40C22×C3×C3⋊D3D412yes—
41C2×C33×C4—22no—
42C2×C33×C4D412yes—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—22no—
50C2×C32×Dic3—22no—
51C3×#397—91no—
52#4992—31no—
53#4674—31no—
54C23×C3⋊D3C22×C311yescharacteristic
55C23×C3×D3—31no—
56C23×C3×D3—31no—
57C23×C3×D3—61no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—32no—
62C22×C32×C4—32no—
63C22×C32×C4—41no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C311yescharacteristic
68C22×C3×Dic3—22no—
69C22×C3×Dic3—22no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—61no—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C2×D312yes—
75C22×C3×Dic3C22×C312yes—
76C22×C3×Dic3C22×C312yes—
77C32×C4⋊C4—62no—
78#3752—22no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×(C2×D3)⋊C4—64no—
82C3×(C2×D3)⋊C4—62no—
83C3×(C2×D3)⋊C4—62no—
84C3×C4⋊Dic3—24no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3—22no—
87#3722—24no—
88C22×C33C2×D411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C22×C33#4111yescharacteristic
91C22×C33#4112yes—
92C22×C33#4112yes—
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3—24no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—42no—
98C33×C4—22no—
99C3×C3⋊Dic3—41no—
100C32×Dic3—42no—
101C32×Dic3—42no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—61no—
105C2×C3×C22⋊C4—92no—
106C2×C3×C22⋊C4—91no—
107C2×(C2×D3)⋊C4—31no—
108C2×(C2×D3)⋊C4—31no—
109C2×(C2×D3)⋊C4—32no—
110C2×(C2×D3)⋊C4—32no—
111C2×(C2×D3)⋊C4—61no—
112C2×C4⋊Dic3C3×D311yescharacteristic, complemented
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—41no—
124C23×C32C22×D311yescharacteristic
125C23×C32C22×D311yescharacteristic
126C23×C32C23×C311yescharacteristic
127C22×C3⋊D3—21no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3—22no—
131C22×C3⋊D3C3×D412yes—
132C22×C3×D3—32no—
133C22×C3×D3—32no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C22×C3×D3—61no—
138C22×C3×D3—62no—
139C22×C3×D3—62no—
140C22×C3×D3—61no—
141C22×C3×D3—62no—
142C22×C3×D3—61no—
143C22×C3×D3—62no—
144C22×C3×D3—61no—
145C22×C3×D3—122no—
146C22×C3×D3—122no—
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—62no—
159C2×C32×C4—62no—
160C2×C32×C4—62no—
161C2×C32×C4—62no—
162C2×C32×C4—62no—
163C2×C32×C4—62no—
164C2×C32×C4D1212yes—
165C2×C32×C4#6712yes—
166C2×C32×C4C3×D412yes—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—22no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—24no—
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—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3—42no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—42no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—122no—
193#397—91noSylow
194C2×C33#12111yescharacteristic
195C2×C33#12312yes—
196C2×C33#12311yescharacteristic
197C2×C33#12411yescharacteristic
198C2×C33#12612yes—
199C3×C3⋊D3—42no—
200C24×C3—91no—
201C23×D3—31no—
202C23×D3—31no—
203C23×D3—61no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—32no—
208C22×C3×C4—32no—
209C22×C3×C4—32no—
210C22×C3×C4—32no—
211C22×C3×C4—41no—
212C22×C3×C4—62no—
213C22×C3×C4—62no—
214C22×C3×C4—91no—
215C22×C3×C4D3211yescharacteristic
216C22×C3×C4C2×C3×D311yescharacteristic
217C22×C3×C4C2×C3×D311yescharacteristic
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—61no—
221C22×Dic3C2×C3×D312yes—
222C22×Dic3C2×C3×D312yes—
223C3×C4⋊C4—64no—
224C3×C4⋊C4—62no—
225C3×C4⋊C4—62no—
226C3×C22⋊C4—184no—
227C3×C22⋊C4—182no—
228(C2×D3)⋊C4—64no—
229(C2×D3)⋊C4—64no—
230(C2×D3)⋊C4—64no—
231(C2×D3)⋊C4—62no—
232(C2×D3)⋊C4—62no—
233C4⋊Dic3—22no—
234C22×C32—21no—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—41no—
251C22×C32—42no—
252C22×C32—41no—
253C22×C32—42no—
254C22×C32C2×D1211yescharacteristic
255C22×C32#23512yes—
256C22×C32#23512yes—
257C22×C32D3×D411yescharacteristic
258C22×C32D3×D411yescharacteristic
259C22×C32#23711yescharacteristic
260C22×C32#23712yes—
261C22×C32#23911yescharacteristic
262C22×C32#23912yes—
263C22×C32C2×#6711yescharacteristic
264C22×C32C2×C3×D411yescharacteristic
265C22×C32C2×C3×D411yescharacteristic
266C22×C32C3×#4112yes—
267C22×C32C3×#4112yes—
268C22×C32C3×#4111yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—24no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—42no—
274C2×C3×D3—64no—
275C2×C3×D3—64no—
276C2×C3×D3—64no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—124no—
280C2×C3×D3—124no—
281C2×C3×D3—122no—
282C2×C3×D3—122no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C32×C4—22no—
290C32×C4—24no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—42no—
294C32×C4—42no—
295C32×C4—44no—
296C32×C4—122no—
297C32×C4—122no—
298C3⋊Dic3—41no—
299C3×Dic3—44no—
300C3×Dic3—44no—
301C3×Dic3—42no—
302C3×Dic3—42no—
303C3×Dic3—42no—
304C3×Dic3—42no—
305C3×Dic3—122no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C2×C4⋊C4—31no—
309C2×C22⋊C4—92no—
310C2×C22⋊C4—91no—
311#95—31no—
312#95—92no—
313C33#39711yescharacteristic, complemented, Sylow
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—41no—
318C23×C3—92no—
319C23×C3—181no—
320C23×C3—182no—
321C23×C3—182no—
322C23×C3—181no—
323C23×C3C2×D3211yescharacteristic
324C23×C3C22×C3×D311yescharacteristic
325C23×C3C22×C3×D311yescharacteristic
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—62no—
329C22×D3—62no—
330C22×D3—62no—
331C22×D3—61no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—61no—
335C22×D3—62no—
336C22×D3—61no—
337C22×D3—62no—
338C22×D3—61no—
339C22×D3—122no—
340C22×D3—122no—
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—64no—
353C2×C3×C4—64no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—122no—
367C2×C3×C4—122no—
368C2×C3×C4—122no—
369C2×C3×C4—122no—
370C2×C3×C4—182no—
371C2×C3×C4—181no—
372C2×C3×C4#62612yes—
373C2×C3×C4C3×D1212yes—
374C2×C3×C4C3×#6712yes—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—61no—
386C2×Dic3—122no—
387C2×C32—22no—
388C2×C32—21no—
389C2×C32—22no—
390C2×C32—21no—
391C2×C32—22no—
392C2×C32—22no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—22no—
397C2×C32—22no—
398C2×C32—21no—
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#87511yescharacteristic
408C2×C32#88312yes—
409C2×C32#88611yescharacteristic
410C2×C32#89412yes—
411C2×C32#89511yescharacteristic
412C2×C32#89812yes—
413C2×C32#89812yes—
414C2×C32#92911yescharacteristic
415C2×C32#93811yescharacteristic
416C2×C32#94711yescharacteristic
417C2×C32C3×#12111yescharacteristic
418C2×C32C3×#12312yes—
419C2×C32C3×#12311yescharacteristic
420C2×C32C3×#12411yescharacteristic
421C2×C32C3×#12612yes—
422C3⋊D3—42no—
423C3×D3—124no—
424C3×D3—122no—
425C3×D3—122no—
426C24—91no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—91no—
430C22×C4C3×D3211yescharacteristic
431C4⋊C4—62no—
432C22⋊C4—184no—
433C22⋊C4—182no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
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—41no—
450C22×C3—42no—
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—41no—
454C22×C3—184no—
455C22×C3—182no—
456C22×C3—362no—
457C22×C3—361no—
458C22×C3—361no—
459C22×C3#379611yescharacteristic
460C22×C3#379912yes—
461C22×C3#380211yescharacteristic
462C22×C3#380412yes—
463C22×C3C2×#62611yescharacteristic
464C22×C3C2×C3×D1211yescharacteristic
465C22×C3C3×#23512yes—
466C22×C3C3×#23512yes—
467C22×C3C3×D3×D411yescharacteristic
468C22×C3C3×D3×D411yescharacteristic
469C22×C3C3×#23711yescharacteristic
470C22×C3C3×#23712yes—
471C22×C3C3×#23911yescharacteristic
472C22×C3C3×#23912yes—
473C22×C3C2×C3×#6711yescharacteristic
474C2×D3—64no—
475C2×D3—64no—
476C2×D3—64no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—124no—
480C2×D3—122no—
481C2×D3—124no—
482C2×D3—122no—
483C2×D3—121no—
484C2×D3—122no—
485C2×D3—122no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—121no—
489C3×C4—24no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—124no—
497C3×C4—124no—
498C3×C4—122no—
499C3×C4—122no—
500C3×C4—122no—
501C3×C4—122no—
502C3×C4—361no—
503Dic3—42no—
504Dic3—42no—
505Dic3—122no—
506Dic3—121no—
507Dic3—121no—
508C32—21no—
509C32—21no—
510C32—21no—
511C32—41no—
512C32#467411yescharacteristic, complemented
513C32#499211yescharacteristic, complemented
514C32C3×#39711yescharacteristic, complemented
515C23—92no—
516C23—181no—
517C23—182no—
518C23—182no—
519C23—181no—
520C23C2×C3×D3211yescharacteristic
521C2×C4—22no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—182no—
529C2×C4—181no—
530C2×C4C3×#62612yes—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—41no—
547C2×C3—42no—
548C2×C3—42no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—362no—
552C2×C3#6384812yes—
553C2×C3#6385412yes—
554C2×C3#6386411yescharacteristic
555C2×C3#6391611yescharacteristic
556C2×C3#6396611yescharacteristic
557C2×C3C3×#87511yescharacteristic
558C2×C3C3×#88312yes—
559C2×C3C3×#88611yescharacteristic
560C2×C3C3×#89412yes—
561C2×C3C3×#89511yescharacteristic
562C2×C3C3×#89812yes—
563C2×C3C3×#89812yes—
564C2×C3C3×#92911yescharacteristic
565C2×C3C3×#93811yescharacteristic
566C2×C3C3×#94711yescharacteristic
567D3—124no—
568D3—122no—
569D3—122no—
570C22—184no—
571C22—182no—
572C22—361no—
573C22—362no—
574C22—361no—
575C22C3×#379611yescharacteristic
576C22C3×#379912yes—
577C22C3×#380211yescharacteristic
578C22C3×#380412yes—
579C22C2×C3×#62611yescharacteristic
580C4—22no—
581C4—122no—
582C4—122no—
583C4—361no—
584C3—21no—
585C3—21no—
586C3—21no—
587C3—41no—
588C3#1059132611yescharacteristic, direct product factor
589C3C3×#467411yescharacteristic, complemented
590C3C3×#499211yescharacteristic, complemented
591C2—362no—
592C2C3×#6384812yes—
593C2C3×#6385412yes—
594C2C3×#6386411yescharacteristic
595C2C3×#6391611yescharacteristic
596C2C3×#6396611yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊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
2. C2×C32×C4⋊Dic3C2semidirect
4. C32×C3⋊#95C2semidirect
5. C3×#63579C2semidirect
8. #10591326C3direct (N × H)
25. C2×C3×C4⋊Dic3D3semidirect
26. C2×C3×C4⋊Dic3C2×C3semidirect
32. C3×C3⋊#95D3semidirect
33. C3×C3⋊#95C2×C3semidirect
34. #63579C2×C3semidirect
112. C2×C4⋊Dic3C3×D3semidirect
119. C3⋊#95C3×D3semidirect
313. C33#397semidirect
512. C32#4674semidirect
513. C32#4992semidirect
514. C32C3×#397semidirect
588. C3#10591326direct (N × H)
589. C3C3×#4674semidirect
590. C3C3×#4992semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37522
C2×C32×C4⋊Dic32
C2×C3×#37222
C32×C3⋊#952
C3×#635792
C3×#49923
C3×#46743
#105913263